combinatorial number theory

Combinatorial NumberTheory: Proceedings of the'Integers Conference 2007'

Edited byB. Landman et al.

Walter de Gruyter

Combinatorial Number Theory

Combinatorial Number TheoryProceedings of the

‘Integers Conference 2007’

Carrollton, Georgia, USAOctober 24�27, 2007

EditorsB. Landman

M. B. NathansonJ. Nesetril

R. J. NowakowskiC. PomeranceA. Robertson

≥Walter de Gruyter · Berlin · New York

EditorsBruce Landman Richard J. NowakowskiDepartment of Mathematics Department of Mathematics and StatisticsUniversity of West Georgia Dalhousie University1601 Maple Street Halifax, Nova Scotia, Canada B3H 3J5Carrollton, GA 30118, USA e-mail: [email protected]: [email protected] Carl PomeranceMelvyn B. Nathanson Department of MathematicsDepartment of Mathematics Dartmouth CollegeLehman College (CUNY) Hanover, NH 03755-3551, USA250 Bedford Park Boulevard West e-mail: [email protected], NY 10468, USA Aaron Robertsone-mail: [email protected] Department of MathematicsJaroslav Nesetril Colgate UniversityDepartment of Applied Mathematics 13 Oak DriveCharles University Hamilton, NY 13346, USAMalostranske nam. 25 e-mail: [email protected] 00 Praha 1, Czech Republice-mail: [email protected]

Keywords: Combinatorics, Number Theory, Primes, Euler Product, Euler Function, Pseudosquares,Pseudonumbers, Perfect Numbers, Theory of Partitions, Ramsey Theory

Mathematics Subject Classification 2000: 11-06

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ISBN 978-3-11-020221-2

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Preface

The Integers Conference 2007 was held October 24–27, 2007, at the University

of West Georgia in Carrollton, Georgia. This was the third Integers Conference,

held bi-annually since 2003.

It featured sixty-four invited talks including six plenary lectures presented

by George Andrews, Vitaly Bergelson, Bryna Kra, Florian Luca, Ken Ono, and

Van Vu.

This volume consists of sixteen refereed articles, which are expanded and re-

vised versions of talks presented at the conference. These sixteen articles will

appear as a special volume of the journal Integers: Electronic Journal of Com-

binatorial Number Theory. They represent a broad range of topics in the ar-

eas of number theory and combinatorics including multiplicative number theory,

additive number theory, Ramsey theory, enumerative combinatorics, elementary

number theory, the theory of partitions, algebraic number theory, and integer se-

quences.

The conference was made possible with the generous support of the National

Science Foundation and the University of West Georgia. The Integers conferences

are organized by the Editors of Integers, which publishes articles in the field of

combinatorial number theory. The conferences are held in order to further support

and strengthen this growing field.

December, 2008 The Editors

Table of contents

Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : v

GEORGE E. ANDREWS

The Finite Heine Transformation : : : : : : : : : : : : : : : : : : : : : : : 1

TSZ HO CHAN

Finding Almost Squares III : : : : : : : : : : : : : : : : : : : : : : : : : : 7

DENNIS EICHHORN, MIZAN R. KHAN, ALAN H. STEIN,

CHRISTIAN L. YANKOV

Sums and Differences of the Coordinates of Points on Modular Hyperbolas : 17

DAVID GARTH, JOSEPH PALMER, HA TA

Self Generating Sets and Numeration Systems : : : : : : : : : : : : : : : : 41

NEIL HINDMAN

Small Sets Satisfying the Central Sets Theorem : : : : : : : : : : : : : : : 57

BRIAN HOPKINS

Column-to-Row Operations on Partitions: The Envelopes : : : : : : : : : : 65

XIAN-JIN LI

On the Euler Product of Some Zeta Functions : : : : : : : : : : : : : : : : 77

FLORIAN LUCA, CARL POMERANCE

On the Range of the Iterated Euler Function : : : : : : : : : : : : : : : : : 101

GRETCHEN L. MATTHEWS

Frobenius Numbers of Generalized Fibonacci Semigroups : : : : : : : : : 117

JAMES MCLAUGHLIN, ANDREW V. SILLS

Combinatorics of Ramanujan–Slater Type Identities : : : : : : : : : : : : : 125

KEN ONO

A Mock Theta Function for the Delta-function : : : : : : : : : : : : : : : 141

RAM KRISHNA PANDEY, AMITABHA TRIPATHI

On the Density of Integral Sets with Missing Differences : : : : : : : : : : 157

viii Table of contents

CARL POMERANCE, IGOR E. SHPARLINSKI

On Pseudosquares and Pseudopowers : : : : : : : : : : : : : : : : : : : : 171

FRANK THORNE

Maier Matrices Beyond Z : : : : : : : : : : : : : : : : : : : : : : : : : : 185

TOMOHIRO YAMADA

Linear Equations Involving Iterates of �.N / : : : : : : : : : : : : : : : : 193

PAUL YIU, K. R. S. SASTRY, SHANZHEN GAO

Heron Sequences and Their Modifications : : : : : : : : : : : : : : : : : 199

List of participants : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 205

Combinatorial Number Theory © de Gruyter 2009

The Finite Heine Transformation

George E. Andrews

Abstract. We shall present finite summations that converge to the Heine 2�1 transfor-

mations in the limit as n!1. We shall investigate their partition-theoretic implications.

Keywords. q-series, partitions, basic hypergeometric series, Heine’s transformation.

AMS classification. 11P81, 11P83, 05A17, 05A19.

1 Introduction

In an expository article describing Euler’s pioneering work on partitions, I was

particularly drawn to Euler’s assertion [6, p. 566, eq. (5.2) corrected]

1Y

nD0

�

q�3n

C 1C q3n

�

D

1X

nD�1

qn; (1.1)

an identity valid only in a formal sense in that neither the series nor the product

converges for any value of q.

This led to my comparisons of the two infinite series identities ([6, p. 567,

eq. (5.5)] and [6, p. 567, eq. (5.6)] respectively):

1X

nD0

qn2

.1 � q/2.1 � q2/2 � � � .1 � qn/2D

1Y

nD1

1

1 � qn; (1.2)

and

1X

nD0

qn

.1 � q/2.1 � q2/2 � � � .1 � qn/2D

1Y

nD1

1

.1 � qn/2

1X

j D0

.�1/j qj.j C1/=2: (1.3)

Each of the left-hand series is analytic inside jqj < 1 with jqj D 1 as a nat-

ural boundary, and the second series is formally transformable into the first by

the mapping q ! 1=q. The fact that jqj D 1 is a natural boundary means we

should not be surprised when the same transformation applied to the right-hand

side produces only nonsense.

Partially supported by National Science Foundation Grant DMS 0200097.

2 George E. Andrews

However, it was observed in [4] that it is sometimes possible to find polynomial

or rational function identities that converge to infinite q-series in the limit. This

observation in [7] was the secret to dealing with Regime II of Baxter’s generalized

hard-hexagon model (cf. [5, Ch. 8]).

So this led to the question: Are there finite identities that would both (A)

simplify to (1.2) and (1.3) in the limit, and (B) allow the mapping q ! 1=q

prior to taking limits?

The answer to this question is yes. In Section 2 we provide q-analogs of the

Heine transformations of the 2�1. In Section 3, we shall derive generalizations of

the following corollaries:

NX

nD0

qn2

.1 � q/2.1 � q2/2 � � � .1 � qn/2D

NY

nD1

1

.1 � qn/

NX

j D0

q.N C1/j

.1 � q/.1 � q2/ � � � .1 � qj /;

(1.4)

and

NX

nD0

qn

.1 � q/2.1 � q2/2 � � � .1 � qn/2D

NY

nD1

1

.1 � qn/

NX

j D0

.�1/j qj.j C1/=2

.1 � q/.1 � q2/ � � � .1 � qN �j /:

(1.5)

Clearly (1.4) and (1.5) converge to (1.2) and (1.3) asN !1, and by reversing

the sum on the right-hand side it is a simple matter to see that (1.4) becomes (1.5)

under the now legitimate mapping q ! 1=q.

In Section 4, we shall note quite transparent combinatorial proofs of (1.4) and

(1.5).

2 Finite Heine Transformations

We shall employ the following standard notation

.a/n D .aI q/n D

n�1Y

j D0

.1 � aqj /; (2.1)

.a1; : : : ; ar I q/n D .a1I q/n.a2I q/n � � � .ar I q/n; (2.2)

and

rC1�r

a0; a1; : : : ; ar I q; t

b1; : : : ; br

!

D

1X

nD0

.a0; a1; : : : ; ar I q/ntn

.q; b1; : : : ; br I q/n: (2.3)

The Finite Heine Transformation 3

Lemma 1. For non-negative integers n,

3�2

q�n; ˛; ˇI q; q

; q1�n=�

!

D.˛� I q/n

.� I q/n3�2

q�n; =ˇ; ˛I q; ˇ�qn

; ˛�

!

: (2.4)

Proof. In (III:13) of [8, p. 242], set b D =ˇ, c D ˛, d D , e D ˛� . The result

after simplification is (2.4).

We note in passing that Lemma 1 is, in fact, a finite version of Jackson’s sum-

mation [9] (cf. [8, p. 11, eq. (1.54)], [2, p. 527, Lemma]).

Theorem 2. For non-negative integers n,

3�2

q�n; ˛; ˇI q; q

; q1�n=�

!

D.ˇ; ˛� I q/n

. ; � I q/n3�2

q�n; =ˇ; � I q; q

˛�; q1�n=ˇ

!

: (2.5)

Remark. When n ! 1, this is Heine’s classic 2�1 transformation [8, p. 9,

eq. (1.4.1)], [3, p. 28, Cor. 2.3].

Proof. If in Lemma 1, we replace ˛, ˇ, , and � by =ˇ, � , ˛� and ˇ respectively,

we find that

3�2

q�n; =ˇ; ˛I q; ˇ�qn

; ˛�

!

D.ˇI q/n

. I q/n3�2

q�n; =ˇ; � I q; q

˛�; q1�n=ˇ

!

: (2.6)

Now substituting the left-hand side of (2.6) into the right-hand side of (2.4) we

deduce (2.5).

Corollary 3. For non-negative integers n,

3�2

q�n; ˛; ˇI q; q

; q1�n=�

!

D. =ˇ; ˇ� I q/n

. ; � I q/n3�2

q�n; ˛ˇ�= ; ˇI q; q

ˇ�; ˇq1�n=

!

: (2.7)

Proof. Apply Theorem 2 (with ˛, ˇ, and � replaced by � , =ˇ, ˛� and ˇ

respectively) to transform the 3�2 on the right-hand side of (2.5).

Corollary 4. For non-negative integers n,

3�2

q�n; ˛; ˇI q; q

; q1�n=�

!

D.˛ˇ�

I q/n

.� I q/n3�2

q�n; ˛;

ˇI q; q

; q1�n=.˛ˇ�/

!

: (2.8)

Proof. Apply Theorem 2 (with ˛, ˇ, and � replaced by ˇ, ˛ˇ�= , ˇ� , =ˇ

respectively) to transform the 3�2 on the right-hand side of (2.7).

Corollaries 3 and 4 reduce to the second and third Heine transformations [8, p.

10] when n!1.

4 George E. Andrews

3 Identities (1.4) and (1.5)


nX

j D0

qj

.q; I q/jD

1

. /n

nX

j D0

.�1/j j qj.j �1/=2

.q/n�j: (3.1)

Proof. Set ˛ D 0, � D q and let ˇ ! 0 in Theorem 2. The desired result follows

after algebraic simplification.


nX

j D0

qj 2

j

.q; qI q/jD

1

. q/n

nX

j D0

j qj.nC1/

.q/j: (3.2)

Proof. Replace q by 1=q and by 1=q in (3.1), then reverse the sum on the

right-hand side and simplify.

Identity (1.5) is Theorem 5 with D q, and (1.4) is Theorem 6 with D 1.

4 Combinatorial Proofs

Replacing q by q2 in Theorem 5 and then replacing with � q, we see that

Theorem 5 is equivalent to the following assertion:

nX

j D0

q2j�

� q2j C1I q2�

n�j

.q2I q2/jD

nX

j D0

j qj 2

.q2I q2/n�j: (4.1)

Proof of (4.1). The left-hand side of (4.1) is the generating function for partitions

in which (1) all parts are 5 2n, (2) odd parts are distinct, and (3) each odd is >

each even. The general two-modular Ferrers graph [3, p. 13] for such partitions is

thus

2 2 � � � 2 � � � 2 � � � 2 1

2 2 � � � 2 � � � � � � 2 1

2 2 � � � 2 � � � 2 1

: : : : : : : : : : : : : : : : : : : : : : : : :

2 2 � � � 2 2

2 2 � � � 2:::

2

The Finite Heine Transformation 5

Now remove the columns that have a 1 at the bottom. In light of the fact that the

odds were distinct, we see that if there were originally j odd parts, then we have

removed 1C3C5C� � �C.2j�1/ (D j 2). The remaining parts are all even and the

largest is at most 2n � 2j . Thus this transformation (which is clearly reversible)

provides the partitions generated by the right-hand side of (4.1) and thus we have

a bijective proof of Theorem 5.

Proof of (3.2). Classical arguments immediately reveal that the left-hand side of

(3.2) is the generating function for partitions with Durfee square of side at most

n. keeps track of the number of parts.

On the other hand, the side of the Durfee square is the largest j such that the

j th part is = j . So we may replicate the partitions generated by the left-hand side

of (3.2) by exhibiting the generating function for partitions in which the parts > n

are at most n in number. If there are j parts greater than n, the generating function

is j qj.nC1/

. q/n.q/j:

Hence summing on j from 0 to n we obtain a new expression for the generating

function for partitions with Durfee square at most n, and this proves (3.2).

5 Conclusion

There are many other corollaries obtainable from the finite Heine transforma-

tions. The q-Pfaff-Saalschütz summation is merely [8, p. 13, eq. (1.7.2)] with

� D =˛ˇ. One can also obtain a finite version of the q-analog of Kummer’s

theorem [2], however, the result does not reduce to the hoped for “sum equals

product” identity. Also it should be possible to provide a fully combinatorial

proof of Theorem 2 along the lines given in [1] for the n!1 case.

References

[1] G. E. Andrews, Enumerative proofs of certain q-identities, Glasgow Math. J. 8

(1967), 33–40.

[2] G. E. Andrews, On the q-analog of Kummer’s theorem and applications, Duke Math.

J. 40 (1973), 525–528.

[3] G. E. Andrews, The Theory of Partitions, Encycl. of Math. and Its Appl., Vol. 2,

Addison-Wesley, Reading, 1976 (Reissued: Cambridge University Press, 1998).

[4] G. E. Andrews, The hard-hexagon model and Rogers–Ramanujan type identities,

Proc. Nat. Acad. Sci. (USA) 78 (1981), 5290–5292.

6 George E. Andrews

[5] G. E. Andrews, q-Series: Their Development and Application in Analysis, Number

Theory, Combinatorics, Physics and Computer Algebra, CBMS Regional Conference

Lecture Series 66, Amer. Math. Soc., Providence, 1986.

[6] G. E. Andrews, Euler’s “De Partitio(ne)” Numerorum, Bull. Amer. Math. Soc. 44

(2007), 561–573.

[7] G. E. Andrews, R. J. Baxter and P. J. Forrester, Eight-vertex SOS model and general-

ized Rogers–Ramanujan-type identities, J. Stat. Phys. 35 (1984), 193–266.

[8] G. Gasper and M. Rahman, Basic Hypergeometric Series, Encycl. of Math. and Its

Appl., Vol. 35, 1st ed., Cambridge University Press, Cambridge, 1990.

[9] F. H. Jackson, Tranformations of q-series, Messenger of Mathematics 39 (1910), 145–

153.

Author information

George E. Andrews, Department of Mathematics, The Pennsylvania State University,

University Park, PA 16802, USA.

E-mail: [email protected]


Finding Almost Squares III

Tsz Ho Chan

Abstract. An almost square of type 2 is an integer n that can be factored in two different

ways as n D a1b1 D a2b2 with a1, a2, b1, b2 �pn. In this paper, we shall improve

upon our previous results on short intervals containing such an almost square. This leads

to another question of independent interest: given some 0 < c < 1, find a short interval

around x which contains an integer divisible by some integer in Œxc ; 2xc �.

Keywords. Almost square, Erdos–Turán inequality, exponent pairs, almost divisible.

AMS classification. 11B75, 11L07, 11N25.

1 Introduction and Main Results

In [1] and [2], the author started studying almost squares which are integers n that

can be factored as n D ab with a; b close topn. For example, n D 9999 D

99 � 101 is an almost square. We say that an integer n is an almost square of

type 2 if it has two different representations, n D a1b1 D a2b2, with a1; b1; a2; b2

close topn. For example n D 99990000 D 9999� 10000 D 9900� 10100 is an

almost square of type 2.

More precisely, for 0 � � � 1=2 and C > 0,

Definition 1. An integer n is a (� , C )-almost square of type 1 if n D ab for some

integers a; b in the interval Œn1=2 � Cn� ; n1=2 C Cn� �.

Definition 2. An integer n is a (� , C )-almost square of type 2 if n D a1b1 D a2b2

for some integers a1 < a2 � b2 < b1 in the interval Œn1=2 � Cn� ; n1=2 C Cn� �.

Let x be a large positive real number. Following [1] and [2], we are interested

in finding almost squares of type 1 or 2 near to x. In particular, given 0 � � � 1=2,

we want to find “admissible” �i � 0 (as small as possible) such that, for some

constants C�;i ;D�;i > 0, the interval Œx �D�;ix�i ; x CD�;ix

�i � contains a (� ,

C�;i )-almost square of type i (i D 1; 2) for all large x.

Definition 3. f .�/ WD inf�1 and g.�/ WD inf�2 where the infima are taken over

all the “admissible” �i (i D 1; 2) respectively.

8 Tsz Ho Chan

Clearly f and g are non-increasing functions of � . Summarizing the results in

[1] and [2], we have:

Theorem 1. For 0 � � � 1=2,

f .�/

8

ˆ

ˆ

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

ˆ

ˆ

:

D 1=2, if 0 � � < 1=4,

D 1=4, if � D 1=4,

D 1=2 � � , if 1=4 � � � 3=10 and a conjectural upper bound on

certain average of twisted incomplete Salie sum is true,

� 1=2 � � , if 1=4 � � � 1=2.

Theorem 2. For 0 � � � 1=2,

g.�/

8

ˆ

<

ˆ

:

does not exist, if 0 � � < 1=4,

� 1 � 2� , if 1=4 � � � 1=2,

� 1 � � , if 1=4 � � � 1=3.

And we gave the following conjecture.

Conjecture 1. For 0 � � � 1=2,

f .�/ D

´

1=2, if 0 � � < 1=4,

1=2 � � , if 1=4 � � � 1=2;

and

g.�/ D

´

does not exist, if 0 � � < 1=4,

1 � 2� , if 1=4 � � � 1=2.

In this paper, we improve Theorem 2:

Theorem 3. For 1=4 � � � 1=2,

(i) g.1=4/ � 5=8,

(ii) g.�/ � 9=16, if 5=16 � � � 1=2,

(iii) g.�/ � 17=32, if 5=16 � � � 1=2,

(iv) g.�/ � 1=2, if 1=3 < � � 1=2,

(v) g.�/ � 1=2, if 743=2306 < � � 1=2.

Clearly (iii) is better than (ii). The reason we keep (ii) is that (ii) and (iii) use

different approaches. Also (v) includes (iv). We keep (iv) because it is a prototype

of (v).

Finding Almost Squares III 9

0

12

1

g.�/

14

12

�516

7432306

58 17

32?

? ?HHHHHHHHHH

6

XXXX? ?

The above picture summarizes Theorems 2 and 3. The thin line segments are

the upper and lower bounds from Theorem 2. The thick horizontal line segments

are the upper bounds from Theorem 3. The next challenge is to beat the 12

upper

bound for g.�/.

Some Notations: Throughout the paper, � denotes a small positive number. Both

f .x/ D O.g.x// and f .x/� g.x/mean that jf .x/j � Cg.x/ for some constant

C > 0. Moreover f .x/ D O�.g.x// and f .x/ �� g.x/ mean that the implicit

constant C D C� may depend on the parameter �. Finally f .x/ � g.x/ means

that f .x/� g.x/ and g.x/� f .x/.

2 Proof of Theorem 3 (i)

Let 1=4 � � � 1=2. From [2], we recall that a (� , C )-almost square of type 2 is

of the form

n D .d1e1/.d2e2/ D .d1e2/.d2e1/;

where a1 D d1e1, b1 D d2e2, a2 D d1e2, b2 D d2e1; n1=2 � Cn� � a1 < a2 �

b2 < b1 � n1=2 C Cn� ;

1

2Cn

12��

1

2� d1; d2; e1; e2 � 2Cn� ; e2 � e1 � 2C

n�

d2

; d2 � d1 � 2Cn�

e2

:

Let 1 � k � 1 be any integer. By the � D 1=4 case in Theorem 1, for some

constant C > 0, we can find integers d; e 2 Œx1=4 � Cx1=8; x1=4 C Cx1=8� such

that

de D x1=2 � 2kx1=4 CO.x1=8/:

Then .d C 2k/.e C 2k/ D de C 2k.d C e/C k2 D x1=2 C 2kx1=4 C O.x1=8/;

and

de.d C 2k/.e C 2k/ D x � 4k2x1=2 CO.x5=8/ D x CO.x5=8/:

This gives g.1=4/ � 5=8.

10 Tsz Ho Chan

3 Proof of Theorem 3 (ii)

The key idea is the identity

ab D�aC b

2

�2

��a � b

2

�2

as used in [1]. Using this identity,

d1e1d2e2 Dh�d2 C d1

2

�2

��d2 � d1

2

�2ih�e2 C e1

2

�2

��e2 � e1

2

�2i

D�d2 C d1

2

�2�e2 C e1

2

�2

��d2 � d1

2

�2�e2 C e1

2

�2

��e2 � e1

2

�2�d2 C d1

2

�2

C�d2 � d1

2

�2�e2 � e1

2

�2

DW G2H 2 � g2H 2 � h2G2 C g2h2

where G D d2Cd1

2, H D e2Ce1

2, g D d2�d1

2and h D e2�e1

2. Now we want

x � d1e1d2e2 D G2H 2 � g2H 2 � h2G2 C g2h2;

G2H 2 � x � g2H 2 C h2G2 � g2h2;

.GH �px/.GH C

px/ � g2H 2 C h2G2 � g2h2: (1)

By the � D 1=4 case in Theorem 1, for some constant C > 0, there exist integers

G;H 2 Œx1=4 � Cx1=16; x1=4 C Cx1=16� such that 0 < GH �px � x1=8. Then

the left hand side of (1) is � x1=2C1=8. As for the right hand side of (1), observe

that, for fixed h (say h D 1), the increment

Œ.i C 1/2H 2 C h2G2 � .i C 1/2h2� � Œi2H 2 C h2G2 � i2h2�

D .2i C 1/H 2 � .2i C 1/h2 � x1=2i:

Now observe that

g2H 2 C h2G2 � g2h2

D h2G2 CX

0�i<g

Œ.i C 1/2H 2 C h2G2 � .i C 1/2h2� � Œi2H 2 C h2G2 � i2h2�

� x1=2X

1�i<g

i � g2x1=2:

Therefore, for some integer 1 � g � x1=16,

jRight hand side of .1/ � Left hand side of .1/j � x1=2g � x1=2C1=16:


This gives

jx � .G2 � g2/.H 2 � h2/j � x1=2C1=16

or

jx � d1d2e1e2j D jx � .G � g/.G C g/.H � h/.H C h/j � x1=2C1=16:

Consequently, with

a1 D d1e1 D .G � g/.H � h/ D x1=2 CO.x1=4C1=16/;

b1 D d2e2 D .G C g/.H C h/ D x1=2 CO.x1=4C1=16/;

a2 D d1e2 D .G � g/.H C h/ D x1=2 CO.x1=4C1=16/;

b2 D d2e1 D .G C g/.H � h/ D x1=2 CO.x1=4C1=16/;

we have a .14C 1

16; C 0/-almost square n D a1b1 D a2b2 of type 2 in the interval

Œx � C 00x1=2C1=16; x C C 00x1=2C1=16� for some C 0; C 00 > 0. This proves that

g.�/ � 916

for � � 516

.

4 Proof of Theorem 3 (iii)

This time we try to approximate the left hand side of (1) by the quadratic form

g2H 2 C h2G2 directly. As in the proof of Theorem 3 (ii), for some C > 0,

there exist integers x1=4 � Cx1=16 � G;H � x1=4 C Cx1=16 such that 0 <

GH �px � x1=8. The left hand side of (1) is � x1=2C1=8. Without loss of

generality, G � H . Then g2H 2 C h2G2 D G2.g2 C h2/ C .H 2 � G2/g2.

Observe that 0 � H 2�G2 D .H �G/.H CG/� x1=4C1=16. By an elementary

argument, for any real number X > 0, we can find a sum of two squares g2 C h2

such that jX � .g2 C h2/j � X1=4. In particular, we can find 1 � g; h � x1=16

such thatˇ

ˇ

ˇ

.GH �px/.GH C

px/

G2� .g2 C h2/

ˇ

ˇ

ˇ� x1=32:

This implies

j.GH �px/.GH C

px/ � .g2H 2 C h2G2 � g2h2/j

� j.GH �px/.GH C

px/ �G2.g2 C h2/j C j.H 2 �G2/g2j C jg2h2j

� x1=2C1=32:

Hence

jx � d1d2e1e2j D jx � .G � g/.G C g/.H � h/.H C h/j � x1=2C1=32:

12 Tsz Ho Chan

Consequently, with

a1 D d1e1 D .G � g/.H � h/ D x1=2 CO.x1=4C1=16/;

b1 D d2e2 D .G C g/.H C h/ D x1=2 CO.x1=4C1=16/;

a2 D d1e2 D .G � g/.H C h/ D x1=2 CO.x1=4C1=16/;

b2 D d2e1 D .G C g/.H � h/ D x1=2 CO.x1=4C1=16/;

there is a . 14C 1


Œx � C 00x1=2C1=32; x C C 00x1=2C1=32� for some C 0; C 00 > 0. This proves that

g.�/ � 1732

for � � 516

.

5 Proof of Theorem 3 (iv)

Let 1=2 � � � 1. Observe that, for large x, the interval Œx C x1�� ; x C 2x1��

contains an integer n which is divisible by an integer a 2 Œx1��=2; x1�� . In

particular n D ab with integer b 2 Œx� ; 3x� �.

Again we use (1). Instead of having G;H close to x1=4 as in the proof of

Theorem 3 (iii), we want

G � x.1��/=2 and H � x�=2 for some1

2< � <

2

3:

By the observation at the beginning of this section, we can findH 2 Œx�=2; 3x�=2�

and G 2 Œx.1��/=2=2; x.1��/=2� such that 0 < GH �px � x.1��/=2. Then the

left hand side of (1) is L WD .GH �px/.GH C

px/ � x1��=2.

Firstly, we approximate L by g2H 2. For some choice of g � x1=2�3�=4, we

have 0 < L � g2H 2 � gH 2 � x1=2C�=4. Note that 12� 3�

4> 0 as � < 2

3.

Secondly, we approximate L � g2H 2 by h2G2. For some choice of h �

x5�=8�1=4, we have jL�g2H 2�h2G2j � hG2 � x3=4�3�=8. Note that5�8� 1

4> 0

as � > 12.

Thirdly, observe that g2h2 � x1=2��=4 � x3=4�3�=8 as � < 2. Therefore,

jL � g2H 2 � h2G2 C g2h2j � x3=4�3�=8 which gives

jx � d1d2e1e2j D jx � .G � g/.G C g/.H � h/.H C h/j � x3=4�3�=8:

Consequently, as 12< � < 2

3, with

a1 D d1e1 D .G � g/.H � h/ D x1=2 CO.x1=2��=4/;

b1 D d2e2 D .G C g/.H C h/ D x1=2 CO.x1=2��=4/;

a2 D d1e2 D .G � g/.H C h/ D x1=2 CO.x1=2��=4/;

b2 D d2e1 D .G C g/.H � h/ D x1=2 CO.x1=2��=4/;


there is a . 12� �


Œx � C 00x3=4�3�=8; x C C 00x3=4�3�=8� for some C 0; C 00 > 0. By picking � close

to 23, we have g.�/ � 1

2for � > 1

3.

6 Integers Almost Divisible by Some Integer

in an Interval

Again let 1=2 � � � 1. In the previous section, we found an interval of length

x1�� around x containing an integer divisible by some integer in the interval

Œx1��=2; x1�� . This is obviously true. Our goal in this section is to find a

shorter interval still containing an integer divisible by some integer in the interval

Œx1��=2; x1�� . We hope that this will give some improvements to Theorem 3

(iv). Let us reformulate the question as follows:

Question 1. Let 0 < ˛ � 1=2 and X > 0 be a large integer. Given 0 < c1 <

c2 � 1, find L, as small as possible, such that the interval ŒX �L;X� contains an

integer that is divisible by some integer in the interval Œc1X˛; c2X

˛�.

One may interpret the above as finding an integer in the interval Œc1X˛; c2X

˛�

that almost divides X (with a remainder less than or equal to L). We suspect that

Conjecture 2, below, is true, but are only able to prove Proposition 1, below.

Conjecture 2. For any � > 0, one can take L D X� in the above question as long

as X is sufficiently large (in terms of �).

Proposition 1. Suppose .p; q/ with 0 � p � 12� q � 1 is an exponent pair for

exponential sums. Then one can take L D X˛.q�p/

1CpC

p

1CpC�

in the above question

for any � > 0 as long as X is sufficiently large (in terms of �).

Our method of proof uses Erdos–Turán inequality in the following form (see

H. L. Montgomery [4, Corollary 1.2] for example):

Lemma 1. Suppose M is a positive integer chosen so that

MX

lD1

ˇ

ˇ

ˇ

JX

j D1

e.lxj /ˇ

ˇ

ˇ�J

10:

Then every arc J D Œ˛; ˇ� � Œ0; 1� of length ˇ � ˛ � 4M C1

contains at least12J.ˇ � ˛/ of the points xj , 1 � j � J . Here jjxjj D minn2Z jx � nj, the

distance from x to the nearest integer, and e.x/ D e2�ix .

14 Tsz Ho Chan

Proof of Proposition 1. Our sequence ¹xj ºJj D1

should be ¹XaW a 2 Z and a 2

Œc1X˛; c2X

˛�º. We want to find some a such that the fractional part of Xa

is small.

For if ¹Xaº2 Œ0; 4

M�, then X

aD k C �

Mfor some integer k and 0 � � � 4. This

gives X D ka C �Ma and X � �

Ma D ka. Hence, with L D 4c2X˛

M, the interval

ŒX � L;X� contains an integer that is divisible by some integer in Œc1X˛; c2X

˛�.

Thus, in view of Lemma 1, it suffices to show

S WD

2KX

lDK

ˇ

ˇ

ˇ

X

c1X˛�a�c2X˛

e� lX

a

�ˇ

ˇ

ˇ� X˛��

for any 2K � M and � > 0 as long as X is sufficiently large in terms of �. Keep

in mind that we want M as large as possible.

By the theory of exponent pairs on exponential sums (for example, see Chap-

ter 3, Section 4 of [4] for an overview),

X

c1X˛�a�c2X˛

e� lX

a

�

� .lX.X˛/�2/p.X˛/q � KpXp�2˛pC˛q (2)

if .p; q/ with 0 � p � 12� q � 1 is an exponent pair. Using (2), we have

2KX

lDK

ˇ

ˇ

ˇ

X

c1X˛�a�c2X˛

e� lX

a

�ˇ

ˇ

ˇ� K1CpXp�2˛pC˛q :

Thus, S � X˛�� provided that, for X large enough,

K1CpXp�2˛pC˛q � X˛�� or K � X˛.1�qC2p/

1Cp�

p

1Cp��:

Therefore, we can pick M D X˛.1�qC2p/

1Cp�

p

1Cp��

, which gives the identity L D

4c2X˛.q�p/

1CpC

p

1CpC�

. This proves Proposition 1 as � is arbitrary.

7 Proof of Theorem 3 (v)

We follow closely the proof of Theorem 3 (iv). Applying Proposition 1 with ˛ D

1�� andX D xC3x.1��/.q�p/

1CpC

p

1CpC�

, the interval ŒxCx.1��/.q�p/

1CpC

p

1CpC�; xC

3x.1��/.q�p/

1CpC

p

1CpC�� contains an integer n D ab with integers a 2 Œx1��=2;

x1�� and b 2 Œx� ; 3x� �. Thus we can find

H 2 Œx�=2; 3x�=2� and G 2 Œx.1��/=2=2; x.1��/=2�


such that

0 < GH �px � x

.1��/.q�p/

2.1Cp/C

p

2.1Cp/C

�

2 :

Then for the left hand side of (1) we have L WD .GH �px/.GH C

px/ �

x1CpCq

2.1Cp/�

q�p

2.1Cp/�C

�

2 .

Firstly, approximateL by g2H 2. For some choice of g � x1CpCq

4.1Cp/�

2CpCq

4.1Cp/�C

�

4 ,

we have

0 < L � g2H 2 � gH 2 � x1CpCq

4.1Cp/C

2C3p�q

4.1Cp/�C

�

4 :

Note that we need1CpCq4.1Cp/

� 2CpCq4.1Cp/

� � 0 which means � � 1CpCq2CpCq

.

Secondly, we approximate L � g2H 2 by h2G2. For some choice of h �

x6C7p�q

8.1Cp/��

3C3p�q

8.1Cp/C

�

8 , we have

jL � g2H 2 � h2G2j � hG2 � x5C5pCq

8.1Cp/�

2CpCq

8.1Cp/�C

�

8 :

Note that6C7p�q8.1Cp/

� � 3C3p�q8.1Cp/

� 0 as � � 1=2 and p; q � 0.

Thirdly, observe that g2h2 � x3q�p�14.1Cp/

�3q�5p�2

4.1Cp/�C

3�

4 � x5C5pCq

8.1Cp/�

2CpCq

8.1Cp/�C

�

8

provided � < 7C7p�5q6C11p�5q

and � is small enough. One can easily check that7C7p�5q6C11p�5q

> 1CpCq2CpCq

. Therefore we have jL � g2H 2 � h2G2 C g2h2j ��

x5C5pCq

8.1Cp/�

2CpCq

8.1Cp/�C

�

8 which gives

jx � d1d2e1e2j D jx � .G � g/.G C g/.H � h/.H C h/j

�� x5C5pCq

8.1Cp/�

2CpCq

8.1Cp/�C

�

8

provided 12� � � 1CpCq

2CpCq. Choose � D 1CpCq

2CpCq, we have, after some simple

algebra,

jx � d1d2e1e2j D jx � .G � g/.G C g/.H � h/.H C h/j �� x12C

�

8 :

Now observe that with � D 1CpCq2CpCq

, after some algebra,

GH � x1=2 � xq

2.1Cp/�

q�p

2.1Cp/�C

�

2 D xpCq

2.2CpCq/C

�

2 ;

gH � x1CpCq

4.1Cp/�

2CpCq

4.1Cp/�C

�

2C

�

4 D x1CpCq

2.2CpCq/C

�

4 ;

hG � x6C7p�q

8.1Cp/��

3C3p�q

8.1Cp/C

1��

2C

�

8 D x1CpCq

2.2CpCq/C

�

8 ;

and

gh� x3q�p�18.1Cp/

�3q�5p�2

8.1Cp/�C

3�

8 D xpCq

2.2CpCq/C

3�

8 :

Therefore a1 D d1e1 D .G � g/.H � h/, b1 D d2e2 D .G C g/.H C h/,

a2 D d1e2 D .G � g/.H C h/ and b2 D d2e1 D .G C g/.H � h/ are all

D x12CO�.x

1CpCq

2.2CpCq/C

�

2 /. Therefore, there is a . 1CpCq2.2CpCq/

C �2; C�/-almost square

16 Tsz Ho Chan

of type 2 in the interval Œx � x12C

�

8 ; x C x12C

�

8 �. This shows that g.�/ � 12

for

� > 1CpCq2.2CpCq/

. Since 1Cu2Cu

is an increasing function of u, we try to find exponent

pairs that make p C q as small as possible.

For example, recently, Huxley [3] proved that .p; q/ D . 32205C �; 1

2C 32

205C �/

is an exponent pair for any � > 0. This gives1CpCq

2.2CpCq/� 743

2306C � for any � > 0

and hence Theorem 3 (v).

Note that 7432306D 0:3222029488 : : : < 1

3. However, we still cannot beat the

12

bound for g.�/. Assuming the exponent pair conjecture that .�; 12C �/ is an

exponent pair, we can push the range for � to � > 0:3 with g.�/ � 12

but this

is still shy of the range � � 14. Furthermore, if one assumes Conjecture 2 in the

previous section and imitates the proof of part (iv) or (v) of Theorem 3, one can

get g.�/ � 12

for � > 14. This comes close to the conjecture g.1

4/ D 1

2.

Acknowledgments. The author would like to thank the American Institute of

Mathematics where the study of almost squares began during a visit from 2004

to 2005. He also thanks Central Michigan University where the main idea of this

paper was worked out during a one-year visiting position (2005–2006). Finally,

he thanks the University of Hong Kong where the Erdos–Turán and exponent pair

part was worked out during a visit there in the summer of 2007.

References

[1] T. H. Chan, Finding almost squares, Acta Arith. 121 (2006), no. 3, 221–232.

[2] T. H. Chan, Finding almost squares II, Integers 5 (2005), no. 1, A23, 4 pp. (elec-

tronic).

[3] M. N. Huxley, Exponential sums and the Riemann zeta function. V., Proc. London

Math. Soc. (3) 90 (2005), no. 1, 1–41.

[4] H. L. Montgomery, Ten Lectures on the Interface Between Analytic Number The-

ory and Harmonic Analysis, CBMS Regional Conference Series in Mathematics 84,

published for the Conference Board of the Mathematical Sciences, Washington, DC,

American Mathematical Society, Providence, RI, 1994.

Author information

Tsz Ho Chan, Department of Mathematical Sciences, University of Memphis,

Memphis, TN 38152, USA.



Sums and Differences of the Coordinates

of Points on Modular Hyperbolas

Dennis Eichhorn, Mizan R. Khan, Alan H. Stein and

Christian L. Yankov

Abstract. The modular hyperbola Hn is ¹.x; y/ Wxy D 1.mod n/; 1 � x; y � n � 1º.

This simply defined set of points has connections to a variety of other mathematical topics

including Kloosterman sums, quasirandomness, and consecutive Farey fractions. These

connections have inspired a closer look at the distribution of the points of Hn, and many

questions remain open. In this paper, we examine the propensity of these points to collect

on lines of slope˙1.

Keywords. Modular hyperbola, arithmetical function, average order.

AMS classification. 11A07, 11A25, 11N37.

1 Introduction

Let Hn denote the modular hyperbola Hn D ¹.x; y/ W xy D 1.mod n/; 1 �

x; y � n � 1º: An important property of these sets is that the sequence ¹n�1Hnº

is uniformly distributed in the unit square. More precisely, if � � Œ0; 1�2 has

piecewise smooth boundary then

limn!1

#.� \ n�1Hn/

'.n/D area.�/: (1)

(We note that the cardinality of Hn is '.n/, where ' denotes the Euler phi func-

tion.) To prove (1) it suffices to only consider rectangles R � Œ0; 1�2. We can

express #.R\n�1Hn/ as an exponential sum and then invoke bounds for Kloost-

erman sums to obtain that

#.R \ n�1Hn/ D area.R/'.n/CO.�2.n/ log2.n/pn/;

where �.n/ is the number of positive divisors of n. The limit (1) is an immedi-

ate consequence of this asymptotic formula. The details of this calculation are

elegantly presented in [2, Lemma 1.7].

18 Dennis Eichhorn, Mizan R. Khan, Alan H. Stein and Christian L. Yankov

Using a computer algebra package, such as MAPLE, we can easily generate

graphs of Hn. A typical example is shown in Figure 1. Such pictures provide

convincing visual evidence of the validity of (1) and we encourage the reader to

generate other examples. The relevant MAPLE code is given in the appendix.

5000

4000

4000

3000

3000

1000

2000

20000 1000 5000

0

Figure 1. The graph H5001

In recent years quantitative forms of (1) have been given in a number of papers,

see [3, 5, 15, 17, 18] and references therein. For example, it follows from general

results of [5] that for primes p,

area.�/ �#.� \ p�1Hp/

p � 1D O

�

p�1=4 logp�

; (2)

where the implied constant depends only on �.

On a whimsical note we observe that from a visual perspective the graphs of

Hn are particularly interesting when n is small. For such integers, the cardinality

of Hn, '.n/, is small and so one can try to identify patterns in the graphs in the

same vein as one looks at clouds in the sky and identifies fanciful shapes! (Once

'.n/ takes on values in the thousands you simply see a mass of points illustrating

the uniform distribution of ¹n�1Hnº.) We give a few of our favorite examples

below.

Sums and Differences of the Coordinates of Points on Modular Hyperbolas 19

40

40

30

10

3010 20

20


80

80

0

6020

60

40

20

40

0




In displaying these images, we are delighted to reveal that a butterfly, a dragonfly,

and a scholar all lie hidden in the arithmetic structure of the integers!

Returning to matters mathematical, letD.n/; S.n/; ND.n/ and NS.n/ be the fol-

lowing sets:

D.n/ D ¹x � y W .x; y/ 2 Hnº;

S.n/ D ¹x C y W .x; y/ 2 Hnº;

ND.n/ D ¹.x � y/ mod n W .x; y/ 2 Hnº;

NS.n/ D ¹.x C y/ mod n W .x; y/ 2 Hnº:

The quantities #D.n/ and #S.n/ count the number of lines, of slope 1 and �1

respectively, that have nonempty intersection with Hn. The central results of this

paper are precise formulas for # ND.n/ and # NS.n/.

Since ¹n�1Hnº is uniformly distributed in the unit square, it is natural to be-

lieve that the ratio #D.n/=#S.n/ should be close to 1 when n is large. Further-

more, it is easy to show that for primes p,

#D.p/

#S.p/D 1 �

1 � .�1=p/

p C 1;

where .a=p/ is the Legendre symbol. (We prove this assertion at the end of this

section.) However whilst looking at some graphs of Hn (typically with n having

several factors of 2), we were quite surprised to see that there seemed to be many


more lines of slope 1 intersecting the graph than lines of slope �1. Two such

“unusual” examples are H1024 and H1728.

0 800

600

800

0

200

200 400 1000

400

600

1000


1600

1600

0

1200

1200400 800

0

400

800


We then used MAPLE to generate some data. In particular we noticed that for

powers of 2, the ratio #D.2k/=#S.2k/, with k � 10, seemed to lie between 4 and

5 (see Table 1). Our numerical work at this juncture suggested the asymptotic

#D.n/

#S.n/� 1I

but we eventually proved that

lim infn!1

#D.n/

#S.n/D 0 and lim sup

n!1

#D.n/

#S.n/D1;

a result completely contrary to our initial intuition and belief!


m 4 8 16 32 64 128 256 512 1024 2048 4096 8192 16384

#D.m/ 1 1 3 5 13 21 53 97 205 393 797 1549 3089

#S.m/ 2 4 4 8 8 12 16 28 44 84 162 328 652

Table 1. Values of #D.m/ and #S.m/ with m D 2t , t D 2; : : : ; 14

In the course of our work we realized that we could apply the Chinese Remain-

der Theorem to the sets ND.n/ and NS.n/ and consequently determine formulas for

# ND.n/ and # NS.n/. This is not the case for the setsD.n/ and S.n/; but our formu-

lae for # ND.n/ and # NS.n/, in conjunction with the inequalities

# ND.n/ � #D.n/ � 2# ND.n/ and # NS.n/ � #S.n/ � 2# NS.n/; (3)

give us upper and lower bounds for #D.n/; #S.n/ and related ratios. For ex-

ample they allow us to prove that 3 � #D.2t /=#S.2t / � 12, for t large. Two

interesting consequences of the formulas are that the mean-value of c.n/, where

c.n/ D # NS.n/=# ND.n/, is approximately 1.3; and for more than 80% of all inte-

gers c.n/ > 1.

We end this section by proving our earlier assertion that for primes

#D.p/

#S.p/D 1 �

1 � .�1=p/

p C 1:

Proposition 1. For primes p > 2,

#S.p/ Dp C 1

2(4)

and

#D.p/ Dp C .�1=p/

2; (5)

where .a=p/ denotes the Legendre symbol.

Proof. Let k 2 S.p/ and let .a; b/ 2 lk \Hp, where lk denotes the line xCy D

k. It is easy to check that a is a root of x2 � kx C 1 D 0 .mod p/. Since any

quadratic congruence modulo a prime has at most two roots, we conclude that

1 � #.lk \ Hp/ � 2. Now x D y is a line of symmetry of Hp and therefore

.b; a/ 2 lk \ Hp. If a D b then lk \ Hp D ¹.a; a/º, and if a ¤ b then

lk \Hp D ¹.a; b/; .b; a/º.

There are two of the former, ¹.1; 1/º and ¹.p � 1; p � 1/º, and .p � 3/=2 of

the latter, so #S.p/ D .p C 1/=2.

The proof of (5) is similar. We look at lines x � y D k, and since x C

y D p is a line of symmetry of Hp the points again come in pairs, .a; b/ and

.p � b; p � a/. If �1 is a quadratic residue, the counting is exactly the same and

#D.p/ D .p C 1/=2. If �1 is not a quadratic residue, there are no singleton sets

and #D.p/ D .p � 1/=2.


The set ¹jx�yj .x; y/ 2 Hpº has been studied in [12] and each result in [12]

has an analogous result for D.p/ and S.p/. In particular, the above result and

proof is essentially [12, Theorem 1]. As mentioned in the abstract, there are many

interesting questions that one can ask about modular hyperbolas. For a discussion

of recent results and open problems on modular hyperbolas we refer the reader to

the survey article [11].

2 General Strategy

From this point on, p will always denote a prime. In this section, we will apply

the Chinese Remainder Theorem to prove that the quantities # ND.n/ and # NS.n/

are multiplicative. We will then translate the problem of counting # ND.pk/ and

# NS.pk/ to one of counting squares.

Proposition 2. Let n DQm

iD1 pei

i be the canonical factorization of n. Then

# ND.n/ D

mY

iD1

# ND.pei

i / (6)

and

# NS.n/ D

mY

iD1

# NS.pei

i /: (7)

Proof. Since the proofs (6) and (7) are identical we only prove the former. The

Chinese Remainder Theorem states that the map

f W Zn !

mY

iD1

Zp

ei

i

via

f .x/ D .x mod pe1

1 ; : : : ; x mod pem

m /

is an isomorphism of rings. Consequently, when we restrict f to ND.n/ we obtain

a map g W ND.n/!Qm

iD1ND.p

ei

i /. We now show that g is a bijection.

The injectivity of g is clear, so we need to only worry about the surjectivity. Let

.k1; : : : ; km/ 2Qm

iD1ND.p

ei

i /: So there exist .ai ; bi / 2 Hp

ei

i

, with i D 1; : : : ; m,

such that .ai � bi / mod pei

i D ki . By the Chinese Remainder Theorem, the two

systems of congruences

x � ai .mod pei

i /; y � bi .mod pei

i /; i D 1; : : : ; m;

have a unique solution x D a; y D b modulo n. Since aibi � 1 .mod pei

i /

for i D 1; : : : ; m, we have that ab � 1 .mod n/. Clearly g..a � b/ mod n/ D

.k1; : : : ; km/.


We now translate the problem of counting # ND.pk/ and # NS.pk/ to one of

counting squares. We start with a minor observation.

Lemma 3. Let .a; b/ 2 Hpt . Then a � b � 2k1 .mod pt / and a C b � 2k2

.mod pt / for some k1; k2 2 Z.

Proof. If p D 2 then a; b are both odd. If p 6D 2, then 2 is invertible modulo

pt :

Theorem 4. Let .a; b/ 2 Hpt . Then

(i) .2k mod pt / 2 ND.pt /” .k2 C 1/ is a square modulo pt :

Furthermore, the map dpt .k/ D 2k mod pt defines a bijection

dpt W ¹k W k2 C 1 is a square modulo pt ; 0 � k < ptº ! ND.pt /;

when p 6D 2.

For the special case p D 2, the map d2t .k/ D 2k mod pt defines a bijec-

tion

d2t W ¹k W k2 C 1 is a square modulo 2t ; 0 � k < 2t�1º ! ND.2t /;

(that is, we restrict the elements of the domain to lie between 0 and 2t�1�1).

(ii) .2k mod pt / 2 NS.pt /” .k2 � 1/ is a square modulo pt :

Furthermore, the map spt .k/ D 2k mod pt defines a bijection

spt W ¹k W k2 � 1 is a square modulo pt ; 0 � k < ptº ! NS.pt /;

when p 6D 2.

For the special case p D 2, the map s2t .k/ D 2k mod pt defines a bijection

s2t W ¹k W k2 � 1 is a square modulo 2t ; 0 � k < 2t�1º ! NS.2t /;

(that is, we restrict the elements of the domain to lie between 0 and 2t�1�1).

Proof. Since the proofs of the two parts are identical, we only prove the result forND.pt /.

Let .a; b/ 2 Hpt . By Lemma 3, a� b � 2k .mod pt / for some k 2 Z. Upon

completing the square, we obtain k2 C 1 � .a � k/2 .mod pt /. Conversely, if

k2 C 1 is a square, then there exists c 2 Z such that c2 � k2 � 1 .mod pt /. It

follows that

.a; b/ D ..c C k/ mod pt ; .c � k/ mod pt / 2 Hpt ;

and a � b � 2k .mod pt /.


If p 6D 2, then 2 is invertible modulo pt , and consequently

d�1pt .x/ D 2�1x mod pt :

The case when p D 2 is slightly more involved. Let k be an integer, with

0 � k < 2t , such that k2 C 1 is a square modulo 2t . It follows immediately

that for the integer k1 D .k � 2t�1/ mod 2t , k21 C 1 is also a square modulo 2t .

The congruence 2x � 2k .mod 2t / has precisely two distinct solutions, which

must be k and k1. Since either k or k1 is less than 2t�1, we conclude that d2t is a

bijection.

From this we see that counting # ND.pt / or # NS.pt / is equivalent to counting

the k’s such that k2 C 1 and k2 � 1 are squares. In this context, we will on two

separate occasions invoke the following formulas of Stangl [13].

Theorem 5 (Stangl). Let p be an odd prime. Then

#¹k2 mod ptº DptC1

2.p C 1/C .�1/t�1 p � 1

4.p C 1/C

3

4: (8)

For the special case p D 2 we have that

#¹k2 mod 2t º D2t�1

3C.�1/t�1

6C

3

2; t � 2: (9)

Finally, we will need the following criteria concerning the solvability of qua-

dratic congruences. (See [8, Propositions 4.2.3, 4.2.4, page 46].)

Proposition 6. For the congruence

x2 � a .mod pt /

where p is prime and a is an integer such that p 6 j a, we have the following:

(i) p 6D 2 W If the congruence x2 � a .mod p/ is solvable, then for every

t � 2 the congruence x2 � a .mod pt / is solvable with precisely 2 distinct

solutions.

(ii) p D 2 W If the congruence x2 � a .mod 23/ is solvable, then for every

t � 3 the congruence x2 � a .mod 2t / is solvable with precisely 4 distinct

solutions.


3 The Formulas for # NS.pt/ and # ND.pt/

3.1 Case n D 2t

In this section we determine the cardinality of ND.2t / and NS.2t /.

Theorem 7. The cardinality of the set ND.2t / is

# ND.2t / D

´

1; 1 � t � 3;

2t�3; t � 4:(10)

Proof. Direct computations show that the result is true for t � 4. So we assume

that t � 5. By Theorem 4,

# ND.2t / D #¹k W k2 C 1 is a square modulo 2t ; 0 � k < 2t�1º:

We claim that

k2 C 1 is a square modulo 2t , k D 4l for some l 2 Z:

We obtain the .)/ direction by reducing modulo 8 and observing that k2 C 1 is

a square modulo 8 if and only if k � 0 .mod 8/ or k � 4 .mod 8/. To obtain

the .(/ direction we note that x2 � 16l2 C 1 .mod 8/ is solvable for any l , and

therefore by the second part of Proposition 6, .4l/2 C 1 D 16l2 C 1 is a square

modulo 2t for all l . Hence,

¹k W k2 C 1 is a square modulo 2t ; 0 � k < 2t�1º D ¹4l W 0 � l < 2t�3º;

and therefore # ND.2t / D 2t�3.

Theorem 8. The cardinality of the set NS.2t / is

# NS.2t / D

8

ˆ

<

ˆ

:

1; t D 1; 2;

2; t D 3; 4;2t�4

3C .�1/t�1

3C 3; t � 5:

(11)

Proof. Direct computations show that the result is true for t � 6 and so we may

assume that t � 7. We will prove that

#¹k W k2 � 1 is a square modulo 2t ; 0 � k < 2t�1º D 2 � #¹k2 mod 2t�4º

and conclude by applying (9).

Let j 2 ¹k W k2 � 1 is a square modulo 2t ; 0 � k < 2t�1º. Since j 2 � 1 is a

square modulo 2t , we have that j D 2l C 1 for some l; 0 � l < 2t�2. (Otherwise


we would have that �1 is a square modulo 4.) It follows that .l2 C l/ is a square

modulo 2t�2 and so we can conclude that

#¹k W k2 � 1 is a square modulo 2t ; 0 � k < 2t�1º

D #¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2º:

The set ¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2º is the union of

¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2; l oddº

and

¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2; l evenº:

Since l2 C l � .2t�2 � 1 � l/2 C .2t�2 � 1 � l/ .mod 2t�2/,

#¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2; l oddº

D #¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2; l evenº:

Therefore,

#¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2º

D 2#¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2; l oddº:

Now

#¹l W l2 C l is a square modulo 2t�2; 0 � l < 2t�2; l oddº

D #¹l�1 C 1 W l�1 C 1 is a square modulo 2t�2; 0 � l < 2t�2; l oddº:

If l�1 C 1 is square modulo 2t�2, then it is a multiple of 4 and consequently

l�1 C 1 � 4m2 .mod 2t�2/

for some m; 0 � m < 2t�4. Consequently,

#¹l�1 C 1 W l�1 C 1 is a square modulo 2t�2; 0 � l < 2t�2º

D #¹k2 mod 2t�4º;

which ends the proof.


3.2 Case n D pt, p an Odd Prime

Proposition 9. If p � 1 .mod 4/, then for all t ,

# ND.pt / D # NS.pt /: (12)

Proof. If p � 1 .mod 4/, then for any value of t the congruence x2 � �1

.mod pt / has a solution, ipt . The map Ipt W Zpt ! Zpt via Ipt .x/ D iptx is a

bijection.

Let k 2 ¹k mod pt W k2 � 1 is a square modulo ptº. Then k2 � 1 � m2

.mod pt / for some m, and consequently

i2ptk

2 C 1 � �.k2 � 1/ � �m2 � .iptm/2 .mod pt /:

The above calculation shows that

Ipt

�

¹k mod pt W k2 � 1 is a square modulo ptº�

D ¹k mod pt W k2 C 1 is a square modulo ptº;

and therefore by Theorem 4, # ND.pt / D # NS.pt /.

For the rest of this section we will use the following notation: Let

S 0.pt / D ¹k mod pt W k2 � 1 is a square modulo pt ; p 6 j .k2 � 1/º;

S 00.pt / D ¹k mod pt W k2 � 1 is a square modulo pt ; p j .k2 � 1/º;

D 0.pt / D ¹k mod pt W k2 C 1 is a square modulo pt ; p 6 j .k2 C 1/º;

and

D 00.pt / D ¹k mod pt W k2 C 1 is a square modulo pt ; p j .k2 C 1/º:

We note that the bijections from Theorem 4 imply

# NS.pt / D #S 0.pt /C #S 00.pt /

and

# ND.pt / D #D 0.pt /C #D 00.pt /;

which explains our notation. In our next theorem we determine #D 0.pt / and

#S 0.pt / by calculating a sum of Legendre symbols. We then determine #D 00.pt /

and #S 00.pt / by applying Stangl’s formula (8).


Theorem 10. Let p be an odd prime, and let .a=p/ denote the Legendre symbol.

Then

#S 0.pt / D.p � 3/pt�1

2(13)

and

#D 0.pt / D

´

.p � 3/pt�1=2; p � 1 .mod 4/;

.p � 1/pt�1=2; p � 3 .mod 4/:(14)

Proof. The proofs of (13) and (14) are identical and so we will only do the second

one. If l 2 D 0.pt /, then the Legendre symbol�

.l2 C 1/=p�

D 1. Therefore,

#D 0.pt / D1

2

pt �1X

lD0; gcd.l2C1;p/D1

��

l2 C 1

p

�

C 1

�

D1

2

pt�1�1X

kD0

p�1X

lD0; l2 6D�1 .mod p/

��

.l C kp/2 C 1

p

�

C 1

�

D

� p�1X


��

l2 C 1

p

�

C 1

��

pt�1

2

D

�

� 1C

p�1X


1

�

pt�1

2;

where the �1 term in the last expression arises by invoking

p�1X

aD0

�

a2 C 1

p

�

D �1;

(see [1, Theorem 2.1.2, page 58]). We complete our proof by noting that

p�1X


1 D

´

p � 2; p � 1 .mod 4/;

p; p � 3 .mod 4/:

Lemma 11. If p � 3 .mod 4/, then

#D 00.pt / D 0;

and consequently

# ND.pt / D #D 0.pt / D'.pt /

2: (15)


Proof. If p � 3 .mod 4/ then the congruence x2 C 1 D 0 .mod p/ has no

solutions and consequently

#D 00.pt / D 0:

Proposition 12. The cardinality of the set S 00.pt / is

#S 00.pt / D

´

2; t � 2;pt�1

pC1C 3

2C .�1/t�3 p�1

2.pC1/; t � 3:

(16)

Proof. Let k 2 S 00.pt /. Then k2 � 1 � p2m2 .mod pt / for some m; 0 � m <

pt�2. We have two cases to consider.

(i) t � 2: In this case we have m D 0, and since the congruence x2 � 1

.mod pt / has exactly 2 solutions, we conclude that #S 00.pt / D 2.

(ii) t � 3: In this case we define a map

S 00.pt /! ¹m2 mod pt�2º

via k 7! m2. By Proposition 6, for each m2 2 ¹m2 mod pt�2º, the con-

gruence x2 � p2m2 C 1 .mod pt / is solvable with precisely two solu-

tions. Hence, the map is surjective with each element in the image hav-

ing exactly two elements in its preimage. Consequently, we can infer that

#S 00.pt / D 2#¹m2 mod pt�2º, and we conclude by invoking (8).

On combining all of the pieces we obtain the following formulas for # ND.pt /

and # NS.pt /.

Theorem 13. The cardinalities of the sets ND.pt / and NS.pt / are

# ND.pt / D

8

ˆ

ˆ

<

ˆ

ˆ

:

.p�3/pt�1

2C 2; p � 1 .mod 4/; t � 2;

.p�3/pt�1

2C pt�1

pC1C 3

2C .�1/t�3 p�1

2.pC1/; p � 1 .mod 4/; t � 3;

.p�1/pt�1

2D '.pt /

2; p � 3 .mod 4/;

(17)

and

# NS.pt / D

´

.p�3/pt�1

2C 2; t � 2;

.p�3/pt�1

2C pt�1

pC1C 3

2C .�1/t�3 p�1

2.pC1/; t � 3:

(18)


4 Some Properties of c.n/ D # NS.n/=# ND.n/

To simplify notational matters, let c.n/ D # NS.n/=# ND.n/. We use our formulas

for # NS.n/ and # ND.n/ to determine some properties of c.n/. We begin with some

elementary, but useful remarks about c.pk/.

Lemma 14. c.pk/ has the following properties.

(i) For p � 1 .mod 4/,

c.pk/ D 1: (19)

(ii) For p � 3 .mod 4/;

c.p/ D

�

1C2

p � 1

�

: (20)

(iii) For p � 3 .mod 4/ and k � 2;

c.pk/ � c.pk�1/ D

´

� 4

pk�1 ; k even;

� 2

pk�1 ; k odd:(21)

(iv) For k � 6,

c.2k/ � c.2k�1/ D

´

� 1

2k�5 ; k even;

� 1

2k�4 ; k odd:(22)

(v) For k � 1,

1=6 � c.2k/ � 2: (23)

(vi) For k � 2,1

4� c.3k/ �

2

3: (24)

(vii) For p � 5; p � 3 .mod 4/ and k � 2,

�

1 �2

p � 1

�

� c.pk/ �

�

1 �2

p � 1C

4

p2 � p

�

< 1: (25)

Proof. These are immediate consequences of formulas (17) and (18). We make a

few remarks: the maximum value of c.2k/ is c.8/ D 2; in (25) the upper bound

is achieved when k D 2; if p � 3 .mod 4/ then for any k � 2, c.pk/ < c.p/.

Finally, the reason we treated the case of 3k separately is that the LHS of (25)

equals 0 when p D 3. We could have merged (24) and (25) by replacing the LHS

of (25) with�

1 �2

p � 1C

2

p2 � 1

�

:


We now prove that the maximal and minimal order of c.n/ is log logn and

1= log logn respectively.

Theorem 15. Let Nk DQk

iD1 pi , where pi is the i-th prime that is congruent to

3 modulo 4.

(1) We have

c.Nk/ � log logNkI (26)

and for any t � 2,

c.N tk/ � .log logNk/

�1: (27)

Consequently,

lim supn!1

c.n/ D1; (28)

and

lim infn!1

c.n/ D 0: (29)

(2) Furthermore,

c.n/� log logn (30)

and

c.n/�1

log logn: (31)

Proof. For (1), we only prove (26) as the proof of (27) is similar. Since

c.Nk/ D

kY

iD1

�

1C2

pi � 1

�

;

we have by the Taylor series of log.1C x/ that

log c.Nk/ D 2

kX

iD1

1

pi � 1C

kX

iD1

1X

mD2

.�1/m�1

m

�

2

pi � 1

�m

: (32)

For p � 5,

ˇ

ˇ

ˇ

ˇ

ˇ

4

.p � 1/2

1X

mD2

.�1/m�1

m

�

2

p � 1

�m�2ˇ

ˇ

ˇ

ˇ

ˇ

�4

.p � 1/2

1X

mD0

2�m D8

.p � 1/2I

and therefore the double series

C D

1X

iD1

1X

mD2

.�1/m�1

m

�

2

pi � 1

�m


converges since the i D 1 term is a convergent alternating series and the remaining

terms are bounded in magnitude byP

1

iD2 8i�2.

Since C < 0, we obtain from (32) the inequality

2

kX

iD1

1

piC C � log c.Nk/ � 1C 2

kX

iD1

1

pi: (33)

We now apply to (33) Mertens’s formula (see [4, Section 2.2]) for primes p � 3

.mod 4/,

limk!1

0

@

kX

iD1

1

pi�

1

2log logpk

1

A D 0:048239 : : : ;

and obtain

log.c.Nk// � log log.pk/ D O.1/: (34)

The prime number theorem for arithmetic progressions for primes p � 3

.mod 4/ implies

limk!1

logNk

pk

D1

2:

Consequently log log logNk�log logpk D o.1/ and so we can replace log logpk

with log log logNk in (34) and obtain the desired conclusion that log.c.Nk// �

log log logNk D O.1/:

The proof of (27) is nearly identical. The main difference is that we start with

the inequality

1

4

kY

iD2

�

1 �2

pi � 1

�

� c.N tk/ �

kY

iD1

�

1 �2

pi � 1C

4

p2 � p

�

;

that we obtain by applying the inequalities (24) and (25).

We next prove item (2). If n has no prime factors congruent to 3 modulo 4,

then we have the inequality 1=6 � c.n/ � 2. So without loss of generality we

may assume that q1; : : : ; qk are the distinct prime factors of n that are congruent

to 3 modulo 4. Clearly n � q1 : : : qk � Nk . From (20) and (26) we get that

c.n/ � 2c.Nk/ � log logNk � log logn:

We now prove (31). An immediate consequence of the asymptotic

2

kX

iD1

1

pi� log log logNk D O.1/

is that

1

4

kY

iD2

�

1 �2

pi � 1

�

�1

log logNk

:


We combine this with the inequality

c.n/ �1

24

kY

iD2

�

1 �2

qi � 1

�

�1

24

kY

iD2

�

1 �2

pi � 1

�

to obtain (31).

Corollary 16. For the sequence #S.n/=#D.n/ we have

lim inf#S.n/

#D.n/D 0 and lim sup

#S.n/

#D.n/D1: (35)

Proof. Since # ND.n/ � #D.n/ � 2# ND.n/ and # NS.n/ � #S.n/ � 2# NS.n/, we

obtain the inequality

0:5c.n/ �#S.n/

#D.n/� 2c.n/:

We now apply (29), (28) to obtain (35).

Corollary 17. The Dirichlet seriesP

1

nD1 c.n/n�s converges absolutely in the

half-plane <.s/ > 1.

Proof. This is an immediate consequence of (30).

We preface our calculation of the mean value of c.n/ with the following ob-

servation about mean values of arithmetical functions. Let

M.g/ D limx!1

1

x

X

n�x

g.n/

denote the mean value of g. If M.g/ exists, then, via partial summation, we have

that

M.g/ D lims!1C

P

1

nD1 g.n/n�s

�.s/:

Thus if g is multiplicative then we can represent M.g/ by the infinite product

M.g/ DY

p prime

�

1 �1

p

�

1X

kD0

g.pk/

pk

!

DY

p prime

1C

1X

kD1

g.pk/ � g.pk�1/

pk

!

:

Since we have explicit formulas for c.pk/ we can easily compute M.c/ provided

we first show that this mean value exists. Arguably the simplest proof of the

existence of M.c/ is to invoke Wintner’s mean-value theorem for multiplicative

functions.


Theorem 18 (Wintner). If g is a multiplicative function satisfying the conditions

X

p prime

jg.p/ � 1j

p<1 and

X

p prime

1X

kD2

jg.pk/ � g.pk�1/j

pk<1;

then M.g/ exists.

This was first proved in [16] – a monograph that is very difficult to find. For a

more recent and accessible reference see [10, II.2, Corollary 2.3]. The proof is a

straightforward convolution argument.

Theorem 19. The mean-value of c.n/, M.c/, is given by the infinite product

M.c/ D limx!1

1

x

X

n�x

c.n/ D337

320

Y

p�3 .mod 4/

�

1C2.p2 � p C 1/

p4 � 1

�

� 1:32:

(36)

Proof. From the properties of c.pk/ listed in Lemma 14 the series

X

p prime

jc.p/ � 1j

pand

X

p prime

1X

kD2

jc.pk/ � c.pk�1/j

pk

are both convergent and therefore by Wintner’s theorem we conclude that M.c/

exists. We now use (10), (11), (19), (20), (21) and (22) to obtain M.c/.

Our final result shows that c.n/ > 1 for over 80% of all integers. We prove it

by applying Wirsing’s mean-value theorem for multiplicative functions [14, III.4,

Theorem 5].

Theorem 20 (Wirsing). If g is a real multiplicative function with jg.n/j � 1 for

all n 2 ZC, then M.g/ exists.

Wirsing’s theorem is a deep theorem. For example, it contains the Prime Num-

ber Theorem in its equivalent form M.�/ D 0, where � is the Möbius function,

see [7, Section 3]. We would have preferred to have used a simpler result such

as Wintner’s theorem; however, the condition thatP

p prime jg.p/� 1jp�1 is con-

vergent is not satisfied in one part of our argument. We will need the following

lemma to justify the use of Wirsing’s theorem.

Lemma 21. For each prime p, let vp.n/ denote the exponent of the prime p in the

canonical factorization of n and let Ap denote a non-empty subset of ZC [ ¹0º.

Then the characteristic function of the set ¹n W vp.n/ 2 Apº[¹1º is multiplica-

tive.


Proof. We construct a function � W ZC ! ¹0; 1º in the following way. Let

�.1/ D 1 and, for k � 1, let

�.pk/ D

´

1; k 2 Ap;

0; k 62 ApI

and

�.n/ DY

l

�.pel

l/;

whereQ

l pel

lis the canonical factorization of n. Clearly, � is both multiplicative

and also the characteristic function of ¹n W vp.n/ 2 Apº[¹1º.

Theorem 22. Let C D ¹n W c.n/ > 1º and let � denote the characteristic function

of C . Then the lower density of C , lim inf x�1P

n�x �.n/, satisfies the inequality

lim infx!1

1

x

X

n�x

�.n/ �63

64

Y

p�3 .mod 4/

�

1 �1

p2

�

� 0:84:

Furthermore, for any positive constant L, the set ¹n W c.n/ � Lº has positive

lower density.

Proof. Let C1;C2;C3 be the sets

C1 D ¹n W v2.n/ � 5; and vp.n/ � 1 if p � 3 .mod 4/º;

C2 D ¹n W n 2 C1; v2.n/ 6D 3; and vp.n/ D 0 if p � 3 .mod 4/º;

and C3 D C1 n C2; and let �1 and �2 be the characteristic functions of C1 and C2

respectively. The conditions on C1 ensure that for any n 2 C1 and for any prime

p, c.pvp.n// � 1. Therefore for any n 2 C1 we have c.n/ � 1 with equality

precisely when n 2 C2, showing that C3 � C .

By Lemma 21, �1 and �2 are multiplicative functions and so we can apply

Wirsing’s theorem to obtain that

density.C3/ DM.�1 � �2/ DM.�1/ �M.�2/

DY

p prime

�

1 �1

p

�

1C

1X

iD1

�1.pi /

pi

!

�Y

p prime

�

1 �1

p

�

1C

1X

iD1

�2.pi /

pi

!

D63

64

Y

p�3 .mod 4/

�

1 �1

p2

�

� 0 D63

64

Y

p�3 .mod 4/

�

1 �1

p2

�

:


(We cannot invoke Wintner’s theorem here as the seriesP

p j�2.p/ � 1jp�1 is

divergent.) By our formulas for # ND.n/ and # NS.n/ we have the inclusion C3 � C ,

and so we can conclude that the lower density of C is greater or equal to the

density of C3.

A slight variation of the above proof gives the second assertion. Recalling the

notation in Theorem 15, let pi denote the i-th prime congruent to 3 modulo 4 and

let Nk DQk

iD1 pi . Since c.Nk/ � log logNk (see asymptotic (26)), we can find

an integer l such that c.Nk/ � L for k � l . Let

L1 D ¹n W v2.n/ D 0 and vpi.n/ � 1 for i D 1; 2; : : : º;

L2 D ¹n W n 2 L1 and vpi.n/ D 0 for i D 1; : : : ; lº;

and L3 D L1 nL2. A slight modification of our earlier calculation of density.C3/

gives that

density.L3/ D1

2

0

@1 �

lY

iD1

1

1C 1pi

1

A

Y

p�3 .mod 4/

�

1 �1

p2

�

:

We conclude by observing that for any n 2 L3, c.n/ � c.Nl / � L.

4.1 Unanswered Questions and Ongoing Work

We have not resolved the following 3 questions.

(i) Does the density of A D ¹n 2 ZC W c.n/ D 1º equal 0? This would follow

if we could prove that for any n 2 A, the odd prime factors of n are all

congruent to 1 mod 4.

(ii) What is the density of ¹n 2 ZC W c.n/ < 1º? Is it non-zero?

(iii) What is the normal order of c.n/?

It is easy to generalize Proposition 2 to arbitrary polynomials in ZŒx; y�. Spe-

cifically, if f 2 ZŒx; y� and we define the map fn W Hn ! Zn via fn..x; y// D

f .x; y/ mod n, then the quantity #Image.fn/ is a multiplicative function of n.

One possible extension of our work is to determine formulas for #Image.fpe / for

some other polynomials f 2 ZŒx; y�, especially for cases where one can apply the

formulas in [13] and this paper. S. Hanrahan, under the supervision of M. Khan,

is currently writing an undergraduate honors thesis on this topic for the quadratic

forms x2 C y2 and x2 � y2.


5 Appendix

This is the MAPLE code that generated the graph of H5001.

n:=5001:

a:=array(1..numtheory[phi](n)):

b:=array(1..numtheory[phi](n)):

count:=1:

for i from 1 to n-1 do;

if gcd(i,n)=1 then

a[count]:=i: b[count]:=(i^(-1)mod n):

count := count+1;

end if;

end do;

printf("n=\%d, no. of points on graph=\%d \n",n,count-1):

points := zip((x,y) -> [x,y],a,b):

p1:=plot(points,style=POINT,symbol=CROSS):

plots[display](p1);

References

[1] B. C. Berndt, R. J. Evans and K. S. Williams, Gauss and Jacobi Sums, Wiley, 1997.

[2] F. Boca, C. Cobeli and A. Zaharescu, Distribution of Lattice Points Visible from the

Origin, Commun. Math. Phys. 213 (2000), 433–470.

[3] C. Cobeli and A. Zaharescu, On the Distribution of the Fp-points on an Affine Curve

in r Dimensions, Acta Arithmetica 99 (2001), 321–329.

[4] S. R. Finch, Mathematical Constants, Encylopedia of Mathematics and its Applica-

tions 94, Cambridge, 2003.

[5] A. Granville, I. E. Shparlinski and A. Zaharescu, On the Distribution of Rational

Functions Along a Curve over Fp and Residue Races, J. Number Theory 112 (2005),

216–237.

[6] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed.,

Oxford, 1985.

[7] A. Hildebrand, Some New Applications of the Large Sieve, in: Number Theory

(New York 1985–1988), pp. 76–88, Lecture Notes in Mathematics 1383, Springer-

Verlag, 1989.

[8] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory,

Springer-Verlag, 1982.

[9] H. L. Montgomery and R. C. Vaughan, Multiplicative Number Theory I. Classical

Theory, Cambridge, 2007.


[10] W. Schwarz and J. Spilker, Arithmetical Functions, Cambridge, 1994.

[11] I. E. Shparlinski, Distribution of Points on Modular Hyperbolas, Sailing on the Sea

of Number Theory: Proc. 4th China-Japan Seminar on Number Theory, Weihai,

2006, World Scientific, 2007, 155–189.

[12] I. E. Shparlinski and A. Winterhof, On the Number of Distances Between the Co-

ordinates of Points on Modular Hyperbolas, J. Number Theory 128 (2008), 1224–

1230.

[13] W. Stangl, Counting squares in Zn, Math. Mag. 69 (1996), 285–289.

[14] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cam-

bridge, 1995.

[15] M. Vajaitu and A. Zaharescu, Distribution of Values of Rational Maps on the Fp-

points on an Affine Curve, Monathsh. Math. 136 (2002), 81–86.

[16] A. Wintner, Eratosthenian Averages, Baltimore, 1943.

[17] W. Zhang, On the Distribution of Inverses Modulo n, J. Number Theory 61 (1996),

301–310.

[18] Z. Zheng, The Distribution of Zeros of an Irreducible Curve over a Finite Field, J.

Number Theory 59 (1996), 106–118.

Author information

Dennis Eichhorn, Department of Mathematics, University of California,

Irvine, CA 92697, USA.


Mizan R. Khan, Department of Mathematics and Computer Science, Eastern Connecticut

State University, Willimantic, CT 06226, USA.


Alan H. Stein, Department of Mathematics, University of Connecticut,

Waterbury, CT 06702, USA.


Christian L. Yankov, Department of Mathematics and Computer Science, Eastern Con-

necticut State University, Willimantic, CT 06226, USA.



Self Generating Sets and

Numeration Systems

David Garth, Joseph Palmer and Ha Ta

Abstract. Kimberling has studied a variety of sets generated in the following way. Let

F be a countable set of functions, and let S D SF be the smallest set containing 0 that

is closed under any function in F . When F D ¹2x; 4x C 1º, the resulting set S is pre-

cisely the set of nonnegative integers whose binary expansion does not contain the block

11. The set of all such binary expansions corresponds to the set of greedy representa-

tions of the natural numbers with respect to the Fibonacci sequence. Other examples of

a similar nature can be found in the literature. In this paper we explore the following

question; which self generating sets consist of integers whose digit expansions in base

two correspond to the digit expansions of the natural numbers with respect to a linearly

recurrent base sequence? We study this problem in the framework of abstract numeration

systems. That is, we consider an abstract numeration system as an infinite language over

a finite alphabet, ordered under a genealogical ordering. We then define a self generating

numeration system as one that can be realized as the set of base two expansions of the

integers in some self generating set. Our first result is to prove a necessary and sufficient

condition for an abstract numeration system to have a base. This result is then used to

prove that certain families of generating functions give rise to self generating numeration

systems that have a base sequence. Finally, we prove that the base sequence in any based

self generating numeration system satisfies a linear recurrence. Many of our results make

use of a natural tree structure that can be put on an abstract numeration system.

Keywords. Numeration systems, greedy expansions, lazy expansions, self generating

sets.

AMS classification. 11B13, (11B85).

1 Introduction

The purpose of this paper is to explore some of the connections between self

generating sets and abstract numeration systems. Kimberling ([9], [10], [11]) has

studied a variety of sets generated in the following way. Let F be a countable set

The work of the first and second authors was supported in part by NSF Grant No. 0431664.

42 David Garth, Joseph Palmer and Ha Ta

of functions, and let S D SF be the smallest set containing 0 that is closed under

all the functions in F . In other words, 0 2 S , and if x 2 S and f 2 F then

f .x/ 2 S . Moreover, no other elements are in S .

As an example, the set S generated by F D ¹2x; 4x C 1º has been consid-

ered in the literature ([1], [10], [8]). Since 2x simply adds a zero to the binary

expansion of x, and since 4x C 1 adds a 01, the ordered sequence

¹0; 1; 2; 4; 5; 8; 9; 10; 16; : : : º

of elements of S consists precisely of those integers whose binary expansions have

no adjacent ones. Allouche, Shallit, and Skordev [2] studied the self generating

set S arising from F D ¹2x C 1; 4x C 2º and showed that in this case

S D ¹0; 1; 2; 3; 5; 6; 7; 10; 11; 13; 14; 15; : : : º;

the set of natural numbers whose binary expansions do not contain the block 00.

The connection with numeration systems becomes apparent when considering

the so-called greedy representations of the natural numbers with respect to some

sequence. More precisely, let ¹biºi�0 be a strictly increasing sequence of natural

numbers, with b0 D 1. The sequence ¹biº serves as a base for a positional numer-

ation system for the natural numbers as follows. For n � 1, let k � 0 be such that

bk � n < bkC1. Using the division algorithm we write

n D qkbk C rk; where 0 � rk < bk :

Thus, qk D bn

bk

c where b�c is the greatest integer function. For i D k � 1; : : : ; 0

let qi D briC1

bi

c and ri D riC1 � biqi . It follows that n D q0b0 C � � � C qkbk ,

and we say that the string qkqk�1 � � � q1q0 is the greedy representation of n with

respect to the sequence ¹biº. We define the greedy representation of 0 to be 0,

although most authors define it to be the empty word.

Suppose ¹biº is the Fibonacci sequence, indexed as b0 D 1; b1 D 2, and

bi D bi�1C bi�2 for i � 2. Using the above algorithm it is not hard to show that

the set of greedy representations of the natural numbers in this case consists of 0

and all words over ¹0; 1º that begin with 1 and do not contain the block 11. This

is precisely the set of binary expansions of the elements of the set S generated by

F D ¹2x; 4x C 1º.

The lazy representation of a natural number n with respect to the Fibonacci

sequence is obtained by successively replacing all occurrences of the string 100

in the greedy Fibonacci representation of n with the string 011 until the resulting

string contains no occurrences of the block 00. Any leading zeros that arise in

this process are disregarded. Thus, for F D ¹2x C 1; 4x C 2º, the set of base

Self Generating Sets and Numeration Systems 43

2 expansions of elements of S is the set of lazy Fibonacci representations of the

natural numbers.

These examples are easy to generalize, and some examples are considered in

[8]. For example, if F D ¹2x; 4x C 1; 8x C 3º, then

SF D ¹0; 1; 2; 4; 8; 9; 10; 16; 17; 18; 32; 33; 34; 36; : : : º

consists of the natural numbers whose binary expansions do not contain the block

111. These binary expansions correspond to the greedy expansions of the natural

numbers with respect to the base ¹biº of Tribonacci numbers, where b0 D 1, b1 D

2, b2 D 4, and bi D bi�1Cbi�2Cbi�3 for i � 3. If F D ¹2xC1; 4xC2; 8xC4º,

then the elements of the resulting self generating set S are precisely those natural

numbers whose binary expansions do not contain the block 000. The set of all

such expansions corresponds to the set of lazy Tribonacci representations of N

that are defined in a manner analogous to the lazy Fibonacci representations of N.

In this paper we will place these results in a more general context. In Section

2 we define the notion of an abstract numeration system. Under this definition,

the set of base two expansions of the elements of any self generating set can be

considered as the set of representations of the natural numbers in such a system.

We will then prove a result that will be useful for determining whether such an

abstract numeration system has a base sequence. In Section 3 we apply this result

to give conditions on the set of generating functions F which guarantee that the

binary expansions of the elements of SF correspond to the digit expansions of the

natural numbers with respect to some base sequence ¹biº. Of natural interest is the

question of whether the base sequence satisfies a linear recurrence. In Section 4

we show that for any set of generating functions F , if the binary expansions of the

elements of SF correspond to the expansions of the natural numbers with respect

to some base sequence, then the base sequence must satisfy a linear recurrence

relation.

2 Numeration Systems

Non-standard numeration systems and more generally the so-called abstract nu-

meration systems have been considered in a variety of settings in the literature

([4], [3], [5], [7], [12], [13], [14], [15]). Generally such a numeration system is

regarded as a set S of words over some alphabet along with a bijection between

that set of words and the natural numbers. The alphabet is the set of allowable

digits, the words in S are the valid representations of the natural numbers, and the

bijection gives a means for obtaining the numerical value of a given representa-

tion. In this paper we take our digit set to be †2 D ¹0; 1º. To define our bijection

we recall a few definitions. Let †�

2 denote the set of all finite words over †2, and


let †C

2 be the set of nonempty words over †2. For w 2 †�

2 , let jwj denote the

length of w. The radix order [13] on †�

2 is the ordering where, for w; v 2 †�

2 ,

w < v if jwj < jvj or if jwj D jvj and w D uaw0 and v D ubv0 with a; b 2 †2

and a < b in the natural order on †2. Enumerating the elements of S under the

radix order induces a natural order preserving bijectionN W S ! N. (We will use

the convention that 0 2 N). We are now ready for our definition of a numeration

system.

Definition 1. A numeration system is an ordered pair .S ; N /, where S is an infi-

nite subset of .†C

2 n 0†�

2 / [ ¹0º and N is the natural order preserving bijection

from S to N that maps the .nC 1/st word of S to n. The map N is referred to as

the evaluation map, and N�1 is the representation map. If N.v/ D n, then v is

the S-representation of n.

A few remarks about this definition are in order. First, while the definition

generalizes naturally to allow for more general digit sets, we will consider only

numeration systems with digit set †2. Also, since the evaluation map will always

be obtained from the radix order, we therefore write S instead of .S ; N /. Sec-

ond, notice that our definition requires that 0 2 S . While this restriction is not

necessary in general, it is natural for our purposes. Finally, our condition that

S � .†C

2 n 0†�

2 / [ ¹0º ensures that for n � 1 the S-representation of n begins

with 1. More general numeration systems that allow for leading zeros have been

considered elsewhere in the literature (e.g. [12], [14]), but will not be considered

here.

Suppose S is a numeration system, and let B D ¹biº1

iD0be an arbitrary

sequence of natural numbers. Let �B W S ! N be the function that assigns

w D wkwk�1 � � �w0 2 S to

�B.w/ D

kX

iD0

biwi : (1)

A numeration system S is based if there exists a strictly increasing sequence B D

¹biº1

iD0of natural numbers, with b0 D 1, such thatN.w/ D �B.w/ for allw 2 S .

In this case, the sequence ¹biº is called the base sequence of S . The following

lemma will be important in Sections 3 and 4.

Lemma 1. If S is a based numeration system, then 1 2 S .

Proof. Let B D ¹biº be the base sequence. By definition b0 D 1. Since S is a

numeration system, there exists a w 2 S for which N.w/ D 1. Since S is based,

�B.w/ D N.w/ D 1. Since ¹biº is an increasing sequence, if jwj > 1 then

�B.w/ > 1. It follows then that jwj D 1, and therefore w D 1. Thus, 1 2 S .


We say a based numeration system S is greedy if whenever w 2 S � ¹0º and

v 2 †�

2 � ¹0º with

N.w/ D �B.w/ D �B.v/;

it follows that w � v under the radix order. In other words, for every n � 1 the S-

representation of n is the largest possible representation under the radix order. The

numeration system S is said to be lazy if for every n � 1 the S-representation of n

is the smallest possible representation. The greedy and lazy Fibonacci representa-

tions mentioned in Section 1 provide examples of these definitions. In general, if

¹biº is the base sequence in a greedy numeration system S and if supi�0biC1

bi

< 2

then the digits in the S-representation for n 2 N can be obtained by the algorithm

given in Section 1.

It is natural to consider numeration systems with the property thatw0 2 Sn¹0º

whenever w 2 S n ¹0º. Such a numeration system is said to be right extendable

(see [13], Section 7.3.2). Similarly, S is a Bertrand numeration system if w 2

S n ¹0º, w0 2 S n ¹0º [4]. The set of greedy Fibonacci representations of

the natural numbers mentioned in Section 1 is a Bertrand numeration system,

while the set of lazy Fibonacci representations is not even right extendable. The

following lemma gives a necessary condition for a numeration system that is also

right extendable to be based.

Lemma 2. Let S be a right extendable based numeration system. If w;w1 2 S

then w01 2 S .

Proof. Let B D ¹biº be the base sequence for S . Since S is right extendable, it

follows that w00 2 S and w10 2 S . It must be true then that

b1 D �B.10/ � �B.0/ D �B.w10/ � �B.w00/ D N.w10/ �N.w00/:

If w01 … S , then N.w10/ � N.w00/ D 1, which implies that b1 D 1. However,

since b1 > 1 we have a contradiction, and so w01 2 S .

To any numeration system S we associate a graph T .S/ with vertex set S and

edge set

¹.v; v�/ W � 2 ¹0; 1º and v; v� 2 Sº:

In other words, we draw an edge between v and v� whenever both are members

of S . This graph structure clearly partitions S into a collection of trees. To every

tree � in T .S/ we define the root of � to be the vertex in � of minimal length

under the radix order. Notice that by definition every vertex in T .S/ has at most

two children. We say that for w 2 S , if w0 2 S , then w0 is the left child of w,

and if w1 2 S , then w1 is the right child of w. Whenever S is right extendable,

every vertex in T .S/ has at least one child. Numeration systems in which T .S/

is a single rooted tree are of particular interest.


Definition 2. Let S be a Bertrand numeration system, with 1 2 S . If T .S/ is such

that w 2 S whenever w1 2 S , then S is treelike.

The conditions of the definition guarantee that T .S/ is a single rooted tree.

These conditions also imply that if w 2 S then every prefix of w is in S . Treelike

numeration systems were introduced in [5]. As an example, Figure 1 shows the

first few levels of the tree for the greedy Fibonacci numeration system mentioned

in the introduction. The vertices in the tree of Figure 1 are the greedy Fibonacci

representations of the natural numbers. The numbers in parentheses are the stan-

dard base 10 values of these representations.

0, (0)

1, (1)

10, (2)

100, (3)

1000, (5)

10000, (8) 10001, (9)

1001, (6)

10010, (10)

101, (4)

1010, (7)

10100, (11) 10101, (12)

Figure 1. The tree T .S/ for the greedy Fibonacci numeration system.

In a given numeration system S , for k � 0 we define Mk to be the word in

S having k digits that is maximal under the radix order. We take M0 to the the

empty word. Similarly, for k � 1 let mk be the minimal word having k digits in

S . If 1 2 S it will be convenient to define m1 to be 1. If S has the property that

Mk is a prefix of MkC1 for every k, we let

M D limk!1

Mk: (2)

This M is then referred to as the maximal word associated with S . We define the

minimal word m of S similarly. It is clear that in a treelike numeration system,

mk D 10k�1.

Our first theorem establishes a necessary and sufficient condition for a nu-

meration system to be based. We point out that the theorem actually resembles

Theorem 5.2 of [5]. That theorem is restricted to treelike numeration systems, and

so the following theorem is more general. We also comment that the hypotheses

of the theorem guarantee that any two words of length at least 2 share a common

nonempty prefix.


Theorem 1. Let S be a right extendable numeration system, and assume 1 2 S .

Then S is based if and only if for any two consecutive words v;w 2 S of length

l � 2 having maximal common prefix p it follows that

v D p0Ml�jpj�1; (3)

w D pml�jpj:

Proof. Suppose first that S is based, with base B D ¹biº. Consider two arbitrary

adjacent words v and w in S of the same length l � 2, with v < w. The property

of the theorem clearly holds if l D 2. Suppose that l � 3, and let p be the

maximal common prefix of v and w. Then we can write

v D p0xk�2 � � � x0 and w D p1yk�2 � � �y0;

where k D l � jpj. Note that k � 1. If k D 1, then v D p0 D p0M0 and

w D p1 D pm1, and the property of the theorem is satisfied. Assume then that

k � 2. Since v andw are adjacent in S , it is clear thatN.w/�N.v/ D 1: Since the

numeration system is based, it follows that N.v/ D �B.v/ and N.w/ D �B.w/,

where �B is as defined in (1). Thus

1 D �B.w/ � �B.v/ D �B.p1yk�2 � � �y0/ � �B.p0xk�2 � � � x0/

D �B.1yk�2 � � �y0/ � �B.xk�2 � � � x0/:

Since the word xk�2 � � � x0 has k� 1 digits and the word 1yk�2 � � �y0 has k digits,

it must be true that

xk�2 � � � x0 DMk�1 DMl�jpj�1 and 1yk�2 � � �y0 D mk D ml�jpj:

Now suppose that S has the property mentioned in the statement of the theo-

rem. For i � 0 let bi D N.10i /. We will show that ¹biº is a base for the numera-

tion system. We need to show that N.w/ D �B.w/ for every w 2 S . We use in-

duction on the length of w. Clearly N.0/ D 0 D �B.0/, and N.1/ D 1 D �B.1/.

Let l � 2, and assume thatN.w/ D �B.w/whenever jwj < l . Letw1; : : : ; wm be

the words in S of length l in increasing lexicographic order under <. We need to

show thatN.wj / D �B.wj / for 1 � j � m. Since S is right extendable it follows

immediately that w1 D 10 � � � 0 D 10l�1. Therefore N.w1/ D bl�1 D �B.w1/:

Assume 1 � j < m; and that

N.w1/ D �B.w1/; N.w2/ D �B.w2/; : : : ; N.wj / D �B.wj /:

We need to show that N.wj C1/ D �B.wj C1/. Let p be the maximal common

prefix for wj and wj C1. If wj D p0 and wj C1 D p1, then �B.wj C1/ D

�B.wj /C 1 D N.wj /C 1 D N.wj C1/. Assume then that

wj D p0x and wj C1 D p1y;


where x; y 2 ¹0; 1º� and 1 � jxj; jyj � l � jpj � 1. Since wj and wj C1 are adja-

cent and since S is right extendable it follows from the property in the statement

of the theorem that

x DMl�jpj�1 and y D 0l�jpj�1:

Thus, by the induction hypothesis,

�B.wj C1/ � �B.wj / D �B.p1y/ � �B.p0x/ D �B.1y/ � �B.x/

D N.Ml�jpj�1/ �N.ml�jpj/ D 1:

Since N.wj C1/ �N.wj / D 1, and N.wj / D �B.wj / it follows that,

N.wj C1/ D �B.wj C1/:

So for all w of length l ,N.w/ D �B.w/. And by induction the result follows.

3 Self Generating Numeration Systems

We now return to self generating sets. For a family of functions F , a self generat-

ing set S D SF is the smallest set containing 0 that is closed under the functions

in F . A numeration system S is self generating if there exists a collection of

functions F for which

S D ¹Œm�2 W m 2 SF º;

where Œm�2 is the base 2 expansion of the integer m 2 SF . As mentioned in

Section 1, the sets of greedy and lazy representations of the natural numbers with

respect to both the Fibonacci and Tribonacci sequences give rise to numeration

systems that are self generating and based.

In light of these examples it is natural to consider the question of which sets

of affine functions produce self generating numeration systems that are based. In

this section we will show how to construct a large class of such functions. These

generating functions are perhaps the most natural generalization of the sets of

functions generating the greedy Fibonacci and Tribonacci numeration systems.

First, we require that 2x 2 F . Since multiplication of x by 2 adds a 0 to the

binary expansion of x, the resulting numeration system will be right extendable.

Furthermore, since multiplication of x by 2n adds n zeros to the binary expansion

of x, it is natural to restrict our attention to those families of functions in which the

coefficient on x is a power of 2. We also require that if f .x/ D 2nxCc 2 F , then

0 � c < 2n. Thus, the binary expansion of f .x/ ends in the binary expansion of

c. Our next lemma gives us a further restriction on F .


Lemma 3. Let F D ¹f0; f1; : : : ; fnº be a family of functions defined as follows.

Let f0 D 2x and for 1 � i � n let f .x/ D 2kix C ci , where ki � 1 and

0 � ci < 2ki . Let SF be the set generated by F , and let S be the numeration

system ¹Œs�2 W s 2 SF º. If S is based then ci D 1 for some i .

Proof. If S is based, then by Lemma 1 S must contain 1. By definition of S it

follows that 1 2 SF . Thus, 1 D fi .0/ for some fi 2 F with 1 � i � n.

In light of Lemma 3 we add the assumption that if 2k is the smallest coefficient

on x for all functions in F that have an odd constant term, then 2kx C 1 2 F .

We will prove that a finite number of iterations of this “minimal” function on the

function 2x produces a family of functions F that generates a based numeration

system S . Our method will be to show that the self generating numeration system

S satisfies the conditions of Theorem 1. Our first step in this process is to show

that S is treelike.

Lemma 4. Let n � 1, k � 2, and let F be the set of functions ¹f0; : : : ; fnº where

f0.x/ D 2x, and

fi .x/ D 2k�1.fi�1.x//C 1 for 1 � i � n: (4)

Let SF be the set generated by F , and let S be the numeration system ¹Œs�2 W s 2

SF º. Then S is treelike.

Proof. Clearly 1 2 S . Since 2x 2 F , S is right extendable. Suppose v0 2 S .

Then there is a function f 2 F and an s 2 SF such that Œf .s/�2 D v0. Since f .s/

is even, it follows that f D f0. Then v D Œs�2 2 S , and therefore S is a Bertrand

numeration system. Now let v 2 ¹0; 1º� be such that v1 2 S . We need to show

that v 2 S . By definition, there exists an s 2 SF and an fi 2 F , where 1 � i � n,

such that Œfi .s/�2 D v1. Thus, Œ2k�1fi�1.s/C1�2 D v1, and so Œ2k�1fi�1.s/�2 D

v0, and Œ2k�2fi�1.s/�2 D v. Notice that v D Œf k�20 .fi�1.s//�2. It follows from

the definition of S that v 2 S . This completes the proof that S is treelike.

The next lemma enables us to establish that the property of Theorem 1 is sat-

isfied. If v 2 S and if there exists an f 2 F and an s 2 SF such that v D Œf .s/�2then we say that v is produced by f . Recall also that for j � 0,Mj is the maximal

word of length j in S .

Lemma 5. Let n � 1, k � 2, and let F be the set of functions ¹f0; : : : ; fnº where

f0.x/ D 2x, and

fi .x/ D 2k�1.fi�1.x//C 1 for 1 � i � n:


Let S be the self generating numeration system generated by F . Let v 2 S , and

assume v1 2 S . If v1 is produced by f1, then for j � 1, vMj is the maximal

sequence in S of length jvj C j having v as a prefix.

Proof. Since S is treelike, a maximal sequence in S having v as a prefix corre-

sponds to a path in T .S/ beginning at v and following the rightmost branches at

each level. Let s 2 S be such that v1 D Œf1.s/�2 D Œ2ks C 1�2 D Œs�20k�11. Let

v0 D Œs�2, and define

vi D Œ2k�1fi�1.s/C 1�2 D Œfi .s/�2 for 1 � i � n:

Also, let vnC1 D Œ2k�1fn.s/�2: Notice that

vi D Œfi .s/�2 D Œ2k�1fi�1.s/C 1�2 D vi�10k�21 for 1 � i � n;

and vnC1 D vn0k�1. It follows from the definition of T .S/ then that for 1 �

i � n, vi is a right child of Œ2k�2fi�1.s/�2 in S . Also, for any integer l � 1,

Œ2lfi�1.s/�2 is a left child of Œ2l�1fi�1.s/�2 in S . Thus, for 1 � i � n it follows

that the vi are connected by a path from v to vnC1 of length .k � 1/.n � 1/C k.

For 1 � i � n, the vi are right children in this path, and all the other vertices in

this path are left children. Let vx1x2 � � � xj be the j th vertex in this path. We show

that this is the maximal word in S of length jvj C j having v as a prefix.

Assume to the contrary that vy1y2 � � �yj 2 S is lexicographically larger than

vx1x2 � � � xj . We will consider two cases. First, suppose that vx1 � � � xj lies on the

path from v to vn. Let l be the smallest integer such that yl ¤ xl . This gives that

x1 D y1; x2 D y2; : : : ; xl�1 D yl�1; xl D 0; and yl D 1: (5)

By Lemma 4, S is treelike, so every prefix of a word in S is also in S . Thus there

exists a y 2 S and an integerm, with 1 � m � n, such that vy1 � � �yl D Œfm.y/�2.

Now, fm.y/ D 2k�1fm�1.y/C 1, and so by (5) we have that Œ2k�1fm�1.y/�2 D

vy1y2 � � �yl�10 D vx1x2 � � � xl . However, since vx1 � � � xl is a left vertex in the

aforementioned path from v to vn, there exist s 2 SF and integers i and t with 1 �

i < n and 1 � t � k�2 such that vx1 � � � xl D Œ2tfi .s/�2. Thus, 2k�1fm�1.y/ D

2tfi .s/. Therefore 2k�t�1fm�1.y/ D fi .s/. Since k � t � 1 � 1, this is a

contradiction, since fi .s/ is odd for i � 1. Thus, vx1 � � � xl is the maximal word

in the path from v to vn.

Now, assume that vx1 � � � xl lies on the path from vn to vnC1. Then the argu-

ment of the previous paragraph holds, except that i D n, and 1 � t � k � 1.

We have shown then that for j � .k � 1/.n � 1/C k, vx1 � � � xj is the maximal

word in S having v as a prefix. Notice that our argument reveals that the xj ’s

depend on the functions in F , and not on v. Thus, if we take v D 0 we see that

x1 � � � xj DMj for 1 � j � .k � 1/.n � 1/C k.


So far we have restricted our attention to the case that j is less than the length

of Œf1.fn.0//�2. Since vnC11 is produced by f1, the above argument can be ex-

tended to arbitrary j .

Theorem 2. Let n � 1, k � 2, and let F be the set of functions ¹f0; : : : ; fnº

where f0.x/ D 2x, and

fi .x/ D 2k�1.fi�1.x//C 1 for 1 � i � n:

Let SF be the set generated by F , and let S be the numeration system ¹Œs�2 W s 2

SF º. Then S is based.

Proof. Let v and w be adjacent words in S of the same length. Assume that

v < w. Let p be the maximal common prefix of v and w in S . Then v D p0x

and w D p1y for some x; y 2 ¹0; 1º�. Since S is treelike, every prefix of a word

in S is also in S . Thus, p1 2 S and since S is self generating, there exists an

s 2 S and an fi 2 F such that Œfi .s/�2 D p1. By definition we have that

p1 D Œfi .s/�2 D Œ2k�1fi�1.s/C 1�2:

Therefore,

Œ2k�1fi�1.s/�2 D p0;

and so

Œf1.fi�1.s//�2 D Œ2k.fi�1.s//C 1�2 D Œ2.2

k�1fi�1.s//C 1�2 D p01:

Therefore p01 2 S and is produced by f1. Thus, by Lemma 5, it follows that

v D p0Mjvj�jpj�1. Since S is right extendable, it is clear that w D pmjwj�jpj.

Thus, by Theorem 1, it follows that S is based.

We conclude this section with a few examples. Note that in each case the

elements in the base sequence are obtained from the indices of the powers of 2 in

the sequence of ordered elements of SF .

Example 1. Let F D ¹2x; 8x C 1º. Then

S D ¹0; 1; 2; 4; 8; 9; 10; 16; 17; 18; 32; 33; 34; 36; : : : º:

The numeration system S is defined as the set of base two representations of

the elements of SF . The elements of S correspond to the greedy expansions

of the natural numbers with respect to the base b0 D 1, b1 D 2, b2 D 3, and

bn D bn�1 C bn�3 for n � 3.


Example 2. In general, if F D ¹2x; 2kx C 1º, then SF is the set of nonnegative

integers whose base two expansions correspond to the greedy expansions of the

natural numbers with respect to the base sequence b0 D 1, b1 D 2; : : : ; bk�1 D

k � 1, and bn D bn�1 C bn�k for n � k.

Example 3. Let F D ¹2x; 8x C 1; 32x C 5º. Then

SF D ¹0; 1; 2; 4; 5; 8; 9; 10; 16; 17; 18; 20; 32; 33; 34; 36; 37; 40; 41; : : : º:

The numeration system S consists of the greedy expansions of the natural num-

bers with respect to the base b0 D 1, b1 D 2, b2 D 3, b3 D 5, b4 D 8, and

bn D bn�1 C bn�3 C bn�5 for n � 5.

The generating functions used in Theorem 2 generate a numeration system that

is treelike. It is possible to construct examples of based self generating numeration

systems for which this is not the case. The following example is perhaps the

simplest such example.

Example 4. Let F D ¹2x; 4x C 1; 4x C 2º. Then

SF D ¹0; 1; 2; 4; 5; 6; 8; 9; 10; 12; 16; 17; 18; 20; 21; 22; 24; 25; 26; : : : º:

The numeration system S is defined as the set of base two representations of the

elements of SF . The first 18 elements of S are shown in Figure 2, where we leave

out the root vertex at 0 for the sake of brevity.

1

10

100

1000

10000 10001

1001

10010

101

1010

10100 10101 10110

110

1100

11000 11001 11010

Figure 2. Tree structure for S generated by F D ¹2x; 4x C 1; 4x C 2º.

We have listed the elements of S in rows, where the elements of a given row

k are all the elements of S of length k. The tree connections between vertices are

shown. The elements in the last row that are not connected to anything are roots

of new trees in T .S/. It is interesting to note that the rooted trees in Figure 2 are

all isomorphic. It is a routine exercise to prove a result similar to Lemma 5 for

this S , and therefore the proof that S is based is similar to the proof of Theorem 2.


We can also compute the base and see that the elements of S correspond to the

greedy expansions of the natural numbers with respect to the base b0 D 1, b1 D 2,

b2 D 3, and bn D bn�1 C 2bn�2 � bn�3 for n � 3.

4 Linearly Recurrent Base Sequences

We now show that the base sequence in any based self generating numeration

system satisfies a linear recurrence. In [5] it was shown that the base sequence

in any based treelike numeration system satisfies a linear recurrence if and only if

the maximal sequence M defined in (2) is periodic. As example 4 shows, a self

generating numeration system need not be treelike. Shallit [15] also established

some rather general conditions which guarantee that the base sequence in a based

numeration system satisfies a linear recurrence. It can be shown that the self

generating numeration systems we are considering satisfy these conditions. In

this section we give a different proof which gives us a relatively simple means for

constructing the linear recurrence. Note also that in the following theorem, we are

no longer restricting our attention to the family F given by (4).

Theorem 3. Let F D ¹f0; f1; : : : ; fnº be a family of functions defined as follows.

Let f0.x/ D 2x, and for 1 � i � m, let fi .x/ D 2kix C ci , where ki � 1 and

0 � ci < 2ki . Let S be the numeration system generated by F . If S is based,

then the base sequence ¹biº1

iD0satisfies a linear recurrence.

Proof. Let ¹siº1

iD0be the sequence of elements of S listed in order. Since S is

based, by Lemma 3 it follows that ci D 1 for some i with 1 � i � n. By

definition it follows that for n � 0, bn is the index of 2n in ¹siº. We need to show

that ¹biº satisfies a linear recurrence.

Let ¹uiº1

iD0be the characteristic sequence of SF . That is, for i � 0, ui D 1

if i 2 SF and ui D 0 otherwise. It was shown in [8] that the characteristic

sequence is 2-automatic (for a definition of automatic sequences see [1]). By

Cobham’s Theorem [6], (see also Theorem 6.3.2 of [1]), ¹uiº is the image under

a coding of the fixed point of a morphism � of constant length 2 over a finite

alphabet ¹a1; : : : ; akº. Let � W ¹a1; a2; : : : ; akº�! ¹a1; a2; : : : ; akº

� be this mor-

phism, and suppose the fixed point is generated by iteration on a1. Recall that

the incidence matrix of � is defined as the n by n matrix A D�

mi;j

�

, where

mij D j�.aj /jai, the number of occurrences of ai in �.aj / (see [1], Chapter 8).

Let A be the incidence matrix of � and let

P.x/ D xk � c1xk�1 � c2x

k�2 � � � � � ck�1x � ck


be the characteristic polynomial of A. Finally, let ¹rnº be the sequence defined by

the recurrence relation

rn D c1rn�1 C c2rn�2 C � � � C ckrn�k : (6)

We show that ¹biº satisfies this recurrence.

Let e1 D Œ1 0 � � � 0�T . We first show that for 1 � i � k the i th coordinate of

Ane1 satisfies (6). By the Cayley–Hamilton Theorem P.A/ D 0, and so

Ak � c1Ak�1 � c2A

k�2 � � � � � ck�1A � ckI D 0:

Therefore P.A/ � e1 D 0; and so

Ake1 � c1A

k�1e1 � c2A

k�2e1 � � � � � ck�1Ae1 � ckI e1 D 0:

For n � k, multiplying both sides of this last equation by An�k gives

Ane1 � c1A

n�1e1 � c2A

n�2e1 � � � � � ck�1A

n�kC1e1 � ckA

n�ke1 D 0:

Now, by definition of A, j�n.a1/jai, the number of occurrences of ai in �n.a1/,

is the i th coordinate of Ane1. Thus, for n > 0, j�n.a1/jai

satisfies the recurrence

(6) with initial conditions j�.a1/jai; j�2.a1/jai

; : : : ; j�k.a1/jai.

Now, for n � 0 we have that bn is the index of 2n in ¹siº. Equivalently, since

¹biº is the base system for the abstract numeration system S , bn is the number of

ones in the sequence ¹unº between u0 and u2n�1. On the other hand, since � has

constant length 2, it follows that

kX

iD1

j�n.a1/jaiD j�n.a1/j D 2n:

Now let ai1; ai2

; : : : ; ail, where 1 � l � k, be the characters that map to 1 under

the coding that maps the fixed point of � to ¹unº. We therefore have that

bn D j�n.a1/jai1

C � � � C j�n.a1/jail

:

Since each of j�n.a1/jaij

satisfies (6), it follows that bn satisfies (6) as well.

5 Further Considerations

We close with some suggestions for further study. There are many examples of

sets of generating functions that do not satisfy the conditions of Theorem 2 that

seem to generate based numeration systems. For example, numerical evidence


seems to suggest that if F D ¹2x; 4x C 1; 16x C 3; 32x C 11º then the result-

ing numeration system is based. In particular, the base two expansions of the

elements of SF seem to correspond to the set of greedy representations of N

with respect to the sequence b0 D 1, b1 D 2, b2 D 4, b3 D 7, b4 D 13, and

bn D bn�1 C bn�2 C bn�4 C bn�5. On the other hand, not every self generating

numeration system is based. For example, it follows from Lemma 2 that the nu-

meration system generated by F D ¹2x; 8x C 3º is not based. In light of these

examples we see that the characterization of all self generating based numeration

systems remains an open problem.

It is also natural to try to extend the results of this paper to numeration systems

with digit sets †k D ¹0; 1; 2; : : : ; k � 1º. Some examples were considered in [8].

For example, if F D ¹3x C 1; 3x C 2; 9x C 3; 9x C 6º then

SF D ¹0; 1; 2; 3; 4; 5; 6; 7; 8; 10; 11; 12; 13; 14; 15; 16; 17; 19; 20; 21; : : : º:

This is the set of all nonnegative integers whose base 3 expansion does not contain

the block 00. If S D ¹Œs�3 W s 2 SF º, where Œs�3 denotes the base 3 expansion

of s, then S is a numeration system with digit set †3. It was noted in [8] that S

corresponds to the set of lazy representations of the natural numbers with respect

to the base sequence b0 D 1, b2 D 3, and bn D 2bn�1 C 2bn�2 for n � 2.

Finally, the reader may have noticed that Theorem 2 does not guarantee that the

numeration systems are greedy with respect to their bases. It is not hard to verify

this for specific examples. However, the question of which based numeration

systems are greedy and which are lazy also remains open.

Acknowledgments. We would like to thank the referee for a thorough reading

of the manuscript and for many helpful suggestions.

References

[1] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, General-

izations, Cambridge University Press, Cambridge, 2003.

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pas entière, Acta Math. Acad. Sci. Hungar. 54 (1989), 237–241.

[4] V. Bruyère and G. Hansel, Bertrand numeration systems and recognizability, Theo-

ret. Comput. Sci. 181 (1997), no. 1, 17–43.

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[6] A. Cobham, Uniform tag sequences, Math. Systems Theory 6 (1972), 164–192.

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(2007), Article 07.1.5.

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Article 00.2.8.

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274 (2004), 147–159.

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Applications of Fibonacci Numbers, vol. 9, pp. 137–144, Kluwer Acacamy Publ.,

Dordrecht, 2004

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bridge, 2002.

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Author information

David Garth, Department of Mathematics and Computer Science,

Truman State University, Kirksville, MO 63501, USA.


Joseph Palmer, Department of Mathematics and Computer Science,



Ha Ta, Department of Mathematics and Computer Science,




Small Sets Satisfying

the Central Sets Theorem

Neil Hindman

Abstract. The Central Sets Theorem is a powerful theorem, one of whose consequences

is that any central set in N contains solutions to any partition regular system of homoge-

neous linear equations. Since at least one set in any finite partition of N must be central,

any of the consequences of the Central Sets Theorem must be valid for any partition of N.

It is a result of Beiglböck, Bergelson, Downarowicz, and Fish that if p is an idempotent

in .ˇN;C/ with the property that any member of p has positive Banach density, then any

member of p satisfies the conclusion of the Central Sets Theorem. Since all central sets

are members of such idempotents, the question naturally arises whether any set satisfy-

ing the conclusion of the Central Sets Theorem must have positive Banach density. We

answer this question here in the negative.

Keywords. Central, density.

AMS classification. 05D10.

1 Introduction

In [6] H. Furstenberg introduced the notion of central subsets of N in terms of

notions from topological dynamics. He showed that one cell of any finite partition

of a N must contain a central set and proved the original Central Sets Theorem.

(Given a set X , we denote by Pf .X/ the set of finite nonempty subsets of X .)

Theorem 1.1. Let C be a central subset of N. Let l 2 N and for each i 2 ¹1; 2;

: : : ; lº, let fi be a sequence in Z. Then there exist sequences hani1

nD1 in N and

hHni1

nD1 in Pf .N/ such that

(1) for all n, maxHn < minHnC1 and

(2) for all F 2 Pf .N/ and all i 2 ¹1; 2; : : : ; lº,P

n2F

�

an CP

t2Hn

fi .t/�

2

C .

The author acknowledges support received from the National Science Foundation via Grant

DMS-0554803.

58 Neil Hindman

Proof. [6, Proposition 8.21].

Furstenberg used central sets to prove Rado’s Theorem [10] by showing that

any central subset of N contains solutions to all partition regular systems of ho-

mogeneous linear equations.

Based on an idea of V. Bergelson, central sets in N were characterized quite

simply [4] as members of minimal idempotents of .ˇN;C/, and this characteri-

zation extended naturally to define central subsets of an arbitrary discrete semi-

group S .

What is currently the most general version of the Central Sets Theorem (for

commutative semigroups) is the following.

Theorem 1.2. Let .S;C/ be a commutative semigroup and let T D NS , the set

of sequences in S . Let C be a central subset of S . There exist functions ˛ W

Pf .T /! S and H W Pf .T /! Pf .N/ such that

(1) if F;G 2 Pf .T / and F �6G, then maxH.F / < minH.G/ and

(2) wheneverm 2 N,G1; G2; : : : ; Gm 2 Pf .T /,G1 �6 G2 �6 : : : �6Gm, and for

each i 2¹1; 2; : : : ; mº,fi 2Gi , one hasPm

iD1

�

˛.Gi /CP

t2H.Gi /fi .t/�

2C .

Proof. [5, Theorem 2.2].

To derive Theorem 1.1 from Theorem 1.2, note that one may assume that the

sequences f1; f2; : : : ; fl in the statement of Theorem 1.1 are distinct. Choose

additionally distinct sequences fk for k > l and let for each n 2 N, Gn D

¹f1; f2; : : : ; fnº. For n 2 N, let an D ˛.Gn/ and let Hn D H.Gn/.

For some of the motivating results that we will present, it is necessary to de-

scribe briefly the algebraic structure of the Stone–Cech compactification. If the

reader is willing to accept that the question of whether every subset of N which

satisfies the conclusion of Theorem 1.2 must have positive Banach density is in-

teresting, she may proceed directly to Section 2 where that question is answered.

Given a discrete semigroup .S;C/, the Stone–Cech compactification ˇS of

S is the set of ultrafilters on S , the principal ultrafilters being identified with the

points of S . Given A � S , c`A D A D ¹p 2 ˇS W A 2 pº. The family

¹A W A � Sº is a basis for the open sets (and a basis for the closed sets) of ˇS .

The operation C extends to ˇS so that .ˇS;C/ is a right topological semigroup

(meaning that for each p 2 ˇS the function �p W ˇS ! ˇS defined by �p.q/ D

q C p is continuous) with S contained in its topological center (meaning that

for each x 2 S the function �x W ˇS ! ˇS defined by �x.q/ D x C q is

continuous). Given p; q 2 ˇS and A � S , one has that A 2 p C q if and only if

¹x 2 S W �x C A 2 qº2 p, where �x C A D ¹y 2 S W x C y 2 Aº.

Small Sets Satisfying the Central Sets Theorem 59

As is true of any compact Hausdorff right topological semigroup, ˇS has a

smallest two sided ideal K.ˇS/ and there are idempotents in K.ˇS/. Such idem-

potents are said to be minimal, and a subset C of S is central if and only if it is a

member of a minimal idempotent. The reader is referred to [8] for an elementary

introduction to the algebra of ˇS .

The following notion was originally introduced by Polya in [9], but it is com-

monly referred to as “Banach density”.

Definition 1.3. Let A � N. Then

d�.A/ D sup¹˛ 2 R W .8k 2 N/.9n � k/.9a 2 N/

.jA \ ¹aC 1; aC 2; : : : ; aC nºj � ˛ � n/º;

�� D ¹p 2 ˇN W .8A 2 p/.d�.A/ > 0/º:

Since �� is a two sided ideal of ˇN, one has that K.ˇN/ � ��, and in par-

ticular, if C is a central subset of N, then d�.C / > 0. The following result of

Beiglböck, Bergelson, Downarowicz, and Fish establishes that a weaker assump-

tion than central yields the conclusion of the original Central Sets Theorem.

Theorem 1.4. Let C � N and assume that C is a member of an idempotent in

��. Let l 2 N and for each i 2 ¹1; 2; : : : ; lº, let fi be a sequence in Z. Then

there exist sequences hani1

nD1 in N and hHni1

nD1 in Pf .N/ such that

(1) for all n, maxHn < minHnC1 and

(2) for all F 2 Pf .N/ and all i 2 ¹1; 2; : : : ; lº,P

n2F

�

an CP

t2Hn

fi .t/�

2

C .

Proof. [2, Theorem 10].

In fact, the proof of [2, Theorem 10] is easily modified to show that any mem-

ber of an idempotent in �� satisfies the conclusion of Theorem 1.2. It is a result

of C. Adams [1, Theorem 2.21] that there is a set C which is a member of an

idempotent in �� but C misses the closure of the smallest ideal of ˇN and in

particular, C is not central.

One is naturally led by the above results to ask whether any subset of N which

satisfies the conclusion of Theorem 1.2 must in fact have positive Banach density.

We show in Section 2 that this is not the case.

We close this introduction with an interesting contrast between members of

idempotents in �� and central sets, that is members of idempotents in K.ˇN/.

Those sets A � N such that A \ K.ˇN/ ¤ ; are exactly the piecewise syndetic

subsets of N by [8, Theorem 4.40] while a set A � N has A \ �� ¤ ; if and

60 Neil Hindman

only if d�.A/ > 0 by [8, Theorem 3.11]. If A is piecewise syndetic, then by [8,

Theorem 4.43] there is some x 2 N such that �x C A is central. On the other

hand, it is a result of Ernst Straus that there exist sets A � N with asymptotic

density arbitrarily close to 1 (and thus d�.A/ arbitrarily close to 1) such that no

translate of A is a member of any idempotent. (See [3, Theorem 2.20].)

2 A Small Subset of N Satisfying the Conclusion of the

Central Sets Theorem

We produce in this section a subset of N with zero Banach density which satisfies

the conclusion of Theorem 1.2 applied to the group .Z;C/. The construction is

based on that of [7, Lemma 5.2]. For x 2 N we denote by supp.x/ the subset of

! D N [ ¹0º such that x DP

t2supp.x/ 2t .

Theorem 2.1. Let T D NZ, the set of sequences in Z. There is a subset A of

N such that d�.A/ D 0 and there exist functions ˛ W Pf .T / ! N and H W

Pf .T /! Pf .N/ such that

(1) if F;G 2 Pf .T / and F �6G, then maxH.F / < minH.G/ and

(2) wheneverm 2 N,G1; G2; : : : ; Gm 2 Pf .T /,G1 �6 G2 �6 : : : �6Gm, and for

each i 2 ¹1; 2; : : : ; mº, fi 2 Gi , one hasPm

iD1

�

˛.Gi /CP

t2H.Gi /fi .t/�

2

A.

Proof. For n 2 N, let an D min¹t 2 N W .2n�12n /t � 1

2º and let sn D

PniD1 ai .

(So s1 D 1 and s2 D 4.) Let b0 D 0, let b1 D 1, and for n 2 N and t 2

¹sn; sn C 1; sn C 2; : : : ; snC1 � 1º, let btC1 D bt C n C 1. For k 2 !, let

Bk D ¹bk; bk C 1; bk C 2; : : : ; bkC1 � 1º. Let

A D ¹x 2 N W .8k 2 !/.Bk n supp.x/ ¤ ;/º

and let A0 D ¹x 2 ! W .8k 2 !/.Bk n supp.x/ ¤ ;/º (so A0 D A [ ¹0º).

We show first that d�.A/ D 0. Notice that for any x and m in N,

jA \ ¹x; x C 1; x C 2; : : : ; x C 2m � 1ºj � jA0 \ ¹0; 1; 2; : : : ; 2m � 1ºj :

Indeed, given any y 2 ¹0; 1; 2; : : : ; 2m � 1ºnA0, there is some k with bkC1 � m

such thatBk � supp.y/ and there is a unique z.y/ 2 ¹x; xC1; xC2; : : : ; xC2m�

1º such that the rightmost m bits in the binary representation of z.y/ are equal to

those of y and so Bk � supp�

z.y/�

. Further, if y ¤ y0, then z.y/ ¤ z.y0/.

Let x;m 2 N, let k D smC1 and let l � 2bk . We shall show that

jA \ ¹x; x C 1; x C 2; : : : ; x C l � 1ºj

l<

�

1

2

�m

:


Pick r 2 N such that 2r�1 � l < 2r . Then

jA \ ¹x; x C 1; : : : ; x C l � 1ºj � jA \ ¹x; x C 1; : : : ; x C 2r � 1ºj

� jA0 \ ¹0; 1; : : : ; 2r � 1ºj

so

jA \ ¹x; x C 1; x C 2; : : : ; x C lºj

l�jA0 \ ¹0; 1; 2; : : : ; 2r � 1ºj

2r�1:

Now

jA0 \ ¹0; 1; 2; : : : ; 2r � 1ºj

DP2r�b

k �1tD0 jA0 \ ¹t2bk ; t2bk C 1; : : : ; .t C 1/2bk � 1ºj

�P2r�b

k �1tD0 jA0 \ ¹0; 1; : : : ; 2bk � 1ºj

D 2r�bk � jA0 \ ¹0; 1; : : : ; 2bk � 1ºj

so

jA0 \ ¹0; 1; 2; : : : ; 2r � 1ºj

2r�1�

2r�bk � jA0 \ ¹0; 1; : : : ; 2bk � 1ºj

2r�1

DjA0 \ ¹0; 1; : : : ; 2bk � 1ºj

2bk�1:

We have that jA0 \ ¹0; 1; : : : ; 2bk � 1ºj DQk�1

tD0.2btC1�bt � 1/ and 2bk�1 D

12

Qk�1tD0 2btC1�bt so

jA0 \ ¹0; 1; : : : ; 2bk � 1ºj

2bk�1D 2 �

Qk�1tD0

�

2b

tC1�bt �1

2b

tC1�bt

�

D 2 �21 � 1

21�Qm

nD1

QsnC1�1tDsn

�

2b

tC1�bt �1

2b

tC1�bt

�

DQm

nD1

�

2nC1�1

2nC1

�anC1

�

�

1

2

�m

:

Now we show that A satisfies the conclusion of Theorem 1.2. First note that

if n; k 2 N and and bkC1 � bk > n, then whenever z1; z2; : : : ; zn 2 N, there

must exist r 2 Bk such that for all t 2 ¹1; 2; : : : ; nº, Bk n supp.2r C zt / ¤ ;.

62 Neil Hindman

Indeed, if r 2 Bk , z 2 N, and Bk � supp.2r C z/ then supp.z/\Bk D Bk n ¹rº.

Consequently

j¹r 2 Bk W there is some i 2 ¹1; 2; : : : ; nº with Bk � supp.2r C zi /ºj � n :

Now we claim that

(�) for each n;m 2 N and each F 2 Pf .T /, there exist d 2 N andH 2 Pf .N/

such that minH > m and for all f 2 F , d CP

t2H f .t/ 2 A \ N2n.

To see this, let r D jF j and pick k such that bkC1 � bk > r and bk > n. Pick

H 2 Pf .N/ such that minH > m and for all f 2 F ,P

t2H f .t/ 2 Z2bk .

(Choose an infinite subset C of N such that for all s; t 2 C and all f 2 F ,

f .s/ � f .t/ .mod 2bk /. Then pick H � C such that minH > m and jH j D

2bk .) Pick c 2 N2bk such that for all f 2 F , c CP

t2H f .t/ > 0.

Let l D maxS®

supp�

c CP

t2H f .t/�

W f 2 F¯

and pick j such that

l < bj . Pick r0 2 Bk such that Bk n supp�

2r0 C c CP

t2H f .t/�

¤ ; for

each f 2 F . Inductively for i 2 ¹1; 2; : : : ; j � kº, pick ri 2 BkCi such that

BkCi n supp�

2ri CPi�1

tD0 2rt C c CP

t2H f .t/�

¤ ; for each f 2 F . Let

d D c CPj �k

iD02ri . Then (�) is established.

Now we define ˛.F / 2 N and H.F / 2 Pf .N/ for F 2 Pf .T / inductively

on jF j. If F D ¹f º, pick ˛.F / 2 N and H.F / 2 Pf .N/ by (�) such that

˛.F /CP

t2H.F / f .t/ 2 A. Now let F 2 Pf .T / with jF j > 1 and assume that

we have defined ˛.G/ and H.G/ for all G such that ; ¤ G �6F so that

(1) ˛.G/CP

t2H.G/ f .t/ 2 A for each f 2 G and

(2) if K �6G, then

(a) maxH.K/ < minH.G/ and

(b) there exists k 2 N such that for all f 2 K and all g 2 G,

max supp�

˛.K/CP

t2H.K/ f .t/�

< bk < min supp�

˛.G/CP

t2H.G/ g.t/�

:

Let m D maxS

¹H.G/ W ; ¤ G �6F º and pick k 2 N such that for all G 2

Pf .T / with G �6F and all f 2 G, max supp

�

˛.G/ CP

t2H.G/ f .t/�

< bk .

Pick by (�) some H.F / 2 Pf .N/ and ˛.F / 2 N such that minH.F / > m and

for all f 2 F , ˛.F /CP

t2H.F / f .t/ 2 A \ N2bkC1.

To verify that ˛ and H are as required for Theorem 1, let m 2 N, let

G1; G2; : : : ; Gm 2 Pf .T / ;

and assume that G1 �6 G2 �6 : : : �6Gm, and for each i 2 ¹1; 2; : : : ; mº, fi 2 Gi .

We claim thatPm

iD1

�

˛.Gi / CP

t2H.Gi / fi .t/�

2 A. Suppose instead one has


some k 2 N such that Bk � supp(

PmiD1

�

˛.Gi / CP

t2H.Gi / fi .t/�)

. Then

there is some i such that Bk � supp�

˛.Gi / CP

t2H.Gi / fi .t/�

, contradicting

hypothesis (1) of the construction.

References

[1] C. Adams, Large finite sums sets with closure missing the smallest ideal of ˇN,

Topology Proceedings, to appear.

[2] M. Beiglböck, V. Bergelson, T. Downarowicz and A. Fish, Solvability of Rado sys-

tems in D-sets, Topology and its Applications, to appear.

[3] V. Bergelson, M. Beiglböck, N. Hindman and D. Strauss, Multiplicative structures

in additively large sets, J. Comb. Theory (Series A) 113 (2006), 1219–1242.

[4] V. Bergelson and N. Hindman, Nonmetrizable topological dynamics and Ramsey

Theory, Trans. Amer. Math. Soc. 320 (1990), 293–320.

[5] D. De, N. Hindman and D. Strauss, A new and stronger Central Sets

Theorem, to appear in Fundamenta Mathematicae (currently available at

http://members.aol.com/nhindman/).

[6] H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory,

Princeton University Press, Princeton, 1981.

[7] N. Hindman, A. Maleki and D. Strauss, Central sets and their combinatorial charac-

terization, J. Comb. Theory (Series A) 74 (1996), 188–208.

[8] N. Hindman and D. Strauss, Algebra in the Stone–Cech Compactification: Theory

and Applications, de Gruyter, Berlin, 1998.

[9] G. Polya, Untersuchungen über Lücken und Singularitäten von Potenzreihen, Math.

Zeit. 29 (1929), 549–640.

[10] R. Rado, Studien zur Kombinatorik, Math. Zeit. 36 (1933), 242–280.

Author information

Neil Hindman, Department of Mathematics, Howard University,

Washington, DC 20059, USA.



Column-to-Row Operations on Partitions:

The Envelopes

Brian Hopkins

Abstract. Conjugation and the Bulgarian solitaire move are considered as extreme cases

of several column-to-row operations on integer partitions. Each operation generates a

state diagram on the partitions of n, which leads to the questions: How many Garden of

Eden states are there? How many cycle states? How many connected components? All of

these questions are answered for partitions of n when at least n�12

columns are switched

to rows.

Keywords. Partitions, Bulgarian solitaire.

AMS classification. 05A17, 37E15.

1 Introduction

Conjugation is the fundamental operation on integer partitions. Write a partition

� as .�1; : : : ; �`.�// where `.�/ denotes the partition’s length, its number of parts.

The conjugate partition �0 is defined as �0 D .�0

1; : : : ; �0s/ where �0

i is the number

of parts ¹�iº greater than or equal to i . This is more easily understood in terms

of the Ferrers diagram: the dots are reflected along the diagonal, so that columns

and rows are swapped; see Figure 3.

We write P.n/ for the set of partitions of n. Since conjugation is an involution,

the state diagram of P.n/ determined by conjugation consists of singletons and

pairs, i.e., self-conjugate partitions and conjugate pairs. See Figure 1 for an ex-

ample, which also introduces the superscript notation for partitions, e.g., writing

213 for .2; 1; 1; 1/.

Consider the effect of conjugation onP.n/ as a state diagram. Notice in Figure

1 that all seven partitions of 5 are in cycles and the diagram has four connected

components.

Bulgarian solitaire is an operation on partitions introduced by Brandt in 1982

[3]. We define it as D1.�/ D .�0

1; �1 � 1; : : : ; �`.�/ � 1/ where any zeros are

removed and the parts may not be in the standard non-increasing order. In terms

of the Ferrers diagram, the operation takes the first (leftmost) column and makes

it a row; see Figure 3. Figure 2 shows the effect of D1 on partitions of 5.

66 Brian Hopkins

15

6

?

5

41

213

6

?

311

?

32

6

?

221

Figure 1. Conjugation on P.5/; all 7 partitions are in cycles and there are 4

components.

15

?

5 -

213

?

41 - 32 -

311

6�

�

��

221

Figure 2. Bulgarian solitaire on P.5/; there are 2 Garden of Eden partitions, 3

cycle partitions, and one component.

Like conjugation, the D1 operation also produces a state diagram on P.n/.

Notice in Figure 2 that three partitions of 5 are in cycles and the diagram con-

sists of a single component. There are also two partitions that have no pre-image

under the operation (15 and 213); these are called Garden of Eden partitions (sub-

sequently abbreviated GE-partitions).

In this article, we introduce a sequence of column-to-row operations; con-

jugation and the Bulgarian solitaire operation are the extreme cases. Bulgarian

solitaire has been the subject of several articles; Hopkins–Jones [4] includes a

fairly complete bibliography. Many of the questions concern state diagram con-

cepts: partitions in cycles, partitions with no preimages, and number of connected

components. In this article, we consider these same questions for all general-

ized column-to-row operations. We determine the number of GE-partitions, the

number of cycle partitions, and the number of connected components for approx-

imately half of all possible cases.

2 General Row-to-Column Operations

Conjugation can be thought of as moving all columns to rows; Bulgarian solitaire

moves one column to a row. We connect these ideas by introducing the sequence

of operations

Dk.�/ D .�0

1; : : : ; �0

k; �1 � k; : : : ; �`.�/ � k/

Column-to-Row Operations on Partitions: The Envelopes 67

where any nonpositive numbers are removed and the parts may not be in the stan-

dard non-increasing order. In terms of the Ferrers diagram, the operation takes the

first k columns and makes them rows. Figure 3 shows a partition and its images

under various Dk .

Figure 3. Ferrers diagrams for � D .4; 1/, D1.�/ D .3; 2/, D2.�/ D .2; 2; 1/,

and D3.�/ D D4.�/ D �0 D .2; 1; 1; 1/, with shaded dots showing which rows

came from columns of �.

These operations all generate state diagrams on P.n/. Figures 4 and 5 show

P.5/ under D2 and D3, respectively.

15

?

5 -

?

311

213

41

?

-

32

6

?

221

Figure 4. The D2 operation on P.5/; there are 2 GE-partitions, 3 total partitions

in cycles, and 2 components.

15

?

5 -

41

213

6

?

311

?

32

6

?

221

Figure 5. The D3 operation on P.5/; there is 1 GE-partition, 5 cycle partitions,

and 3 components.

Notice that for P.5/, conjugation (Figure 1) is equivalent to the operation D4.

This is an example of a general fact.

Lemma 1. For a partition � with �1 � k or with �1 D k C 1 and �2 � k, the

operation Dk is equivalent to conjugation. In particular, Dn�1 is equivalent to

conjugation on P.n/.

68 Brian Hopkins

Proof. For � with k or fewer columns, the claim is evident. Assume �1 D k C 1

and �2 � k, i.e., that � has k C 1 columns with .k C 1/-st having height 1.

Moving k columns to rows leaves a single row of length 1, so that the effect of

Dk is equivalent to moving all columns to rows. Every � 2 P.n/ has �1 � n� 1

except the single-part partition .n/, which satisfies the other condition since the

second part of .n/ is 0. Therefore, for all partitions of n, Dn�1.�/ D �0.

3 Results on Partitions with Many Parts

This section consists of results about partitions of n with at least .n � 1/=2 or

n=2 parts. One result is well known and others are particular to the purposes

of this article. First, we introduce some notation. Capital letters signify sets,

corresponding lower-case letters the number of elements in the set, e.g., p.5/ D 7.

Recall the convention that p.0/ D 1. We will use part-wise addition on partitions,

e.g., .3; 1; 1/C .2; 2/ D .5; 3; 1/.

Let P.n; j / denote the set of partitions of n with exactly j parts; from the

examples, we see p.5; 3/ D 2. Table 1 shows the p.n; j / values for 1 � n; j �

12. Notice that, reading right to left, roughly half of each row are initial values of

p.n/, i.e., 1; 1; 2; 3; 5; 7; : : : . We call that portion of the triangle the envelope.

nnk 1 2 3 4 5 6 7 8 9 10 11 12

1 1

2 1 1

3 1 1 1

4 1 2 1 1

5 1 2 2 1 1

6 1 3 3 2 1 1

7 1 3 4 3 2 1 1

8 1 4 5 5 3 2 1 1

9 1 4 7 6 5 3 2 1 1

10 1 5 8 9 7 5 3 2 1 1

11 1 5 10 11 10 7 5 3 2 1 1

12 1 6 12 15 13 11 7 5 3 2 1 1

Table 1. p.n; k/, the number of partitions of n with k parts.

Lemma 2. Let a positive integer n be given. For each integer j � n2

, p.n; j / D

p.n � j /.


Proof. We demonstrate a bijection between P.n; j / and P.n � j /. Any � 2

P.n; j / can be written as � D 1j C � where � 2 P.n � j /; let � 7! �. Any

� 2 P.n � j / has at most j parts, since the restriction on j implies n � j � j ,

so that � D 1j C � 2 P.n; j /; let � 7! �. Clearly these are inverse maps.

This result shows that, in some sense, the difficulty of studying partitions lies

in the partitions with fewer than n=2 parts, which correspond to roughly the left-

hand half of each row in Table 1. There are direct formulas for p.n; j / with

j � 5 (see, e.g., [2] and [6]), but they quickly become complicated. Notice that

the � 7! � relation determined by � D 1j C � is equivalent to removing the

first column of the Ferrers diagram of �, i.e., the Bulgarian solitaire D1 operation

without including the part �0

1.

Lemma 3. Let a positive integer n be given. For each integer j � n�12

, the

following hold.

(a) All � 2 P.n; k/ with k � j C 2 have �j D 1.

(b) All � 2 P.n/ with `.�/ � �0

1 have �1 � n � `.�/C 1.

(c) All � 2 P.n/ with �1 D j C 1 have �2 � j . That is, all � 2 P.n/ with

�1 � j C 1 satisfy one of the conditions of Lemma 1.

Proof. (a) Assume that � 2 P.n; jC2/. Since � has jC2 parts, �j ¤ 0. Suppose

that �j � 2. Then the sum of the parts of � would be at least 2j C 1C 1 � nC 1,

a contradiction. For k > j C 2, the sum of the parts has a higher lower bound, so

the result follows.

(b) The first row and the first column share a dot in the Ferrers diagram, so

their sum is at most nC 1.

(c) If �1 D �2 D j C 1, then �1 C �2 > n, contradicting � 2 P.n/.

Note that (a) is “sharp” in the sense that, for example, every � 2 P.9; 6/ has

�4 D 1 but P.9; 5/ includes 32211 and 241 whose third parts are 2, not 1.

4 Garden of Eden Partitions

Let GE.n; k/ denote the GE-partitions of n under the operation Dk . From the

examples of the previous section, we know ge.5; 1/ D ge.5; 2/ D 2, ge.5; 3/ D 1,

and ge.5; 4/ D 0. Those data correspond to the n D 5 row of Table 2.

The diagonal of zeros corresponds to the fact that every partition has a pre-

image under conjugation. Notice that other diagonals seem to eventually stabilize

at some value; these limiting values comprise the envelope.

70 Brian Hopkins

1 2 3 4 5 6 7 8 9 10

2 0

3 1 0

4 1 1 0

5 2 2 1 0

6 3 3 2 1 0

7 5 5 4 2 1 0

8 7 8 6 4 2 1 0

9 10 12 10 7 4 2 1 0

10 14 18 15 11 7 4 2 1 0

11 20 25 23 17 12 7 4 2 1 0

12 27 35 33 26 18 12 7 4 2 1

13 37 48 47 38 28 19 12 7 4 2

14 49 66 65 55 41 29 19 12 7 4

15 66 88 89 77 60 43 30 19 12 7

16 86 118 120 107 85 63 44 30 19 12

17 113 155 161 145 119 90 65 45 30 19

18 147 203 213 196 163 127 93 66 45 30

19 190 263 280 260 222 175 132 95 67 45

20 243 340 364 344 297 240 183 135 96 67

21 311 435 471 449 394 323 252 188 137 97

Table 2. ge.n; k/, the number of GE-partitions in P.n/ under Dk .

What is the sequence of values 0; 1; 2; 4; 7; 12; 19; 30; 34; 67; 97; : : : in the en-

velope? One possibility is the partial sum of partition numbers (A000070 in [8]).

Let

s.n/ D

nX

iD0

p.i/

and s.�1/ D 0. Before we can verify that this sequence describes the envelope,

we need to characterize GE-partitions.

Lemma 4. A partition � D .�1; : : : ; �`.�// 2 P.n/ is in GE.n; k/ precisely when

�k � `.�/ � �1 � k:

Proof. In terms of the Ferrers diagram, � has a pre-image under Dk for every

set of k rows each greater than or equal to `.�/ � k, i.e., long enough to be


moved to become columns to the left side of the remaining dots. This fails when

�k < `.�/ � k.

This generalizes the initial lemma and corollary of Hopkins–Jones [4] for Bul-

garian solitaire (D1).

We now show that, in the envelope, the GE-partitions are precisely the parti-

tions with many parts.

Theorem 1. Let a positive integer n be given. For each integer n�12� j � n� 1,

we have ge.n; j / D s.n � j � 2/.

Proof. If j D n � 1, then the operation is equivalent to conjugation and there

are no GE-partitions, matching ge.n; n � 1/ D s.�1/ D 0. So assume thatn�1

2� j � n � 2. By the preceding lemma, GE.n; j / consists of all � 2 P.n/

with �j � `.�/ � �1 � j . This means that any GE-partition � must have `.�/ �

�j C j C 1, so �j ¤ 0 and in fact `.�/ � j C 2.

But by Lemma 3a, all � 2 P.n/ with j C 2 or more parts have �j D 1, so

that �j � `.�/ D 1� `.�/ � 1� .j C 2/ D �1� j . That is, GE.n; j / is exactly

P.n; j C 2/ [ � � � [ P.n; n/. By Lemma 2, p.n; j C 2/ D p.n � j � 2/, . . . ,

p.n; n/ D p.0/. We conclude that ge.n; j / DP

p.i/ D s.n � j � 2/.

As with the p.n; j / values of Table 1, one would like to have formulas for

the columns of Table 2. Hopkins–Sellers [5] provides two proofs of the following

result.

Theorem 2.

ge.n; 1/ D p.n � 3/ � p.n � 9/C p.n � 18/ � � � �

DX

j �1

.�1/j C1p

�

n �3j 2 C 3j

2

�

:

We make the following conjectures about the next few columns.

Conjectures

ge.n; 2/ D p.n � 4/C p.n � 5/ � p.n � 11/ � p.n � 12/ � p.n � 13/

C p.n � 21/C p.n � 22/C p.n � 23/C p.n � 24/ � � � �

DX

j �1

jX

kD0

.�1/j C1p

�

n �3j 2 C 3j

2� j � k

�

;

ge.n; 3/ D p.n � 5/C p.n � 6/C p.n � 7/ � p.n � 13/ � p.n � 14/

� 2p.n � 15/ � p.n � 16/ � p.n � 17/C p.n � 24/C � � � ;

72 Brian Hopkins

ge.n; 4/ D p.n � 6/C p.n � 7/C p.n � 8/C p.n � 9/ � p.n � 15/

� p.n � 16/ � 2p.n � 17/ � 2p.n � 18/ � 2p.n � 19/ � p.n � 20/

� p.n � 21/C p.n � 27/C p.n � 28/C 2p.n � 29/C � � �

where complete expressions for ge.n; 3/ and ge.n; 4/ involve q-binomial coeffi-

cients. These are consistent with Theorem 1, since only the initial positive terms

arise in the envelope. These conjectures will be considered in future work with

Louis Kolitsch.

5 Cycle Partitions

Let CP.n; k/ denote the partitions of n in cycles under the operationDk . From the

examples of the previous section, we know cp.5; 1/ D cp.5; 2/ D 3, cp.5; 3/ D 5,

and cp.5; 4/ D 7. Those data correspond to the n D 5 row on the left-hand side

of Table 3.

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

2 2 0

3 1 3 2 0

4 3 3 5 2 2 0

5 3 3 5 7 4 4 2 0

6 1 5 7 9 11 10 6 4 2 0

7 4 6 7 11 13 15 11 9 8 4 2 0

8 6 8 10 14 18 20 22 16 14 12 8 4 2 0

9 4 6 11 16 22 26 28 30 26 24 19 14 8 4 2 0

10 1 5 15 20 28 34 38 40 41 37 27 22 14 8 4 2

11 5 5 15 23 32 42 48 52 51 51 41 33 24 14 8 4

12 10 10 20 28 41 53 63 69 67 67 57 49 36 24 14 8

13 10 16 22 32 46 63 77 87 91 85 79 69 55 38 24 14

14 5 23 29 37 56 77 97 111 130 112 106 98 79 58 38 24

15 1 28 35 42 63 91 116 138 175 148 141 134 113 85 60 38

16 6 33 41 49 75 108 143 171 225 198 190 182 156 123 88 60

17 15 35 45 57 83 124 168 207 282 262 252 240 214 173 129 90

18 20 42 48 68 98 145 202 253 365 343 337 317 287 240 183 132

19 15 39 45 79 107 166 233 301 475 451 445 411 383 324 257 189

20 6 41 43 93 126 190 275 360 621 586 584 534 501 437 352 267

21 1 46 42 108 142 215 314 423 791 746 750 684 650 577 478 369

Table 3. On the left, cp.n; k/, the number of cycle partitions. On the right,

p.n/ � cp.n; k/, the number of partitions not in cycles.


Under conjugation, we know every partition is in a cycle, either self-conjugate

or half of a conjugate pair. By Lemma 1, then, cp.n; n � 1/ D p.n/. The right-

hand side of Table 3 shows p.n/ � cp.n; k/. The envelope of this triangle of

differences appears to be 2s.n/ for s.n/ defined in Section 4.


cp.n; j / D p.n/ � 2s.n � j � 2/.

Proof. Given � 2 GE.n; j /, we claim that its iterated images under Dj have the

form

� K�!�0 �! � K !�0

where K! indicates that the operationDj coincides with conjugation and we allow

the possibility � D �0.

First, we know from the proof of Theorem 1 that � 2 GE.n; j / has at least

jC2 parts, and then �1 � j by Lemma 3b. Therefore, by Lemma 1,Dj .�/ D �0.

Let Dj .�0/ D �. Since � is a Garden of Eden partition, � ¤ � and applying Dj

to �0 is not equivalent to conjugation. By the definition of Dj ,

� D Dj .Dj .�// D Dj .�0/ D .�1; : : : ; �j ; �0

1 � j /

with �0

2�j and subsequent terms removed since �0

2 � j by Lemma 3c. Therefore

�0

1 D j C 1 and, by Lemma 3b, �1 � j C 1. By Lemmas 3c and 1 we conclude

that Dj .�/ D �0. Likewise, Dj .�0/ D �. Since conjugation is an involution,

the sets ¹�; �0º and ¹�;�0º are disjoint, completing the claim.

We can now complete the proof of the theorem. If some partition � 2 P.n/

is not in a cycle, it is either a GE-partition or between a GE-partition and a cycle

partition. By the claim above, we know that a GE-partition maps to its conjugate,

which maps to a cycle partition. From Theorem 1, we know that there are s.n �

j � 2/ GE-partitions, which are not in cycles. Their conjugates are the other

s.n � j � 2/ partitions not in cycles.

It is important to realize that the structural results of the proof do not imply

that every component of P.n/ under Dj in the envelope contains at most four

partitions. It is true that � 2 GE.n; j / has iterates �! �0 ! � for� 2 CP.n; j /,

but multiple GE-partitions can lead to the same �, e.g.,

D3.D3..215// D D3.61/ D 3211; D3.D3..314// D D3.511/ D 3211:

Also, the structural results of the proof do not hold in general outside the enve-

lope. For instance, D2.D2.513// D 3221 which does not have k C 1 D 3 parts.

Also, Figure 1 shows that D1.D1.15// D 41 … CP.5; 1/, so two steps from a

74 Brian Hopkins

GE-partition is not always a cycle partition. Lengths from GE-partitions to cycle

partitions for D1 are among the data tabulated in [4].

A formula for the column cp.n; 1/ is proven in [3].

Theorem 4. Write n D�

mC12

�

� a where 0 � a � m � 1. Then cp.n; 1/ D�

ma

�

.

Formulas for other columns would seem to require generalizing the charac-

terization of cycle partition for D1 found in [3]. The proof in the next section

describes cycle partitions in the envelope, but does not apply for smaller j .

6 Connected Components

Let cc.n; k/ denote the number of connected components in the state diagram of

P.n/ under the operation Dk . From the examples of the previous section, we

know cc.5; 1/ D 1, cc.5; 2/ D 2, cc.5; 3/ D 3, and cc.5; 4/ D 4. Those data

correspond to the n D 5 row on the left-hand side of Table 4.

1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8

2 1 0

3 1 2 1 0

4 1 2 3 2 1 0

5 1 2 3 4 3 2 1 0

6 1 2 4 5 6 5 4 2 1 0

7 1 2 4 6 7 8 7 6 4 2 1 0

8 2 3 5 8 10 11 12 10 9 7 4 2 1 0

9 1 2 5 9 12 14 15 16 15 14 11 7 4 2 1 0

10 1 2 6 10 15 18 20 21 21 20 16 12 7 4 2 1

11 1 2 6 11 17 22 25 27 28 27 23 18 12 7 4 2

12 2 3 7 13 21 28 33 36 38 37 33 27 19 12 7 4

13 2 4 7 14 23 33 40 45 50 48 45 38 29 19 12 7

14 1 5 9 15 27 39 50 57 68 64 60 54 42 30 19 12

15 1 6 11 17 30 46 60 71 89 84 79 73 60 44 30 19

16 1 7 12 18 34 54 73 88 117 111 106 100 84 64 45 30

17 3 8 13 20 37 61 85 106 148 143 138 131 114 90 66 45

18 4 9 14 23 41 69 101 128 191 186 181 172 154 126 94 67

19 3 9 13 25 44 78 116 152 245 239 235 223 204 170 132 96

20 1 8 12 29 49 87 135 181 316 309 305 288 268 230 182 136

21 1 7 12 33 53 97 153 212 399 393 388 367 347 303 247 188

Table 4. On the left, cc.n; k/, the number of connected components. On the right,

cc.n/ � cc.n; k/.


The numbers of self-conjugate partitions and conjugate pairs were studied by

Osima [7]. It follows that the total number of components of P.n/ under conju-

gation (equivalently Dn�1) is given by

cc.n/ D p.n/ � p.n � 2/C p.n � 8/ � p.n � 18/C p.n � 32/ � � � �

DX

k�0

.�1/kp.n � 2k2/

with initial terms 1; 1; 2; 3; 4; 6; 8; 12; 16; 22; 29 : : : (A046682 in [8]). The right-

hand side of Table 4 shows cc.n/ � cc.n; k/. The envelope of this triangle of

differences appears once again to be s.n/ defined in Section 4.


we have cc.n; j / D cc.n/ � s.n � j � 2/.

Proof. Recall from the proof of Theorem 3 that a partition of P.n/ not in a cycle

under Dj is either a GE-partition or the conjugate of a GE-partition. For each

� 2 GE.n; j /, the conjugate pair ¹�; �0º counted in cc.n/ is part of another com-

ponent. The discussion of � D Dj .Dj .�// in the proof of Theorem 3 established

that the conjugate pair ¹�;�0º is still a 2-cycle under Dj or the self-conjugate �

is still self-conjugate under Dj .

We show that no partitions in CP.n; j / are part of larger cycles. Recall from

the proof of Theorem 1 that GE.n; j / D P.n; jC2/[� � �[P.n; n/. It follows that

¹conjugates of GE-partitionsº is the set of � 2 P.n/ with �1 � j C 2. Therefore

CP.n; j / is the remainder of P.n/, namely, the � 2 P.n/ with �1 � j C 1 and

� 0

1 � j C 1, i.e., the partitions of n that fit inside a .j C 1/ � .j C 1/ square.

By Lemmas 3c and 1, the operationDj is equivalent to conjugation for CP.n; j /,

which means that it consists of conjugate pairs and self-conjugate partitions.

Of the singletons and pairs counted by cc.n/, exactly ge.n; j / pairs are no

longer components in P.n/ under Dj . By Theorem 1, we conclude cc.n; j / D

cc.n/ � s.n � j � 2/.

Viewing P.n/ dynamically underDn�1, thenDn�2, . . . , some conjugate pairs

are “opened” into “�! �0 !” fragments that attach to preserved conjugate pairs

or self-conjugate partitions. To the left of the envelope, larger cycles develop and

there are longer paths from GE-partitions to cycle partitions, as in Figure 1.

A formula for the column cc.n; 1/ is proven in [3].

Theorem 6. Write n D�

mC12

�

� a where 0 � a � m � 1. Then

cc.n; 1/ D1

m

X

d j.m;a/

'.d/

m=d

a=d

!

76 Brian Hopkins

where the summation is over all divisors of the greatest common divisor of m and

a, and ' is the Euler phi function.

It would be very interesting to determine formulas for other columns, as they

would transition between the number-theoretic formula for cc.n; 1/ and the for-

mulas involving p.n/ for cc.n; j / in the envelope.

Acknowledgments. This article was developed from a talk given at the 2007

Integers conference at the University of West Georgia. Thanks to Bruce Landman

for coordinating the conference, where I enjoyed fruitful discussions on this mate-

rial with James Sellers and Louis Kolitsch. Thanks also to the anonymous referee

for close reading and helpful suggestions. Antonio Pane (SPC ’08) prepared some

of the figures. Helpful data were derived using Mathematica.

References

[1] G. Andrews, The Theory of Partitions, Cambridge University Press, 1984.

[2] G. Andrews and K. Eriksson, Integer Partitions, Cambridge University Press, 2004.

[3] J. Brandt, Cycles of partitions, Proc. Amer. Math. Soc. 85 (1982), 483–486.

[4] B. Hopkins and M. A. Jones, Shift-induced dynamical systems on partitions and com-

positions, Electron. J. Combin. 13 (2006), R80.

[5] B. Hopkins and J. A. Sellers, Exact enumeration of Garden of Eden partitions, Inte-

gers 7 (2007), no. 2, A19.

[6] A. Munagi, Computation of q-partial fractions, Integers 7 (2007), no. 2, A25.

[7] M. Osima, On the irreducible representations of the symmetric group, Canad. J.

Math. 4 (1952), 381–384.

[8] N. Sloane, The On-line Encyclopedia of Integer Sequences, published electronically

at www.research.att.com/�njas/sequences/.

Author information

Brian Hopkins, Department of Mathematics, Saint Peter’s College,

Jersey City, NJ 07306, USA.



On the Euler Product of

Some Zeta Functions

Xian-Jin Li

Abstract. It is well known that the Euler product for the Riemann zeta function �.s/ is

still valid for <.s/ D 1 and s ¤ 1. In this paper, we extend this result to zeta functions of

number fields. In particular, the Dedekind zeta function �k.s/ for any algebraic number

field k and the Hecke zeta function �.s; �/ for the rational number field are shown to have

the Euler product on the line<.s/ D 1 except at s D 1. A functional equation is obtained

for the finite Euler product, which is the product of first finite number of factors in the

Euler product of �.s; �/.

Keywords. Dirichlet series, Euler product, zeta-fuction.

AMS classification. 11M41, 30B40.

1 Introduction

The gamma function �.s=2/ can be regarded as a factor for the infinity place of

the rational number field in the Euler product of the Riemann zeta function �.s/ as

shown in [11], and is required for the functional equation of �.s/. A spectral the-

ory of the gamma function was discovered by Sonine [10]. Remarkable examples

of functions were presented for which the analogue of the Riemann hypothesis

is true (cf [6]). Sonine’s theory was further developed by de Branges [1] whose

theory can be regarded as a generalization of the part of Fourier analysis involving

Fourier transforms and the Plancherel formula.

Fourier analysis over number fields is a fundamental tool for studying zeta-

and L-functions. It started in Tate’s thesis [11], which had a deep influence on

number theory as a piece of clandestine literature. Therefore, it is natural for us

to generalize de Branges’ theory to number fields. In trying to do so, we need

the Euler product for the Dedekind zeta function �k.s/ and for the Hecke zeta

function �.s; �/ to be still valid for <.s/ D 1 and s ¤ 1.

The way, in which the convergence of the Euler product of zeta functions on

the line <s D 1 is needed in my current work in progress, goes briefly as the

Research supported by National Security Agency H98230-06-1-0061.

78 Xian-Jin Li

following: For the Hecke zeta function we expect a certain relation between �.1�

iz; �/ and �.1 C iz; N�/ to be true for all complex z. This relation is similar to

the statement of Theorem 1.3 with S being the set of all places of the rational

number field. Since the Euler product for �.1 � i t; �/ is valid for all nonzero

real t as shown in Theorem 1.2, we can prove that the expected relation between

�.1 � i t; �/ and �.1C i t; N�/ holds for all nonzero real t . Because one side of the

expected relation is analytic in the upper half-plane and is continuous in the closed

upper half-plane, and because the other side of the expected relation is analytic in

the lower half-plane and is continuous in the closed lower half-plane, by analytic

continuation we see that the expected relation holds for all complex z. The results

obtained may be important for studying the analyticity of Artin L-functions in the

whole complex plane.

For the Fourier transformation, we have a similar construction. But, the cor-

responding relation turns out to be between �.1=2 � iz; �/ and �.1=2 C iz; N�/.

In other words, the line where both sides of the relation are extended across by

analytic continuation is the line 1=2C i t , t 2 R instead of the line 1C i t , t 2 R.

Since the Hecke zeta function �.s; �/ does not have Euler product on the line

<s D 1=2, we don’t have a similar result. The result obtained is only for finite

Euler products. It will be described after the statement of Theorem 1.2.

Let k be any algebraic number field. The Dedekind zeta function �k.s/ of k is

defined by

�k.s/ DY

p

1

1 �Np�s

for � > 1 with s D � C i t , where the product is over all prime ideals p of k.

For the convergence of the Euler product of the Dedekind zeta function �k.s/

on the line <s D 1, we prove the following theorem in Section 2.

Theorem 1.1. We have the Euler product

�k.s/ DY

p

1

1 �Np�s

for � D 1 and s ¤ 1.

A Hecke character � of the rational number field Q is a character of the mul-

tiplicative group generated by primes of Q not in some set P and has value 0 on

finite primes in P , where P is a finite set of primes including the infinite prime of

Q and is called the exceptional set of �.

The Hecke zeta-function �.s; �/ is defined by

�.s; �/ DY

p 62P

1

1 � �.p/p�s

for � > 1, s D � C i t .

On the Euler Product of Some Zeta Functions 79

For the convergence of the Euler product of the Hecke zeta function �.s; �/ on

the line <s D 1, we obtain the following theorem which is proved in Section 3.

Theorem 1.2. Let � be a Hecke character of Q. Then

�.s; �/ DY

p 62P

1

1 � �.p/p�s

for � D 1 and s ¤ 1, where the product is over all rational primes p 62 P .

For every place v, we denote by Qv, Ov , and Pv the completion of Q at

v, the maximal compact subring of Qv, and the unique maximal ideal of Ov ,

respectively. We denote by j � jv the valuation of Q normalized so that j � jv is

the ordinary absolute value if v is real, and j�vjv D 1=p if Ov=Pv contains p

elements where Pv D �vOv .

The idele group J of Q is the restricted direct product of the multiplicative

groups Q�v relative to subgroups O�

v of units in Qv. Let J 1 be the set of ideles

˛ D .˛v/ such thatQ

j˛vjv D 1.

Let S be a finite set of places of Q containing the infinite place. We define

AS DQ

v2S Qv. Let

v W x ! e2�i�v.x/

be the character on the additive group Qv given in Section 2.2 of [11]. It is trivial

on Ov , and is nontrivial on P�1v for finite places v. We have order. v/ D 0 and

d�1v D Ov for all finite places v of Q. Let

.˛/ DY

v2S

v.˛v/

for ˛ 2 AS . Then is a character on AS .

For any place v of Q, we select a fixed Haar measure d˛v on the additive

group Qv as follows: d˛v WD the ordinary Lebesgue measure on the real line if

v is real, and d˛v WD that measure for which Ov gets measure 1 if v is finite.

Then d˛ DQ

v2S d˛v is the unique Haar measure on AS such that the inversion

formula

f .˛/ D

Z

AS

bf .ˇ/ .˛ˇ/ dˇ

holds if f is continuous and bf 2 L1.AS /, where

bf .ˇ/ D

Z

AS

f .˛/ .�˛ˇ/ d˛;

for f 2 L1.AS /; see Section 3.3 in [11].

80 Xian-Jin Li

The Schwartz space S.R/ is the space of all smooth functions f , all of whose

derivatives are of rapid decay; that is

@kf

@xk.x/ D O..1C jxj/�N /

for all integers k � 0 and N > 0. Let S.AS / be the Schwartz–Bruhat space on

AS , whose functions are finite linear combinations of functions of the form

f .˛/ DY

v2S

fv.˛v/

where

(1) fv is in the Schwartz space S.R/ if v is the infinite place of Q, and

(2) fv belongs to S.Qv/, the space of locally constant and compactly supported

functions on Qv, if v is finite.

Let

c.˛/ DY

all places v of Q

cv.˛v/ D

0

@

Y

p2P

Qcv. Qv/j˛vjitv

v

1

A�.'P .˛// (1.1)

be a character of J 1 given in Section 4.5 of [11] with t1 D 0, where � is a Hecke

character of Q with exceptional set P and

'P .˛/ DY

p 62P

pordv˛v :

In particular, cv is unramified for every v 62 P , and c.�/ D 1 for all � 2 Q�. We

denote P 0 D P � ¹1º.

The ramification degree of Qcv is denoted by ev for p 2 P 0. We always assume

that ev � 1. Otherwise, if ev D 0 we do not include this prime p in the set P . By

Corollary 2.4.1 in [11] and the remark after Theorem 4 in [9],

cv.�ev

v /pev=2

Z

jxjD1

v.��ev

v x/ Ncv.x/dx D p�i�v (1.2)

for some real number �v when v 2 P 0.


S is always chosen so that it contains P . We define a finite Euler product �Sby

�S .s; �/ DY

v2S;v¤1

1

1 � �.pv/p�sv

:

A functional equation is obtained in the following for the finite Euler product

�S .s; �/.

Theorem 1.3. Assume that c, � and P are given as in (1.1) with Qc1.�1/ D 1.

Let f be a function in S.AS / such that f .˛/ and bf .˛/ vanish for j˛j � ı for

a positive number ı and such that f .˛0/ D f .˛/ for all ˛ 2 AS , where ˛0 is

obtained from ˛ by replacing ˛1 by �˛1. Then

�

Y

p2P 0

pi�vCevs�ev=2

�

��s

2�.s

2/�S .s; �/

Z

AS

bf .ˇ/ Nc.ˇ/jˇj�sdˇ

D ��1�s

2 �.1 � s

2/�S .1 � s; N�/

Z

AS

f .˛/c.˛/j˛js�1d˛ (1.3)

for all complex s, where the left side for<s < 1=2 and the right side for<s > 1=2

are defined by analytic continuation.

2 Proofs of Theorem 1.1

Let k be an algebraic number field with r1 real places and r2 imaginary places.

Put G1.s/ D ��s=2�.s=2/ and G2.s/ D .2�/

1�s�.s/. Let

�k.s/ D s.s � 1/jdjs

2G1.s/r1G2.s/

r2�k.s/; (2.1)

where d is the discriminant of k. By Theorem 3 in Chapter VII of Weil [14], �k.s/

is an entire function.

Define �v to be one when v is a real place of k and to be two when v is an

imaginary place of k. Let x DQ

xv be the variable in the half space Rr1Cr2

C.

Denote by jxj the productQ

x�v

v taken over all infinite places of k. Let N D

r1 C 2r2. The Hecke theta function ‚k.x/ is defined by

‚k.x/ DX

b

exp

�

� �jdj�1

N .Nb/2

N

X

v

�vxv

�

(2.2)

where the sum on b runs over all nonzero integral ideals of k and the sum on v

is over all infinite places of k. Put dx DQ

dxv. It follows from Theorem 3 in

82 Xian-Jin Li

Chapter XIII of Lang [5] that

�k.s/ D 2r1.2�/r2hR=e C s.s � 1/

Z

jxj>1

‚k.x/.jxjs

2 C jxj1�s

2 /dx

x(2.3)

for all complex s, where h, R and e are respectively the number of ideal classes

of k, the regulator of k and the number of roots of unity in k.

Lemma 2.1 (Satz 184 in Landau [4]). �k.s/ ¤ 0 on the line � D 1.

Lemma 2.2 (Satz 186 in Landau [4]). There are positive constants c0; t0 depend-

ing on k with t0 > 1 such that

�0

k.s/

�k.s/D O.ln3 jt j/

for � � 1 � 1c0 ln jt j

and jt j > t0.

Lemma 2.3 (Lemma 3.12 in Titchmarsh [13]). Let

f .s/ D

1X

nD1

an

ns

for � > 1, where an D O. .n// with being non-decreasing. Assume that

1X

nD1

janj

n�D O

²

1

.� � 1/˛

³

as � ! 1. If c > 0 and � C c > 1, x is not an integer, and N is the integer

nearest to x, then

X

n<x

an

nsD

1

2�i

Z cCiT

c�iT

f .s C w/xw

wdw CO

²

xc

T .� C c � 1/˛

³

CO

²

.2x/x1�� ln x

T

³

CO

²

.N/x1��

T jx �N j

³

for T � 0.

Lemma 2.4. Let�0

k

�k.s/ D

1X

nD1

An

ns

for � > 1. Then

An D O�

.lnn/1C��

as n!1, where � is a small positive number.


Proof. For � > 1,

��0

k

�k.s/ D

X

p

1X

mD1

lnNp

.Np/ms

where the sum on p is over all prime ideals of k. For each prime ideal p of k there

is exactly one rational prime p which is divisible by p. Thus Np D pf for some

positive integer f less than or equal to the degree of k over Q; see Theorem 108

in [3].

Let .Np/m D p� . We can write

X

p

1X

mD1

lnNp

.Np/msDX

p

1X

�D1

�

X

p;.N p/mDp�

lnNp

�

p��s;

where the sum on p is over all rational primes. We have

X

p;.N p/mDp�

lnNp � �1.�/ lnp;

where

�1.�/ DX

d j�

d � d.�/� � �1C�

with d.�/ being the number of positive divisors of �. Thus

X

p;.N p/mDp�

lnNp� �1C� lnp:

If we denote p� D n, then

An D �X

p;.N p/mDp�

lnNp:

It follows that

An � .lnn/1C�

as n!1.

This completes the proof of the lemma.

Lemma 2.5. Let An be given as in Lemma 2.4. Then

1X

nD1

jAnj

n�D O

²

1

� � 1

³

as � ! 1C.

84 Xian-Jin Li

Proof. We write

1X

nD1

jAnj

n�DX

p

1X

mD1

lnNp

.Np/m�D �

�0

k

�k.�/

for � > 1. By (2.1) and (2.2)

� 0

k

�k.�/ D

1

�C

1

� � 1C

ln jdj

2� .r2C

r1

2/ ln� C

r1

2

� 0

�.�

2/C r2

� 0

�.�/C

�0

k

�k.�/:

Since �k.1/ D ck and since poles of �.s/ are at s D �1;�2; : : : , multiplying

both sides of the above identity by � � 1 and letting � ! 1C we find that

lim�!1C

.� � 1/�0

k

�k.�/ D �1:

It follows that1X

nD1

jAnj

n�D O

²

1

� � 1

³

as � ! 1C.


Lemma 2.6. Let An be given as Lemma 2.4, and let s D 1 C i t for any fixed

nonzero real number t . Then the partial sums

X

n<x

An

ns

are bounded as x !1.

Proof. By Lemma 2.3, Lemma 2.4, and Lemma 2.5

X

n<x

An

nsD

1

2�i

Z cCiT

c�iT

�0

k

�k.s C w/

xw

wdw CO

²

xc

Tc

³

CO

²

.ln x/2C�

T

³

(2.4)

for c > 0 and T � 0.

Let

ı D1

c0 ln.jt j C T /:


By Lemma 2.2, Lemma 2.1, and (2.3), we can choose T to be sufficiently large

so that �k.s C w/ has no zeros for <.w/ � �ı and j=.s C w/j � jt j C T . By

Cauchy’s residue theorem,

1

2�i

Z cCiT

c�iT

�0

k

�k.s C w/

xw

wdw

D�0

k

�k.s/ �

x1�s

1 � sC

1

2�i

Z cCiT

�ıCiT

C

Z

�ıCiT

�ı�iT

C

Z

�ı�iT

c�iT

!

�0

k

�k.s C w/

xw

wdw:

(2.5)

By Lemma 2.2 and the choice of ı,

1

2�i

Z cCiT

�ıCiT

�0

k

�k.s C w/

xw

wdw �

ln3 T

Txc : (2.6)

Similarly, we have

1

2�i

Z

�ı�iT

c�iT

�0

k

�k.s C w/

xw

wdw �

ln3 T

Txc :

In this paragraph we assume that jtC=.w/j � t0,<.w/ D �ı, and s D 1Ci t .

Thenˇ

ˇ

ˇ

ˇ

1

s C wC

1

s C w � 1C

ln jdj

2� .r2 C

r1

2/ ln�

ˇ

ˇ

ˇ

ˇ

�1

ı:

By using the identity

� 0.z/

�.z/D �

1

zC z

1X

nD1

1

n.nC z/�

where is Euler’s constant, we get thatˇ

ˇ

ˇ

ˇ

r1

2

� 0

�.s C w

2/C r2

� 0

�.s C w/

ˇ

ˇ

ˇ

ˇ

� 1:

By (2.3),ˇ

ˇ

ˇ

ˇ

� 0

k

�k.s C w/

ˇ

ˇ

ˇ

ˇ

� 1:

Thus it follows from the identity

� 0

k

�k.s C w/ D

1

s C wC

1

s C w � 1C

ln jdj

2� .r2 C

r1

2/ ln�

Cr1

2

� 0

�.s C w

2/C r2

� 0

�.s C w/C

�0

k

�k.s C w/

86 Xian-Jin Li

thatˇ

ˇ

ˇ

ˇ

�0

k

�k.s C w/

ˇ

ˇ

ˇ

ˇ

�1

ı(2.7)

when jt C=.w/j � t0.

When jt C=.w/j > t0, as <.s C w/ D 1 � ı by Lemma 2.2 we find that

ˇ

ˇ

ˇ

ˇ

�0

k

�k.s C w/

ˇ

ˇ

ˇ

ˇ

� ln3 T:

Note that t is a fixed real number. Thus by (2.7) we obtain that

1

2�i

Z

�ıCiT

�ı�iT

�0

k

�k.s C w/

xw

wdw � x�ı max

²

1

ı; ln3 T

³Z T

0

1pı2 C u2

du:

SinceZ T

0

1pı2 C u2

du D ln�

T=ı C

q

.T=ı/2 C 1�

and

ı D1

c0 ln.jt j C T /;

we have1

2�i

Z

�ıCiT

�ı�iT

�0

k

�k.s C w/

xw

wdw � x�ı ln4 T: (2.8)

We can take c D 1=ln x and T D exp�p

ln x�

with x being sufficiently large.

Then xc D e and

x�ı ln4 T Dln2 x

eln x=c0 ln.jt jCT /�

ln2 x

eln x=2c0 ln TD

ln2 x

ep

ln x=2c0

D o.1/

as x !1. It follows from (2.5)–(2.8) that

1

2�i

Z cCiT

c�iT

�0

k

�k.s C w/

xw

wdw D

�0

k

�k.s/ �

x1�s

1 � sC o.1/ (2.9)

as x !1.

Sincexc

TcDe ln x

ep

ln xD o.1/

and.ln x/2C�

TD.ln x/2C�

ep

ln xD o.1/;


by (2.4) and (2.9)X

n<x

An

nsD�0

k

�k.s/ �

x1�s

1 � sC o.1/

as x !1. By Lemma 2.1, the stated result then follows.


Proof of Theorem 1.1. For � > 1,

��0

k

�k.s/ D

X

p

1X

mD1

lnNp

.Np/ms

and

ln �k.s/ DX

p

1X

mD1

1

m.Np/ms

where the sum on p is over all prime ideals of k. We write

ln �k.s/ DX

p

1X

mD1

lnNp

.Np/ms�

1

ln ¹.Np/mº:

Let An’s be given as in Lemma 2.4. Then

ln �k.s/ D �

1X

nD2

An

ns�

1

lnn(2.10)

for � > 1, because the series is absolutely convergent.

For any fixed non-zero real number t , by Lemma 2.6 the partial sums

NX

nD2

An

n1Cit

are bounded for all sufficiently large integers N . Since 1=lnn tends steadily to 0

as n!1, by Dirichlet’s test the series

1X

nD2

An

ns�

1

lnn

converges for � D 1 and s ¤ 1.

Since ln �k.s/ is continuous for � � 1 and s ¤ 1 by Lemma 2.1, and since the

right side of (2.10) represents an analytic function of s in the half-plane � > 1

88 Xian-Jin Li

and is convergent for � D 1 and s ¤ 1, by the continuity theorem for Dirichlet

series (see Section 9.12 of Titchmarsh [12])

ln �k.s/ D �

1X

nD2

An

ns�

1

lnn


By the proof of Lemma 2.4, the series

X

p

1X

mD2

1

m.Np/ms

is absolutely convergent for � D 1. This implies that

ln �k.s/ DX

p

1X

mD1

1

m.Np/msDX

p

ln1

1 �Np�s (2.11)

for � D 1 and s ¤ 1. By taking exponentials of both sides of (2.11), we find that

�k.s/ DY

p

.1 �Np�s/�1


This completes the proof of Theorem 1.1.

3 Proof of Theorem 1.2

The proof of Theorem 1.2 is a minor modification of that for Theorem 1.1. We

present it for the convenience of readers.

Lemma 3.1 (Theorem 7.15 in [7]). �.s; �/ ¤ 0 on the line � D 1.

Lemma 3.2 (Theorem 7.20 and its proof in [7]). There are positive constants t0 >

e and c0, depending on �, such that

�0.s; �/

�.s; �/� lnM jt j

for � � 1 � 1c0 ln jt j

and jt j > t0, where M is some constant greater than 1.


Lemma 3.3. Let

�0

�.s; �/ D

1X

nD1

Bn

ns

for � > 1. Then jBnj � lnn for all n.

Proof. For � > 1,

��0

�.s; �/ D

X

p 62P

1X

mD1

�.pm/ lnp

pms:

The stated assertion then follows.

Lemma 3.4. Let Bn be given as in Lemma 3.3. Then

1X

nD1

jBnj

n�D O

²

1

� � 1

³

as � ! 1C.

Proof. We have1X

nD1

jBnj

n��X

p

1X

mD1

lnp

pm�D �

�0

�.�/

for � > 1, where � is the Riemann zeta-function.

By (2.12.7) in [13],

��0.�/

�.�/D

1

� � 1C 1C

2� ln 2� C

1

2

� 0

�.1C

�

2/C

X

�

�

�.� � �/

where the sum is over all complex zeros � of �.s/. This implies that

��0.�/

�.�/�

1

� � 1

as � ! 1C. It follows that

1X

nD1

jBnj

n�D O

²

1

� � 1

³

as � ! 1C.


90 Xian-Jin Li

In the next two paragraphs, we review some analytic properties for �.s; �/; see

(3.3) and (3.5). They will be needed for the proof of Lemma 3.5. Let

f DY

all places v of Q

fv

where

fv.˛v/ D

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

1Ov.˛v/ if p 62 P

v.˛v/1P�ev

v

.˛v/ if p 2 P 0

e��˛2v if v D1; cv.�1/ D 1

˛ve��˛2

v if v D1; cv.�1/ D �1:

(3.1)

We have

bf v.ˇv/ D

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

1Ov.ˇv/ if p 62 P

pev 11CPev

v

.ˇv/ if p 2 P 0

e��ˇ 2v if v D1; cv.�1/ D 1

iˇve��ˇ 2

v if v D1; cv.�1/ D �1:

(3.2)

We define �P D 0 if P contains at least one finite prime of k and �P D 1 if P

contains only the infinite place of k. By the proof of Theorem 4.4.1 in [11], Tate’s

zeta-function �.f; cjjs/ equals

�.f; cjjs/ D

Z

1

1

�t .f; cjjs/dt

tC

Z

1

1

�t .bf ; Ncjj1�s/

dt

tC �P

°

bf .0/

s � 1�f .0/

s

±

;

(3.3)

where

�t .f; cjjs/ D

Z

J 1

f .tb/c.b/tsdb

with db being given as in [11].

The two integrals on the right side of (3.3) are entire functions of s. By the

argument in Section 4.5 and Section 2.5 of [11], we can write

�.f; cjjs/ D

�

Y

p2P

�p.fv; cvjjsv/

�

�.s; �/; (3.4)

where

�p.fv; cvjjsv/ D p

evs

Z

j˛jvDpev

v.˛/cv.˛/d�˛

with d�˛ being the standard Haar measure on Q�v given as in Section 2.3 of

Tate [11] for p 2 P 0, and where

�p.fv; cvjjsv/ D

´

��s=2�.s=2/ for v D1; c1.�1/ D 1

��sC1

2 �. sC12/ for v D1; c1.�1/ D �1:


Let �.s; �/ D Œs.s � 1/��P �.f; cjjs/: By (3.3), �.s; �/ is an entire function of s.

We have

� 0

�.s; �/ D

�P

sC

�P

s � 1C

1

2

� 0

�

�2s C 1 � c1.�1/

4

�

�ln�

2

CX

p2P 0

ev lnp C�0

�.s; �/:

(3.5)

Note that (3.5) also follows from Theorem 8.5 in Chapter VII of Neukirch [8],

which is proved using the classical language. The author wishes to thank the

referee for pointing out this to him.

Lemma 3.5. Let Bn be given as Lemma 3.3, and let s D 1 C i t for any fixed

nonzero real number t . Then the partial sums

X

n<x

Bn

ns

are bounded as x !1.

Proof. By Lemma 2.3, Lemma 3.3, and Lemma 3.4

X

n<x

Bn

nsD

1

2�i

Z cCiT

c�iT

�0

�.s C w;�/

xw

wdw CO

²

xc

Tc

³

CO

´

ln2 x

T

µ

(3.6)

for c > 0 and T � 0.

Let

ı D1

c0 ln.jt j C T /:

By Lemma 3.2 and Lemma 3.1, we can choose T to be sufficiently large so that

�.sCw;�/ has no zeros for <.w/ � �ı and j=.sCw/j � jt j CT . By Cauchy’s

residue theorem and (3.5), we find that

1

2�i

Z cCiT

c�iT

�0

�.s C w;�/

xw

wdw D

�0

�.s; �/ � �P

x1�s

1 � s

C1

2�i

Z cCiT

�ıCiT

C

Z

�ıCiT

�ı�iT

C

Z

�ı�iT

c�iT

!

�0

�.s C w;�/

xw

wdw: (3.7)

By Lemma 3.2 and the choice of ı,

1

2�i

Z cCiT

�ıCiT

�0

�.s C w;�/

xw

wdw �

lnM T

Txc : (3.8)

92 Xian-Jin Li

Similarly, we also find that

1

2�i

Z

�ı�iT

c�iT

�0

�.s C w;�/

xw

wdw �

lnM T

Txc : (3.9)

By Lemma 3.1, �.s; �/ ¤ for <s D 1. Since �.s; �/ is an entire function, we

can choose T large enough so that j�.� C i t; �/j � �0 for a positive number �0

depending only on � and t0 when jt j � t0 and 1 � ı � � � 1. Thus, if we use

(3.5) when jt C =.w/j � t0 and use Lemma 3.2 when jt C =.w/j > t0, we find

that

1

2�i

Z

�ıCiT

�ı�iT

�0

�.s C w;�/

xw

wdw � x�ı max¹

1

ı; lnM T º

Z T

0

1pı2 C u2

du:

SinceZ T

0

1pı2 C u2

du D ln�

T=ı C

q

.T=ı/2 C 1�

and

ı D1

c0 ln.jt j C T /;

we have1

2�i

Z

�ıCiT

�ı�iT

�0

�.s C w;�/

xw

wdw � x�ı lnM C1 T: (3.10)

We can take c D 1=ln x and T D exp�p

ln x�

with x being sufficiently large.

Then xc D e and

x�ı lnM C1 T Dln

MC12 x

eln x=c0 ln.jt jCT /�

.p

ln x/M C1

eln x=2c0 ln TD.p

ln x/M C1

ep

ln x=2c0

D o.1/

as x !1. It follows from (3.7)–(3.10) that

1

2�i

Z cCiT

c�iT

�0

�.s C w;�/

xw

wdw D

�0

�.s; �/ � �P

x1�s

1 � sC o.1/ (3.11)

as x !1.

Sincexc

TcDe ln x

ep

ln xD o.1/

andln2 x

TD

ln2 x

ep

ln xD o.1/;


by (3.6) and (3.11)

X

n<x

Bn

nsD�0

�.s; �/ � �P

x1�s

1 � sC o.1/

as x !1. By Lemma 3.1, the stated result then follows.


Proof of Theorem 1.2. For � > 1,

��0

�.s; �/ D

X

p 62P

1X

mD1

�.pm/ lnp

pms

and

ln �.s; �/ DX

p 62P

1X

mD1

�.pm/

mpms:

We write

ln �.s; �/ DX

p 62P

1X

mD1

�.pm/ lnp

pms�

1

ln.pm/:

Let Bn’s be given as in Lemma 3.3. Then

ln �.s; �/ D �

1X

nD2

Bn

ns�

1

lnn(3.12)

for � > 1, because the series is absolutely convergent.

For any fixed non-zero real number t , by Lemma 3.5 the partial sums

NX

nD2

Bn

n1Cit

are bounded for all sufficiently large integers N . Since 1=lnn tends steadily to 0

as n!1, by Dirichlet’s test the series

1X

nD2

Bn

ns�

1

lnn

converges for � D 1 and s ¤ 1.

Since ln �.s; �/ is continuous for � � 1 and s ¤ 1 by Lemma 3.1, and since

the right side of (3.12) represents an analytic function of s in the half-plane � > 1

94 Xian-Jin Li

and is convergent for � D 1 and s ¤ 1, by the continuity theorem for Dirichlet

series (see Section 9.12 of Titchmarsh [12])

ln �.s; �/ D �

1X

nD2

Bn

ns�

1

lnn


The seriesX

p 62P

1X

mD2

�.pm/

mpms

is absolutely convergent for � D 1. It follows that

ln �.s; �/ DX

p 62P

1X

mD1

�.pm/

mpmsDX

p 62P

ln1

1 � �.p/p�s(3.13)

for � D 1 and s ¤ 1. By taking exponentials of both sides of (3.13), we find that

�.s; �/ DY

p

.1 � �.p/p�s/�1



4 Proofs of Theorem 1.3

Let f be an even function in S.R/ such that f and bf vanish outside a finite

interval .�a; a/. For existence of such functions f , see Sonine [10]. We define

F.z/ D ��1=2�iz

2 ��1=2 � iz

2

�

Z

1

0

f .t/t�1=2Ciz dt

and

G.z/ D ��1=2�iz

2 ��1=2 � iz

2

�

Z

1

0

bf .t/t�1=2Ciz dt

for =z � 0.

Lemma 4.1. F andG have analytic extensions to the lower half-plane and satisfy

the identity

G.z/ D F.�z/ (4.1)


for all complex z. In particular,

��1=2�ix

2 ��1=2 � ix

2

�

Z

1

0

bf .t/t�1=2Cix dt

D ��1=2Cix

2 ��1=2C ix

2

�

Z

1

0

f .t/t�1=2�ix dt

(4.2)

for all real x.

Proof. Let y be any fixed positive number. If A.t/ D e��y2t2

, then

bA.t/ D y�1e��y�2t2

:

By Plancherel’s formula

Z

1

0

f .t/A.t/ dt D

Z

1

0

bf .t/bA.t/ dt I

that is,Z

1

0

f .t/e��y2t2

dt D

Z

1

0

bf .t/y�1e��y�2t2

dt

for all y > 0. It follows that

Z

1

0

y�1=2Ciz dy

Z

1

0

f .t/e��.yt/2

dt

D

Z

1

0

y�1=2Ciz dy

Z

1

0

bf .t/y�1e��.y�1t/2

dt

(4.3)

for real z. The left side of (4.3) is equal to

Z

1

0

f .t/t�1=2�iz dt

Z

1

0

y�1=2Cize��y2

dy

D1

2��

1=2Ciz

2 ��1=2C iz

2

�

Z

1

0

f .t/t�1=2�iz dt D1

2F.�z/

for real z, and the right side of (4.3) equals

Z

1

0

bf .t/t�1=2Ciz dt

Z

1

0

y�1=2�ize��y2

dy

D1

2��

1=2�iz

2 ��1=2 � iz

2

�

Z

1

0

bf .t/t�1=2Ciz dt D1

2G.z/

for real z. Therefore

G.z/ D F.�z/

96 Xian-Jin Li

for all real z. Since F and G are analytic and continuous in the closed upper half-

plane, F and G have analytic extensions to the lower half-plane. The extended F

and G satisfy the identity

G.z/ D F.�z/

for all complex z.


Lemma 4.2. Assume that cv is an unramified character of Q�v . Let f be a function

in S.Qv/. Then

1

1 � cv.�v/p�1=2Ciz

Z

Qv

bf .ˇ/jˇj�1=2Ciz Ncv.ˇ/dˇ

D1

1 � Ncv.�v/p�1=2�iz

Z

Qv

f .˛/j˛j�1=2�izcv.˛/d˛

for all complex z.

Proof. We have

Z

Qv


D

Z

Qv

f .˛/d˛

Z

Qv

v.�˛ˇ/jˇj�1=2Ciz Ncv.ˇ/dˇ

D

Z

Qv

f .˛/j˛j�1=2�izcv.˛/d˛

Z

Qv

v.x/ Ncv.x/jxj�1=2Cizdx

for all complex z.

By Theorem 1 in [9],

Z

Qv

v.x/ Ncv.x/jxj�1=2Cizdx D

1 � cv.�v/p�1=2Ciz

1 � Ncv.�v/p�1=2�iz(4.4)

for all complex z. It follows that

Z

Qv


D1 � cv.�v/p

�1=2Ciz

1 � Ncv.�v/p�1=2�iz

Z

Qv

f .˛/j˛j�1=2�izcv.˛/d˛

for all complex z. The stated identity then follows.



Lemma 4.3. Assume that cv is a unitary character on Q�v and has ramification

degree ev > 0. Let fv be a function in S.Qv/. Then

pi�v�ievz

Z

Qv

bf .ˇ/jˇj�1=2Ciz Ncv.ˇ/dˇ D

Z

Qv

f .˛/j˛j�1=2�izcv.˛/d˛

for all complex z, where �v is given in (1.2).

Proof. As in the proof of Lemma 4.2 we have

Z

Qv


D

Z

Qv

f .˛/j˛j�1=2�izcv.˛/d˛

Z

Qv


for all complex z.

By Lemma 1 in Sally and Taibleson [9],

Z

Qv


D pev.1=2Ciz/cv.�ev

v /

Z

jxjD1

v.��ev

v x/ Ncv.x/dx:

(4.5)

By (1.2), we can write

Z

Qv

v.x/ Ncv.x/jxj�1=2Cizdx D pievzp�i�v :

It follows from (4.5) that

pi�v�ievz

Z

Qv

bf .ˇ/jˇj�1=2Ciz Ncv.ˇ/dˇ D

Z

Qv

f .˛/j˛j�1=2�izcv.˛/d˛

for all complex z.


Let c, � and P be given as in (1.1). Then cv.�v/ D �.p/ for all p 62 P and

�.p/ D 0 for p 2 P 0.

Proof of Theorem 1.3. Since elements in S.AS / are finite linear combinations of

functions of the formY

v2S

fv.˛v/;

without loss of generality we can assume that f is of this form.

98 Xian-Jin Li

Because f .˛/ belongs to S.AS / and vanishes for j˛j � ı, the function

F.z/ D ��1=2�iz

2 ��1=2 � iz

2

�

�S .1=2 � iz; N�/

Z

AS

f .˛/c.˛/j˛j�1=2Cizd˛

(4.6)

is analytic in the upper half-plane =.z/ � 0 and is continuous in the closed upper

half-plane.

Similarly, we find that the expression

G.z/ D ��1=2�iz

2 ��1=2 � iz

2

�

�

Y

p2P 0

pi�v�eviz

�

� �S .1=2 � iz; �/

Z

AS

bf .ˇ/ Nc.ˇ/jˇj�1=2Cizdˇ

represents an analytic function of z in the upper half-plane. It is continuous in the

closed upper half-plane.

By Lemma 4.1, Lemma 4.2 and Lemma 4.3,

��1=2�it

2 ��1=2 � i t

2

�

�

Y

p2P 0

pi�v�evit

�

� �S .1=2 � i t; �/

Z

AS

bf .ˇ/ Nc.ˇ/jˇj�1=2Citdˇ

D ��1=2Cit

2 ��1=2C i t

2

�

�S .1=2C i t; N�/

Z

AS

f .˛/c.˛/j˛j�1=2�itd˛

for all real t ; that is, G.t/ D F.�t / for all real t . By analytic continuation, we see

that F and G can be extended to become analytic functions in the whole complex

plane and satisfy the identity

G.z/ D F.�z/ (4.7)

for all complex z. If we let s D 1=2 � iz, then the stated identity follows from

(4.7).


Remark. By Lemma 1 in Sally and Taibleson [9] and Sonine [10], there exist

functions f in S.AS / such that f .˛/ and bf .˛/ vanish for j˛j � ı for a positive

number ı and such that f .˛0/ D f .˛/ for all ˛ 2 AS , where ˛0 is obtained from

˛ by replacing ˛1 by �˛1.


Acknowledgments. The author wishes to thank the referee for his helpful sug-

gestions of improving the presentation of the original manuscript.

References

[1] L. de Branges, Hilbert Spaces of Entire Functions, Prentice-Hall, N.J., 1968.

[2] H. Davenport, Multiplicative Number Theory, Third Edition, Revised by Hugh

L. Montgomery, Springer-Verlag, New York, 2000.

[3] E. Hecke, Lectures on the Theory of Algebraic Numbers, Springer-Verlag, New

York, 1981.

[4] E. Landau, Einführung in die elementare und analytische Theorie der algebraischen

Zahlen und der Ideale, 2. Auflage, Chelsea Publishing Company, New York, 1949.

[5] S. Lang, Algebraic Number Theory, Second Edition, Springer-Verlag, New York,

1994.

[6] Xian-Jin Li, On zeros of defining functions for some Hilbert spaces of polynomi-

als, in: Operator Theory and Interpolation, edited by C. Foias and H. Bercovici,

pp. 235–243, Birkhäuser-Verlag, Basel, 2000.

[7] W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Third Edi-

tion, Springer-Verlag, Berlin, 2004.

[8] J. Neukirch, Algebraic Number Theory, Springer-Verlag, Heidelberg, 1999.

[9] P. J. Sally, Jr. and M. H. Taibleson, Special functions on locally compact fields, Acta

Math. 116 (1966), 279–309.

[10] N. Sonine Recherches sur les fonctions cylindriques et le développement des fonc-

tions continues en séries, Math. Ann. 16 (1880), 1–80.

[11] J. T. Tate, Fourier analysis in number fields and Hecke’s zeta-functions, in: Alge-

braic Number Theory, edited by J. W. S. Cassels and A. Fröhlich, pp. 305–347,

Academic Press, New York, 1967.

[12] E. C. Titchmarsh, The Theory of Functions, Second Edition, Oxford University

Press, 1958.

[13] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Second Edition, edited

by D. R. Heath-Brown, Oxford University Press, New York, 1986.

[14] A. Weil, Basic Number Theory, Springer-Verlag, Heidelberg, 1967.

Author information

Xian-Jin Li, Department of Mathematics, Brigham Young University,

Provo, Utah 84602, USA.



On the Range of the

Iterated Euler Function

Florian Luca and Carl Pomerance

Abstract. For a positive integer k let �k be the k-fold composition of the Euler function

�. In this paper, we study the size of the set ¹�k.n/ � xº as x tends to infinity.

Keywords. Iterations of Euler’s function, applications of sieve methods.

AMS classification. 11N36, 11N56.

1 Introduction

Let � be Euler’s function. For a positive integer k, let �k be the k-fold composi-

tion of �. In this paper, we study the range Vk of �k . For a positive real number

x we put

Vk.x/ D ¹�k.n/ � xº:

In 1935, Erdos [7] showed that #V1.x/ D x=.log x/1Co.1/. (Stronger estimates

are known for #V1.x/, see [10], [17].) In 1977, Erdos and Hall [8] considered the

more general problem of estimating #Vk.x/, suggesting that it is x=.log x/kCo.1/

for each fixed integer k � 1. They were able to prove that

#V2.x/ �x

.log x/2Co.1/;

and in fact, they were able to establish a somewhat more explicit form for this

inequality. Our first result is the following general upper bound on #Vk.x/ which

is uniform in k.

Theorem 1. The estimate

#Vk.x/ �x

.log x/kexp

�

13k3=2.log log x log log log x/1=2�

(1)

holds uniformly in k � 1 once x is sufficiently large.

Work by the first author was done in Spring of 2006 while he visited Williams College. The

second author was supported in part by NSF grants DMS-0401422 and DMS-0703850.

102 Florian Luca and Carl Pomerance

As a corollary we have, when x !1,

#Vk.x/ �x

.log x/kCo.1/(2)

when k D o..log log x= log log log x/1=3/, and

#Vk.x/ �x

.log x/.1Co.1//k

when k D o.log log x= log log log x/. Note that (1) is somewhat stronger than the

explicit upper bound in [8] for the case k D 2.

Let k � 1 be fixed. Letm > 2 be such thatm; 2mC1; : : : ; 2k�1mC2k�1�1

are all prime numbers. Then �k.2k�1mC 2k�1 � 1/ D m � 1. The quantitative

version of the Prime k-tuples Conjecture of Bateman and Horn [2] implies that the

number of such values m � x should be � ckx=.log x/k for x sufficiently large,

where ck > 0 is a constant depending on k. Thus, we see that up to the factor of

size .log x/o.1/ appearing on the right hand side of estimate (2), it is likely that

#Vk.x/ D x=.log x/kCo.1/ holds when k is fixed as x ! 1, thus verifying the

surmise of Erdos and Hall.

Next, we prove a lower bound on #V2.x/ comparable to the one predicted by

the above heuristic construction.

Theorem 2. There exists an absolute constant c2 > 0 such that the inequality

#V2.x/ � c2

x

.log x/2

holds for all x � 2.

In [8], Erdos and Hall assert that they were able to prove such a lower bound with

the exponent 2 replaced by any larger real number.

In the last section we study the integers that are in every Vk and we also discuss

analogous problems for Carmichael’s universal exponent function �.n/.

In what follows, we use the Vinogradov symbols � and � and the Landau

symbols O and o with their usual meaning. The constants and convergence im-

plied by them might depend on some other parameters such as k; K; ", etc. We

use p and q with or without subscripts for prime numbers. We use !.n/ for the

number of distinct prime factors of n, �.n/ for the number of prime power divi-

sors (> 1) of n, p.n/ and P.n/ for the smallest and largest prime divisors of n,

respectively, and v2.n/ for the exponent of 2 in the factorization of n. We write

log1 x D max¹1; log xº, and for k � 2 we put logk x for the k-fold iterate of the

function log1 evaluated at x. For a subset A of positive integers and a positive

real number x we write A.x/ for the set A \ Œ1; x�.

On the Range of the Iterated Euler Function 103

2 The Proof of Theorem 1

Let x be large. By a result of Pillai [18], we may assume that k � log x= log 2,

since otherwise Vk.x/ D ¹1º. Furthermore, we may in fact assume that k �

10�2 log2 x= log3 x, since otherwise the upper bound on #Vk.x/ appearing in es-

timate (1) exceeds x. We may also assume that n � x=.log x/k , since other-

wise there are at most x=.log x/k possibilities for n, and, in particular, at most

x=.log x/k possibilities for �k.n/ also.

By the minimal order of the Euler function, there exists a constant c0 > 0 such

that the inequality �.m/=m � c0m= log logm holds for all m � 3. From this it is

easy to prove by induction on k that if x is sufficiently large and �k.n/ � x, then

n � x.2c0 log2 x/k for all k in our stated range. Let X WD x.log2 x/

2k , so that

for large x, we may assume that n � X .

Let y D x1=.log2 x/2

and write n D pm, where p D P.n/. By familiar esti-

mates (see, for example, [3]), the number of n � X such that p � y is at most,

for large x,

X

.log x/log2 xDx.log2 x/

2k

.log x/log2 x�

x

.log x/k;

so we need only deal with the case p > y. Assume that �.�k.n// � 2:9k log2 x.

Lemma 13 in [15] shows that the number of such possibilities for �k.n/ � x is

�kx log x log2 x

22:9k log2 x�

x.log2 x/2

.log x/2:9k log 2�1�

x

.log x/k

for all k in our range. It follows that we may assume that

�.�k.n// � 2:9k log2 x:

It is easy to see that�.�.a// � �.a/� 1 for every natural number a. Thus, since

�k.m/ j �k.n/, we have

�.�.m// � 2:9k log2 x C k � 1 � 3k log2 x (3)

for all x sufficiently large.

Since also �k.p/ j �k.n/, we may assume that

�.�k.p// � 2:9k log2 x:

Since p > y, we have log2 p > log2 x � 2 log3 x, so that �.�k.p// � 3k log2 p

for x large. Since p � X=m, we thus have, in the notation of Lemma 4 below,

that p 2 Ak;3k.X=m/, and that result shows that the number of such possibilities

is at most

#Ak;3k.X=m/ �X

m.log.X=m//kexp

�

3k.6k log2X log3X/1=2 C 3k2 log3X

�

:


Observe further that with our bound on k,

3k.6k log2X log3X/1=2 C 3k2 log3X

D k3=2.log3X/�

3.6 log2X= log3X/1=2 C 3k1=2

�

� k3=2.log2X log3X/1=2.3p

6C 3=10/:

Since 3p

6C 3=10 < 7:7, it thus follows that if we put

U.x/ D exp.7:7k3=2.log2 x log3 x/1=2/;

then for large x,

#Ak;3k.X=m/ �xU.x/.log2 x/

2k

m.logy/k�xU.x/.log2 x/

4k

m.log x/k

uniformly in m and k. Thus, the number of such possibilities for n � X is

�xU.x/.log2 x/

4k

.log x/k

X

m2M

1

m;

where M is the set of all possible values of m. Such m satisfy, in particular, the

inequality (3). Lemma 3 below shows that if x is sufficiently large then

X

m2M

1

m� exp

�

2:9.3k log2X log3X/1=2�

;

which together with the fact that 2:9p

3 < 5:1 and the previous estimate shows

that the count on the set of our n � X is

�x

.log x/kexp

�

13k3=2.log2 x log3 x/1=2�

for large values of x. We thus finish the proof of Theorem 1 and it remains to

prove Lemmas 3 and 4.

Lemma 3. Let x be large, K be any positive integer and let N .K; x/ denote the

set of natural numbers n � x with �.�.n// � K log2 x. Then

X

n2N .K;x/

1

n� exp.2:9.K log2 x log3 x/

1=2/

holds for large values of x uniformly in K.


Proof. We assume that K � log2 x= log3 x since otherwise the right hand side

above exceeds .log x/2:9, while the left hand side is at most log x CO.1/, so the

desired inequality holds anyway.

Let z be a parameter that we will choose shortly. For each integer n � x write

n D n0n1, where each prime q j n0 has �.q � 1/ < log z and each prime q j n1

has �.q � 1/ � log z. For n 2 N .K; x/ we have that �.n1/ � K log2 x= log z.

Let N0.x/ denote the set of numbers n0 � x divisible only by primes q with

�.q � 1/ < log z and let N1.x/ denote the set of numbers n1 � x with �.n1/ �

K log2 x= log z. We thus have

X

n2N .K;x/

1

n�

�

X

n02N0.x/

1

n0

��

X

n12N1.x/

1

n1

�

: (4)

Note that

X

n02N0.x/

1

n0

�

1X

j D0

1

j Š

X

q�x�.q�1/<log z

1

qC

1

q2C � � �

!j

D exp

X


1

q � 1

!

:

It follows from Erdos [7] that there is some c > 0 such that the number of primes

q � t with !.q�1/ � 12

log2 q isO.t=.log t /1Cc/. Since !.q�1/ � �.q�1/, the

same O-estimate holds for the distribution of primes q with �.q � 1/ � 12

log2 q.

In particular the sum of their reciprocals is convergent, so that

X

ez2<q�x

�.q�1/<log z

1

q � 1�

X

ez2<q

�.q�1/< 12

log2 q

1

q � 1� 1:

Thus,

X


1

q � 1�

X

q�ez2

1

q � 1C

X

ez2<q�x

�.q�1/<log z

1

q � 1� 2 log z CO.1/;

and soX

n02N0.x/

1

n0

� z2: (5)


For the sum over N1.x/, we have

X

n12N1.x/

1

n1

�X

j �K log2 x= log z

1

j Š

X

q�x

1

q � 1

!j

�X

j �K log2 x= log z

1

j Š.log2 x CO.1//

j :

We choose z D exp..12K log2 x log3 x/

1=2/. Observe that the inequalities

K log2 x= log z D .2K log2 x= log3 x/1=2 < 21=2 log2 x= log3 x < log2 x

hold for large values of x. Thus,

X

n12N1.x/

1

n1

� .2 log2 x/K log2 x= log z : (6)

Putting (5) and (6) into (4) and using the fact that 2p

2 < 2:9, we have

X

n2N .K;x/

1

n� exp.2:9.K log2 x log3 x/

1=2//

for all sufficiently large x. This proves the lemma.

Remark 1. The above proof uses ideas from Erdos [7] and is also similar to

Lemma 4 in Luca [14].

Lemma 4. Let k, K be positive integers not exceeding 12

log2 x. Put

Ak;K D ¹p W �.�k.p// � K log2 pº:

We have

#Ak;K.x/ �x

.log x/kexp

�

3k.2K log2 x log3 x/1=2 C 3k2 log3 x

�

for all sufficiently large values of x, independent of the choices of k;K.

Proof. When k D 1, this trivially follows from the Prime Number Theorem. We

assume that k > 1. We let p 2 Ak;K.x/ and assume that p � x=.log x/k because


there are only �.x=.log x/k/ � x=.log x/k primes p failing this condition. Let

p0 D p and write

p0 � 1 D p1m1;

p1 � 1 D p2m2;

:::

pk�2 � 1 D pk�1mk�1;

where pi D P.pi�1 � 1/ for all i D 1; : : : ; k � 1. Since �.�.n// � �.n/ � 1,

we have that

�.pi�1 � 1/ � �.�i .p// � �.�k.p//C k � 2K log2 x

for all i D 1; 2; : : : ; k � 1 if x is sufficiently large. In particular

pi � p1=.2K log2 x/

i�1� p

1=.log2 x/2

i�1;

so that for x sufficiently large we have

pi � p1=.log2 x/2i

0 � yi WD1

2x1=.log2 x/2i

for i D 1; 2; : : : ; k � 1.

Consider the k linear functions Lj .x/ D Ajx C Bj for j D k; k � 1; : : : ; 1

given by Lk.x/ D x and

Lk�1.x/ D mk�1x C 1;

Lk�2.x/ D mk�2mk�1x Cmk�2 C 1;

:::

L1.x/ D m1 � � �mk�1x C .m1 � � �mk�2 Cm1 � � �mk�3 C � � � Cm1 C 1/:

Note that pk�1 � x=.m1 � � �mk�1/ is such that Lj .pk�1/ is a prime for all j D

1; : : : ; k. If some .Ai ; Bi / > 1, then there is at most one prime pk�1 for which

all of Lj .pk�1/ are prime. Further, since 0 D Bk < Bk�1 < � � � < B1, it follows

that if some AjBi D AiBj for some 0 � j < i � k � 1, then 1 < Ai=Aj j Bi

so that .Ai ; Bi / > 1. Thus, we may assume that each AjBi � AiBj ¤ 0. The

following result allows us to use something like a traditional sieve upper bound

for prime k-tuples, where it is not assumed that k is bounded. Note that a stronger

form of this lemma will appear in [11].


Lemma 5. Let Li .n/ D Ain C Bi be linear functions for i D 1; : : : ; k with

integer coefficients such that each Ai > 0, each .Ai ; Bi / D 1, and

E WD A1 � � �Ak

Y

1�j <i�k

.AjBi � AiBj /

is nonzero. Put F.n/ DQk

iD1Li .n/ and for each p let �.p/ be the number of

congruence classes n mod p such that F.n/ � 0 .mod p/. Assume that for

each p, we have �.p/ < p. If N � 2 and k � logN=.10 log2N/2, then the

number of n � N such that each Li .n/ is prime is at most

.ck log1 k/k

�

�

�.�/

�k N.log2N/k

.logN/k;

where c is an absolute constant and � is the product of the distinct primes p j E

with p > k.

Proof. We may assume that N is large since the constant c may be adjusted for

smaller values. Let Z denote the number of n � N with each Li .n/ prime. We

first show

Z � NY

k<p�N 1=.100k log2 N /

�

1 ��.p/

p

�

CO

�

N

.logN/10k

�

: (7)

For the proof, let �.m/ be the number of solutions n modulo m of the congru-

ence F.n/ � 0 .mod m/. Clearly, � is a multiplicative function. Put N1 D

N 1=.100k log2 N /. Noting that �.p/ � k, it follows that �.d/ � k!.d/ holds for all

squarefree positive integers d . Taking M to be the first even integer exceeding

10k log2N , we get, by the Principle of Inclusion and Exclusion and the Bonfer-

roni upper-bound inequality, that

Z � N 1=2 CX

k<p.d/�P.d/�N1

!.d/�M

�

N�.d/�.d/

dCO.k!.d//

�

� NY

k<p�N1

�

1 ��.p/

p

�

CO

N 1=2 CX

d W P.d/�N1

!.d/�M

k!.d/ CNX

d W �.d/¤0;P.d/�N1

!.d/>M

k!.d/

d

!

:


It remains to look at the O-terms. For the first sum, we have that

k!.d/ � k10k log2 N C2 D exp..10k log2N C 2/ log k/ < N 1=9

for all large values ofN uniformly in our range for k. The number of possibilities

for d is � NM1 � N .10k log2 N C2/=.100k log2 N / < N 1=9 for large values of N .

Hence, the first sum is < N 2=9. The second one is

�X

j >M

N

j Š

0

@

X

p�N1

k

p

1

A

j

�X

j >M

N

j Š.k log2N CO.k//

j

� NX

j >M

�

ek log2N CO.k/

j

�j

� NX

j >M

�e

9

�j

�N

eM�

N

.logN/10k

for large values of N . Note that in our range for k, this last error estimate domi-

nates the other two. Thus, we have (7).

To finish the proof of the lemma, we estimate the main term in (7). We have

log

�

Y

k<p�N1

�

1 ��.p/

p

��

� �X

k<p�N1

�.p/

p� �

X

k<p�N1

k

pCX

pj�

k

p

D �k log2N1 C k log2 k � kX

pj�

log.1 � 1=p/CO.k/:

Since the last sum above is � log.�=�.�// and log2N1 D log2N � log3N �

log1 k CO.1/, the main term in (7) is at most

.ck log1 k/k

�

�

�.�/

�k N.log2N/k

.logN/k

for some absolute constant c. Thus, by adjusting the constant c if necessary, we

have the lemma.

We apply Lemma 5 to our system of linear functions with

N D x=.m1 : : : mk�1/ � yk�1:

Thus, the number of choices for pk�1 � N with each Li .pk�1/ prime is at most

x.log log x/k

m1 : : : mk�1.logyk�1/k

�

c�

�.�/k log k

�k

:


We need an estimate for �=�.�/. For this, note that each AjBi in our setting is

at most x2, so that � � xO.k2/, therefore by the minimal order of �, we have

�=�.�/� log1 k C log2 x � log2 x: (8)

With our choice for yk�1, our upper bound for k in the lemma, and the estimate

(8), our count for the number of choices for pk�1 is now at most

x

m1 : : : mk�1.log x/kexp.3k2 log3 x/;

for x sufficiently large.

Observe that �.�k�j .mj // � K log log x holds for all j D 1; : : : ; k � 1,

so that �.�.mj // � 2K log log x for each j D 1; : : : ; k � 1 if x is sufficiently

large. It then follows, by Lemma 3, that summing up over all possibilities for

m1; : : : ; mk�1 (positive integers m � x such that �.�.m// � 2K log2 x), we

have

#Ak;K.x/ �x exp.3k2 log3 x/

.log x/k

X

1�m�x�.�.m//�2K log log x

1

m

!k�1

�x

.log x/kexp

�

3k.2K log2 x log3 x/1=2 C 3k2 log3 x

�

once x is large. This completes the proof of Lemma 4.

3 The Proof of Theorem 2

Here, we use the following theorem essentially due to Chen [5, 6].

Lemma 6. There exists x0 such that if x > x0 the interval Œx=2; x� contains �

x=.log x/2 primes p such that .p�1/=2 is either prime or a product of two primes

each of them exceeding x1=10.

Let

C1.x/ D ¹p 2 Œx=2; x� W .p � 1/=2 is primeº

and let

C2.x/ D ¹p 2 Œx=2; x� W .p � 1/=2 D q1q2; qi > x1=10 is prime for i D 1; 2º:

We distinguish two cases.


Case 1. #C1.x/ � #C2.x/.

In this case, for large x, �2.p/ D .p � 3/=2 is injective when restricted to

C1.x/. Hence,

#V2.x/ � #C1.x/�x

.log x/2;

where the last inequality follows from Lemma 6.

Case 2. #C1.x/ < #C2.x/:

Let p 2 C2.x/ and write p � 1 D 2q1q2, where x1=10 < q1 � q2. Put

y D exp..log x/4=5/. Let C3.x/ be the subset of C2.x/ such that q1 > x1=2=y.

Since q1q2 < x, we get that q2 < x=q1 < x1=2y. We find an upper bound on

#C3.x/. Let q1 2 Œx1=2=y; x1=2� be a fixed prime. By Brun’s sieve, the number of

primes q2 � x=q1 such that 2q1q2 C 1 is a prime is

�x

�.q1/.log.x=q1//2�

x

q1.log x/2:

Summing the above bound for all q1 2 Œx1=2=y; x1=2�, we get that

#C3.x/ �x

.log x/2

X

x1=2=y�q1�x1=2

1

q1

�x

.log x/2�

log y

log x

Dx

.log x/11=5D o .#C2.x//

as x !1, where the last estimate follows again from Lemma 6.

We now look at primes p 2 C2.x/nC3.x/ and we let C4.x/ be the set of

such primes with the property that �2.p/ D �2.p0/ for some p0 ¤ p also in

C2.x/nC3.x/. Writing p�1 D 2q1q2 and p0�1 D 2q0

1q0

2, we have .q1�1/.q2�

1/ D .q0

1 � 1/.q0

2 � 1/. Fix q1 and q0

1. If q1 D q0

1, we then get that q2 D q0

2,

therefore p D p0, which is false. So, q1 ¤ q0

1 and they are both < x1=2=y. Let

D D gcd.q1 � 1; q0

1 � 1/. Then the equation

.q1 � 1/.q2 � 1/ D .q0

1 � 1/.q0

2 � 1/

can be rewritten as

q2

�

q1 � 1

D

�

Cq0

1 � q1

DD q0

2

�

q0

1 � 1

D

�

:

LetA D .q1�1/=D; B D .q0

1�q1/=D; C D .q0

1�1/=D. Then q2ACB D Cq0

2

and A and C are coprime. This puts q2 into a fixed class modulo C , namely the

congruence class of �BA�1 modulo C . Let this class be C0, where 1 � C0 �


C � 1. Then q2 D C` C C0 for some ` � 0. We have q2 � x=q1, therefore

` � x=.q1C/. To count such `’s for a given choice of q1; q0

1, note that

C`C C0 D q2; 2q1C`C 2q1C0 C 1 D 2q1q2 C 1 D p;

A`CAC0 C B

CD q0

2; 2q0

1A`C 2q0

1

�

AC0 C B

C

�

C 1 D 2q0

1q0

2 C 1 D p0

are all four prime numbers. By the Brun sieve (it is easy to see that since B ¤ 0,

the four forms above satisfy the hypothesis from the Brun sieve for large x), it

follows that if we put

� D 2q1q0

1AC0.2q1C0 C 1/.AC0 C B/.2q0

1.AC0 C B/=C C 1/;

then the number of ` � x=.q1C/ with the above property is bounded by

�x

.q1C/.log.x=q1C//4

�

�

�.�/

�4

�xD

q1q0

1

.log log x/4

.logy/4DxD.log log x/4

q1q0

1.log x/16=5;

by the minimal order of the Euler function. Keeping now D fixed and summing

the above inequality over all pairs of primes q1; q0

1 � x1=2 which are congruent

to 1 modulo D we get, by the Brun–Titchmarsh theorem, that the number of such

primes p once D is fixed is

�xD.log log x/4

.log x/16=5

X

1�q�x1=2

q�1 .mod D/

1

q

!2

�xD.log log x/6

�.D/2.log x/16=5�

x.log log x/8

D.log x/16=5;

where we again used the minimal order of the Euler function. Summing up over

all the values for D, we finally get that

#C4.x/�x.log log x/8

.log x/16=5

X

D�x1=2

1

D�

x.log log x/8

.log x/11=5D o.#C2.x//

as x ! 1. Thus, putting C5.x/ D C2.x/n.C3.x/ [ C4.x//, we have, by the

above calculations and Lemma 6, that #C5.x/ � x=.log x/2. Certainly, �2 is

injective when restricted to C5.x/. This takes care of the desired lower bound.

4 Further Problems

Observe that Vk � Vk�1 for all k � 2. Put V1 DT

k�1 Vk . The following

result, which was conjectured by A. Chakrabarti [4], characterizes V1.


Theorem 7. The set V1 is equal to the set of positive integers n whose largest

squarefree divisor is 1; 2, or 6.

Proof. It is clear that such numbers n are in V1, since if the largest squarefree

divisor of n is 1 or 2, then �k.2kn/ D n for every k, while if the largest squarefree

divisor of n is 6, then �k.3kn/ D n.

Suppose that n 2 V1. There is thus a sequence n D n0; n1; n2; : : : such that

�.ni / D ni�1 for each i � 1. Note that v2.'.m// � v2.m/ for m not a power

of 2. In addition, if we have equality, then m D 2cpb where b; c are positive

and p is a prime that is 3 (mod 4). Assume that n0 is not a power of 2, so that

v2.n0/ � v2.n1/ � � � � . Thus, starting at some point, say nk , we have equality;

that is, v2.nk/ D v2.nkC1/ D � � � . Thus, for i � 1 we have

nkCi D 2cpbi

i ; pi � 3 .mod 4/:

We may assume that all pi > 3 for otherwise the theorem holds. If some bi > 1,

then nkCi�1 D '.nkCi / is divisible by two different odd primes, namely pi and

an odd prime factor of pi � 1. Thus, we may assume that each bi D 1 for i � 2.

We have

nkCi D 2cpi ; i � 2; pi D 2pi�1 C 1; i � 2:

We can solve this last recurrence, getting pi D 2i�1.p1C1/�1; i � 2: But note

then since 2p1�1 � 1 .mod p1/, we have pp1� .p1 C 1/ � 1 � 0 .mod p1/:

Thus, pp1cannot be prime, a contradiction which proves the theorem.

Remark 2. Note that the numbers n with largest squarefree divisor 1, 2, or 6 are

precisely those n with �.n/ j n. Note too that from the counting function up to x

of the integers whose largest squarefree factor is 1, 2, or 6, we have

#V1.x/ D1

log 3 log 4.log x/2 CO.log x/: (9)

It is possible to use the proof of Theorem 7 to show that there is a number k D

k.n/ such that if n 2 Vk , then the largest squarefree divisor of n is 1, 2, or 6.

That is, if n is not of this form, not only does there not exist an infinite “reverse

Euler chain" starting at n, there also cannot exist arbitrarily long finite reverse

Euler chains starting at n. It is an interesting question to estimate k.n/; in [11] it

is shown on the generalized Riemann hypothesis that k.n/� logn for n > 1.

Let �.n/ be the Carmichael function of n; that is, the universal exponent mod-

ulo n. This is the largest possible multiplicative order of invertible elements mod-

ulo n. For k � 1 let �k.n/ be the k-fold iterate of � evaluated at n. It would

be interesting to study Lk D ¹�.k/.n/º. For k D 1, an upper bound of the


shape #L1.x/� x=.log x/c1 with an inexplicit positive constant c1 was outlined

in [9], and an actual numerical value for c1 was established in [12]. Trivially,

#L1.x/ � x= log x. A slightly stronger lower bound appears in [1]. Stronger

upper and lower bounds on #L1.x/ will appear in [16]. While #Lk.x/ seems dif-

ficult to study for larger values of k, it is easy to see that the method of the present

paper shows that uniformly for x large,

#¹�k.n/ W n � xº�x

.log x/kexp

�

16k3=2.log2 x log3 x/1=2�

: (10)

Indeed, to see this, assume in the notation of the proof of Theorem 1, that n D

pm � x, and that p > y. Further, we may assume that �k.n/ � x=.log x/k , since

there are at most x=.log x/k positive integers failing this condition. We assume

that �.�k.n// � 2:9k log2 x, since otherwise Lemma 13 in [15] tells us again

that there are at most O.x=.log x/k/ possibilities for the number of such positive

integers �k.n/. We now note that �k.n/ j �k.n/ and that �k.n/ � x, therefore

�k.n/=�k.n/ � .log x/k . Hence,

�.�k.n// D �.�k.n//C�.�k.n/=�k.n//

� 2:9k log log x C

�

k

log 2

�

log log x < 4:5k log log x:

In particular, both �.�k.p// and �.�k.m// are at most 4:5k log log x. The ar-

gument from the end of the proof of Theorem 1 combined with the fact that

3p

9 C 3=10 C 2:9p

4:5 < 16 shows that the number of possibilities for such

n � x is at most what is shown in the right hand side of inequality (10). The

conditional argument from the introduction suggests that ckx=.log x/k should be

a lower bound on the cardinality of the above set.

Finally we remark that if n has the property that �.n/ j n, then n is in every set

Lk , as is easy to see. It is not clear if the converse holds; for example, is n D 10

in every Lk? It is not so easy to find values of � that are not values of �2, but

in fact, one can use Brun’s method to show most shifted primes p � 1 have this

property. By using the basic argument at the end of [7] plus the latest results on

the distribution of primes p with P.p � 1/ small, one can prove that for large x

there are at least x0:7067 numbers n � x with �.n/ j n. Thus, there are at least

this many numbers n � x which are in every Lk , a result which stands in stark

contrast to (9).

Acknowledgments. The first author would like to thank Williams College for

their hospitality and Professor Igor Shparlinski for enlightening conversations.

The second author would like to thank Bob Vaughan for helpful correspondence.


References

[1] W. Banks, J. Friedlander, F. Luca, F. Pappalardi and I. E. Shparlinksi, Coincidences

in the values of the Euler and Carmichael functions, Acta Arith. 122 (2006), 207–

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bution of prime numbers, Math. Comp. 16 (1962), 363–367.

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> y, Nederl. Acad. Wetensch. Proc. Ser. A 54 (1951), 50–60.

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product of at most two primes, Kexue Tongbao 17 (1966), 385–386.

[6] J. R. Chen, On the representation of a large even integer as a sum of a prime and a

product of at most two primes, Sci. Sinica 16 (1973), 157–176.

[7] P. Erdos, On the normal number of prime factors of p�1 and some related problems

concerning Euler’s �-function, Quart. J. Math. (Oxford Series) 6 (1935), 205–213.

[8] P. Erdos and R. R. Hall, Euler’s �-function and its iterates, Mathematika 24 (1977),

173–177.

[9] P. Erdos, C. Pomerance and E. Schmutz, Carmichael’s lambda function, Acta Arith.,

58 (1991), 363–385.

[10] K. Ford, The distribution of totients, Ramanujan J. 2 (1998), 67–151.

[11] K. Ford, S. Konyagin and F. Luca, Prime chains and Pratt trees, in preparation.

[12] J. Friedlander and F. Luca, On the value set of the Carmichael �-function, J. Austral.

Math. Soc. 82 (2007), 123–131.

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Nowakowski, Pomerance, pp. 351–362, Proceedings of the INTEGERS Conference

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Author information

Florian Luca, Instituto de Matemáticas, Universidad Nacional Autonoma de México,

C.P. 58089, Morelia, Michoacán, México.


Carl Pomerance, Department of Mathematics, Dartmouth College,

Hanover, NH 03755–3551, USA.



Frobenius Numbers of Generalized

Fibonacci Semigroups

Gretchen L. Matthews

Abstract. The numerical semigroup generated by relatively prime positive integers

a1; : : : ; an is the set S of all linear combinations of a1; : : : ; an with nonnegative inte-

gral coefficients. The largest integer which is not an element of S is called the Frobenius

number of S . Recently, J. M. Marín, J. L. Ramírez Alfonsín, and M. P. Revuelta deter-

mined the Frobenius number of a Fibonacci semigroup, that is, a numerical semigroup

generated by a certain set of Fibonacci numbers. In this paper, we consider numerical

semigroups generated by certain generalized Fibonacci numbers. Using a technique of

S. M. Johnson, we find the Frobenius numbers of such semigroups obtaining the result of

Marín et al. as a special case. In addition, we determine the duals of such semigroups and

relate them to the associated Lipman semigroups.

Keywords. Fibonacci semigroup, Frobenius number, Frobenius problem, numerical

semigroup.

AMS classification. 20M99, 20M14.

1 Introduction

Given a set of relatively prime positive integers a1; : : : ; an, let S denote the set

of linear combinations of a1; : : : ; an with nonnegative integral coefficients. Since

a1; : : : ; an are relatively prime, every sufficiently large integer N is an element of

S . The largest integer which is not an element of S is called the Frobenius number

of S and is denoted by g.S/. The Frobenius problem is to determine g.S/. An

excellent general reference on the Frobenius problem is [11].

In discussing the Frobenius problem, it is convenient to use the terminology of

numerical semigroups. The set S defined above is called the numerical semigroup

generated by a1; : : : ; an and is denoted by S D ha1; : : : ; ani; that is,

ha1; : : : ; ani WD

´

nX

iD1

ciai W ci 2 N

µ

This work was supported in part by NSA H-98230-06-1-0008.

118 Gretchen L. Matthews

where N denotes the set of nonnegative integers. Typically, we assume that

ai … ha1; : : : ; ai�1; aiC1; : : : ; ani

for all i , 1 � i � n. Then we say that S is an n-generated semigroup. General

references on numerical semigroups include [1, 7, 6, 8].

The Frobenius problem takes its name from the fact that Frobenius is said to

have mentioned it repeatedly in his lectures [3]. However, the first published work

on this problem appears to be due to Sylvester [12] where he determined the num-

ber of elements of N n ha; bi where a and b are relatively prime. Though not

stated explicitly in [12] (or in the often cited [13]), it is suspected that Sylvester

knew that g .ha; bi/ D ab�a�b and this fact is typically attributed to him. Given

such a simple formula for the Frobenius number of a two-generated semigroup,

it is natural to try to find the Frobenius number of an n-generated semigroup for

other small values of n. However, Curtis proved that such a closed-form expres-

sion cannot be given for the Frobenius number of a general n-generated semigroup

for n > 2 [4]. For this reason, Frobenius problem enthusiasts often consider semi-

groups whose generators are of a particular form.

In [10], the authors determine the Frobenius numbers of so-called Fibonacci

semigroups which are numerical semigroups of the form hFi ; FiC2; FiCki where

Fj denotes the j th Fibonacci number. In studying these semigroups, it is useful

to recall the convolution property of Fibonacci numbers:

Fn D FmFn�mC1 C Fm�1Fn�m

for all m; n 2 ZC (where Z

C denotes the set of positive integers). A Fibonacci

semigroup is three-generated if and only if 3 � k < i (equivalently, Fk < Fi ).

To see this, we consider two cases depending on the value of k. If k D i , then

FiCk D F2i D LiFi 2 hFi ; FiC2i

where Li denotes the i th Lucas number. If k > i , then

FiCk � F2iC1 D FiFiC2 C Fi�1FiC1 > g .hFi ; FiC2i/

and so FiCk 2 hFi ; FiC2i.

In this paper, we consider semigroups of the form

S D ha; aC b; aFk�1 C bFki

where a > Fk and gcd.a; b/ D 1. Such a semigroup will be called a generalized

Fibonacci semigroup. Notice that if a D Fi and b D FiC1, then a C b D FiC2

and

aFk�1 C bFk D FiFk�1 C FiC1Fk D FiCk

Frobenius Numbers of Generalized Fibonacci Semigroups 119

and so every Fibonacci semigroup is a generalized Fibonacci semigroup. Using

a method of S. M. Johnson [9], we find the dual of a generalized Fibonacci semi-

group. Recall that the dual of a numerical semigroup S is defined to be

B.S/ WD ¹x 2 N W x C .S n ¹0º/ � Sº :

It is immediate that g.S/ 2 B.S/ for any numerical semigroup S ¤ N as g.S/C

s > g.S/ for all s 2 ZC. Moreover,

g.S/ D max ¹x 2 B.S/ W x … Sº

provided S ¤ N. Hence, we determine the Frobenius number of a generalized

Fibonacci semigroup obtaining the result of [10] as a corollary.

This paper is organized as follows. Section 2 outlines Johnson’s method and

applies it to find the dual of a generalized Fibonacci semigroup. Section 3 con-

tains results relating the dual and Lipman semigroups. The paper concludes with

Section 4 where several open problems are posed.

2 Johnson’s Method

We begin this section with a review of S. M. Johnson’s method [9] for determining

the dual of a semigroup generated by three relatively prime positive integers.

Let S WD ha1; a2; a3i where a1, a2, and a3 are pairwise relatively prime. Sup-

pose N 2 B.S/ n S . Then N C ai 2 S for i D 1; 2; 3; that is,

N D yijaj C yikak � ai

for some yij ; yik 2 N. Since the ai are relatively prime, the semigroup generated

by any two of them has a Frobenius number. Hence, any sufficiently large integer

will be contained in such a semigroup. Let

Li WD min®

c W cai 2˝

aj ; ak

˛¯

:

Then there exist xij ; xik 2 N such that

Liai WD xijaj C xikak :

According to [9, Theorem 3], xij and xik are positive integers and are unique.

Recall that

N D y21a1 C y23a3 � a2

D y31a1 C y32a2 � a3:


Then

N D

´

.L2 � 1/ a2 C .x13 � 1/ a3 � a1 if y21 < y31

.x21 � 1/ a2 C .L3 � 1/ a3 � a1 if y31 < y21

and so

B.S/ n S D

´

.L2 � 1/ a2 C .x13 � 1/ a3 � a1;

.x21 � 1/ a2 C .L3 � 1/ a3 � a1

µ

:

Next, we apply this method to a generalized Fibonacci semigroup. Consider

S WD ha; aC b; aFk�1 C bFki ;

where a > Fk and the generators of S are pairwise relatively prime. Note that

Fk.aC b/ D Fk�2aC .Fk�1aC Fkb/ :

From this and the argument [9, p. 395–396], it follows that

L2 D Fk; x21 D Fk�2; and x23 D 1:

In addition, we find that

L1 D .aC b/ � Fk�2

�

a

Fk

�

; x12 D a � Fk

�

a

Fk

�

; x13 D

�

a

Fk

�

;

L2 D Fk; x21 D Fk�2; x23 D 1;

L3 D

�

a

Fk

�

C 1:

As a consequence,

B.S/ n S D

²�

aC b � Fk�2

�

a

Fk

�

� 2

�

a � b;

.Fk�2 � 2/ aC

�

a

Fk

�

.aFk�1 C bFk/ � b

³

:

This proves the next two results.

Proposition 1. Assume a > Fk . If S D ha; aC b; aFk�1 C bFki is generated by

three pairwise relatively prime integers, then the dual of S is

B.S/ D S [

²�

aC b � Fk�2

�

a

Fk

�

� 2

�

a � b;

.Fk�2 � 2/ aC

�

a

Fk

�


³

:


Theorem 2. The Frobenius number of S D ha; aC b; aFk�1 C bFki where a >

Fk and the generators of S are pairwise relatively prime is

g.S/ D max

²�

aC b � Fk�2

�

a

Fk

�

� 2

�

a � b;

.Fk�2 � 2/ aC

�

a

Fk

�


³

:

This theorem gives a formula for g .Fi ; FiC2; FiCk/, due to Marin et al. [10],

when a D Fi and b D FiC1. However, we should point out that the technique

used in [10], different from ours, allowed the authors not only to state explicitly

when the maximum is obtained in each case but also to give a formula for the

genus (meaning jN n S j) of such Fibonacci semigroups.

3 Duals and Lipman Semigroups

In this section, we compare two chains of semigroups. One of the chains is based

on the dual construction. The other chain arises by taking Lipman semigroups. We

first describe the Lipman semigroup. Then we relate it to the dual of S . Finally,

we consider these for Fibonacci semigroups.

Suppose S D ha1; a2; : : : ; ani is a numerical semigroup with a1 < � � � < an.

The Lipman semigroup of S is defined as

L.S/ WD ha1; a2 � a1; : : : ; an � a1i :

Clearly, S � L.S/. Moreover, B.S/ � L.S/ since x 2 B.S/ implies x C a1 DPn

iD1 ciai for some ci 2 N.

Given a numerical semigroup S , both its dual B.S/ and its Lipman semigroup

L.S/ are numerical semigroups. Hence, one may iterate the B and L construc-

tions to obtain two ascending chains of numerical semigroups

B0.S/ WD S � B1.S/ WD B.B0.S// � � � � � BhC1.S/ WD B.Bh.S// � � � �

and

L0.S/ WD S � L1.S/ WD L.L0.S// � � � � � LhC1.S/ WD L.Lh.S// � � � �

as in [1]. We will refer to these as the B- and L-chains. Notice that for S ¤ N,

S ¤ B.S/ and S ¤ L.S/. This together with the fact that N n S is finite

implies that there exist smallest non-negative integers ˇ.S/ and �.S/ such that


Bˇ.S/.S/ D N0 D L�.S/.S/. Since

B0.S/ D S D L0.S/;

B1.S/ � L1.S/;

and

Bˇ.S/.S/ D N0 D L�.S/.S/;

it is natural to compare the two chains. In [1] the authors suggest that Bj .S/ �

Lj .S/ for all 0 � j � ˇ.S/. While true for two-generated semigroups, this

containment may fail in general; there are examples of four-generated semigroups

T for which B2.T / ª L2.T /; see [5]. This prompts the question of whether or

not Bj .S/ � Lj .S/ for all j � 0 for a three-generated semigroup S . Here, we

consider this question for Fibonacci semigroups.

Suppose that S D ha; aC b; aFk�1 C bFki where a D Fi and b D FiC1;

that is, suppose

S D hFi ; FiC2; FiCki :

Then the Lipman semigroup of S is L1.S/ D ha; bi. Since a < b, L2.S/ D

ha; b � ai. Continuing this process yields the following result.

Proposition 3. Given a Fibonacci semigroup S D hFi ; FiC2; FiCki,

Lj .S/ D˝

Fi�j C1; Fi�j C2

˛

for all j , 1 � j � i � 3. In particular, �.S/ D i � 3.

Of course, one may obtain similar results for generalized Fibonacci semi-

groups. Because the description depends on sizes of a and b, we omit this here

and leave the details to the reader. Since Lj .S/ is a two-generated semigroup,

Lj .S/ is symmetric for all 1 � j < �.S/. Therefore, Lj .S/ is maximal in the

set of all numerical semigroups with Frobenius number g�

Lj .S/�

. In particular,

we notice that if

Fi D min

²

x 2 BFi �

j

Fi

�1

Fk

k

Fk�2�2.S/ W x ¤ 0

³

; (1)

then

g

�

BFi �

j

Fi

�1

Fk

k

Fk�2�1.S/

�

D g.S/ �

�

Fi �

�

Fi � 1

Fk

�

Fk�2 � 2

�

Fi

D g .L1.S//


[1, Proposition I.1.11]. Hence, if (1) holds and

FiC1 2 BFi �

j

Fi

�1

Fk

k

Fk�2�1.S/; (2)

then

BFi �

j

Fi

�1

Fk

k

Fk�2�1.S/ D L1.S/

and Bj .S/ � Lj .S/ for all nonnegative integers j would follow from [5, Theo-

rem 2.6]. Unfortunately, because B1.S/ is not three-generated, B2.S/may not be

computed using the method in Section 2. Instead, one may compute this directly

from the definition and obtain the next result.

Proposition 4. Given a Fibonacci semigroup S D hFi ; FiC2; FiCki

B2.S/ D

*

Fi ; FiC2; FiCk; l � FiC2; h � FiCk if x32 D 1;

l � Fi if k > 4 or x12 D 1; l � FiCk if x13 � 2 or x13 D x31 D 1;

h � Fi if x31 � 2 or x31 D x13 D 1;

h � FiC2 if x12 � 2 or x12 D x21 D 1

+

where

h D .Fk � 1/ FiC2 C

�

FiC2 � Fk�2

��

Fi

Fk

�

C 1

�

� 1

�

Fi � FiCk

and

l D .Fk � 1/ FiC2 C

��

Fi

Fk

C 1

��

Fk � Fi � 1

�

FiCk � Fi :

Consequently,

B2.S/ � L1.S/ � L2.S/:

In light of Proposition 4, determining Bj .S/ for j > 3 will not be an imme-

diate consequence of Johnson’s method. We leave this (and settling when (1) and

(2) hold) as a problem for further study.

4 Conclusion

In this paper, we determined the Frobenius number of generalized Fibonacci semi-

groups. In addition, we also obtained the dual of such a semigroup. We leave as

open problems to

(i) determine if Bj .S/ � Lj .S/ for each j � 0 for Fibonacci semigroups S

(or, more generally, three-generated semigroups), and

(ii) find the Frobenius number and dual of other semigroups generated by gener-

alized Fibonacci numbers.


We conclude by mentioning another interesting problem relating numerical semi-

groups and Fibonacci numbers. M. Bras-Amoros conjectures that the number of

semigroups with a particular genus g behaves asymptotically as the Fibonacci

sequence [2].

References

[1] V. Barucci, D. E. Dobbs and M. Fontana, Maximality properties in numerical semi-

groups and applications to one-dimensional analytically irreducible local domains,

Memoirs Amer. Math. Soc. 125/598 (1997).

[2] M. Bras-Amoros, Fibonacci-like behavior of the number of numerical semigroups

of a given genus, to appear in Semigroup Forum.

[3] A. Brauer, On a problem of partitions, Amer. J. Math. 64 (1942), 299–312.

[4] F. Curtis, On formulas for the Frobenius numer of a numerical semigroup, Math

Scand. 67 (1990), 190–192.

[5] D. E. Dobbs and G. L. Matthews, On comparing two chains of numerical semigroups

and detecting Arf semigroups, Semigroup Forum 63 (2001), 237–246.

[6] R. Fröberg, C. Gottlieb and R. Häggkvist, Semigroups, semigroup rings and analyt-

ically irreducible rings, Reports Dept. Math. Univ. Stockholm no. 1 (1986).

[7] R. Fröberg, C. Gottlieb and R. Häggkvist, On numerical semigroups, Semigroup

Forum 35 (1987), no. 1, 63–83.

[8] R. Gilmer, Commutative Semigroup Rings, Chicago Lectures in Mathematics, Uni-

versity of Chicago Press, Chicago, IL, 1984.

[9] S. M. Johnson, A linear Diophantine problem, Canad. J. Math. 12 (1960), 390–398.

[10] J. M. Marín, J. L. Ramírez Alfonsín and M. P. Revuelta, On the Frobenius number

of Fibonacci numerical semigroups, Integers 7 (2007), A14.

[11] J. L. Ramírez Alfonsín, The Diophantine Frobenius Problem, Oxford Lecture Series

in Mathematics and its Applications 30, Oxford University Press, 2005.

[12] J. J. Sylvester, On Subvariants, i.e. semi-invariants to binary quantics of an unlimited

order, Amer. J. Math. 5 (1882), no. 1–4, 79–136.

[13] J. J. Sylvester, Mathematical questions with their solutions, Educational Times 41

(1884), 21.

Author information

Gretchen L. Matthews, Department of Mathematical Sciences, Clemson University,

Clemson, SC 29634-0975, USA.



Combinatorics of

Ramanujan–Slater Type Identities

James McLaughlin and Andrew V. Sills

Abstract. We provide the missing member of a family of four q-series identities related

to the modulus 36, the other members having been found by Ramanujan and Slater. We

examine combinatorial implications of the identities in this family, and of some of the

identities we considered in Identities of the Ramanujan-Slater type related to the moduli

18 and 24.

Keywords. Rogers–Ramanujan type identities, partitions, signed partitions.

AMS classification. 11B65, 05A17, 11P55, 05A19.

1 Introduction

The Rogers–Ramanujan identities,

1X

j D0

qj 2

.qI q/jD

Y

k=1

k�˙1.mod 5/

1

1 � qk(1.1)

and1

X

j D0

qj 2Cj

.qI q/jD

Y

k=1

k�˙2.mod 5/

1

1 � qk; (1.2)

where

.aI q/j WD

j �1Y

kD0

.1 � aqk/

were first proved by L. J. Rogers [13] in 1894 and later independently redis-

covered (without proof) by S. Ramanujan [12, Vol II, p. 33]. Many additional

“q-series = infinite product” identities were found by Ramanujan and recorded in

his lost notebook [5], [6]. A large collection of such identities was produced by

L. J. Slater [17].

126 James McLaughlin and Andrew V. Sills

Just as the Rogers–Ramanujan identities (1.1), (1.2) are a family of two similar

identities where the infinite products are related to the modulus 5, most Rogers–

Ramanujan type identities exist in a family of several similar identities where the

sum sides are similar and the product sides involve some common modulus.

In most cases Ramanujan and Slater found all members a given family, but in a

few cases they found just one or two members of a family of four or five identities.

In [11], we found some “missing” members of families of identities related to the

moduli 18 and 24 where Ramanujan and/or Slater had found one or two of the

family members, as well as two new complete families.

In this paper, we find the missing member in a family of four identities related

to the modulus 36. We examine combinatorial implications of the identities in this

family, and of some of the identities we considered in [11].

2 Combinatorial Definitions

Informally, a partition of an integer n is a representation of n as a sum of positive

integers where the order of the summands is considered irrelevant. Thus the five

partitions of 4 are 4 itself, 3 C 1, 2 C 2, 2 C 1 C 1, and 1 C 1 C 1 C 1. The

summands are called the “parts” of the partition, and since the order of the parts

is irrelevant, 2 C 1 C 1, 1 C 2 C 1, and 1 C 1 C 2 are all considered to be the

same partition of 4. It is often convenient to impose a canonical ordering for the

parts and to separate parts with commas instead of plus signs, and so we make the

following definitions:

A partition � of an integer n into ` parts is an `-tuple of positive integers

.�1; �2; : : : ; �`/ where

�i = �iC1 for 1 5 i 5 ` � 1, and

X

iD1

�i D n:

The number of parts ` D `.�/ of � is also called the length of �. The sum of the

parts of � is called the weight of � and is denoted j�j.

Thus in this notation, the five partitions of 4 are .4/, .3; 1/, .2; 1; 1/, and

.1; 1; 1; 1/.

In [4], G. Andrews considers some of the implications of generalizing the no-

tion of partition to include the possibility of some negative integers as parts. We

may formalize this idea with the following definitions:

A signed partition � of an integer n is a partition pair .�; �/ where n D j�j �

j�j: We may call � (resp., �) the positive (resp., negative) subpartition of � and

�1; �2; : : : ; �`.�/ (resp. �1; �2; : : : ; �`.�/) the positive (resp. negative) parts of � .

Combinatorics of Ramanujan–Slater Type Identities 127

Thus ..6; 3; 3; 1/; .4; 2; 1; 1//, which represents 6C 3C 3C 1� 1� 1� 2� 4,

is an example of a signed partition of 5. Of course, there are infinitely many

unrestricted signed partitions of any integer, but when we place restrictions on

how parts may appear, signed partitions arise naturally in the study of certain q-

series.

Remark 2.1. Notice that the way we have defined signed partitions, the “nega-

tive parts” are positive numbers (which count negatively toward the weight of the

signed partition), much as the “imaginary part” of a complex number is real.

3 Partitions and q-series Identities of Ramanujan

and Slater

Using ideas that originated with Euler, MacMahon [12, vol. II, Ch. III] and

Schur [14] independently realized that (1.1) and (1.2) imply the following par-

tition identities:

Theorem 3.1 (First Rogers–Ramanujan identity – combinatorial version). For all

integers n, the number of partitions � of n where

�i � �iC1 = 2 for 1 5 i 5 `.�/ � 1 (3.1)

equals the number of partitions of n into parts congruent to˙1 .mod 5/.

Theorem 3.2 (Second Rogers–Ramanujan identity – combinatorial version). For

all integers n, the number of partitions � of n where

�i � �iC1 = 2 for 1 5 i 5 `.�/ � 1 (3.2)

and

�`.�/ > 1; (3.3)

equals the number of partitions of n into parts congruent to˙2 .mod 5/.

When studying sets of partitions where the appearance or exclusion of parts is

governed by difference conditions such as (3.1), it is often useful to introduce a

second parameter a. The exponent on a indicates the length of a partition being

enumerated, while the exponent on q indicates the weight of the partition.


For example, it is standard to generalize (1.2) and (1.1) as follows:

F1.a; q/ WD

1X

j D0

aj qj 2Cj

.qI q/j

D1

.aqI q/1

1X

j D0

.�1/ja2j qj.5j C3/=2.aI q/j .1 � aq2j C1/

.qI q/j

(3.4)

F2.a; q/ WD

1X

j D0

aj qj 2

.qI q/j

D1

.aqI q/1

1X

j D0

.�1/ja2j qj.5j �1/=2.aI q/j .1 � aq2j /

.1 � a/.qI q/j;

(3.5)

where

.aI q/1 WD

1Y

kD0

.1 � aqk/:

It is then easily seen that F1.a; q/ and F2.a; q/ satisfy the following system of

q-difference equations:

F1.a; q/ D F2.aq; q/; (3.6)

F2.a; q/ D F1.a; q/C aqF1.aq; q/: (3.7)

Notice that there are straightforward combinatorial interpretations to (3.6) and

(3.7). Equation (3.6) states that if we start with the collection of partitions satis-

fying (3.1) and add 1 to each part (i.e., replace a by aq), then we obtain the set of

partitions that satsify (3.2) and (3.3); the difference condition is maintained, but

the new partitions will have no ones. The left-hand side of (3.7) generates par-

titions that satisfy (3.1) while the right-hand side segregates these partitions into

two classes: those where no ones appear (generated by F1.a; q/) and those where

a unique one appears (generated by aqF1.aq; q/).

Remark 3.3. It may seem awkward to have the a-generalization of the first (resp.

second) Rogers–Ramanujan identity labeled F2.a; q/ (resp., F1.a; q/), but this is

actually standard practice (see, e.g., Andrews [2, Ch. 7]). Here and in certain

generalizations, the subscript on F corresponds to one more than the maximum

number of ones which can appear in the partitions enumerated by the function.

Remark 3.4. While the a-generalizations are useful for studying the relevant par-

titions, the price paid for generalizing (1.1) to (3.5) and (1.2) to (3.4) is that the a-

generalizations no longer have infinite product representations; only in the a D 1


cases will Jacobi’s triple product identity [10, p. 15, Eq. (1.6.1)] allow the right-

hand sides of (1.2) and (1.1) to be transformed into infinite products.

An exception to Remark 3.4 may be found in one of the identities in Ramanu-

jan’s lost notebook [6, Entry 5.3.9]; cf. [11, Eq. (1.16)]:

1X

j D0

qj 2

.q3I q6/j

.qI q2/2j .q4I q4/j

DY

j =1

j �1.mod 2/ or j �˙2.mod 12/

1

1 � qj: (3.8)

Equation (3.8) admits an a-generalization with an infinite product:

1X

j D0

aj qj 2

.q3I q6/j

.qI q2/j .aqI q2/j .q4I q4/jD

Y

j =1

1C aq4j �2 C a2q8j �4

1 � aq2j �1: (3.9)

Notice that the right-hand side of (3.9) is easily seen to be equal to

X

n;`=0

s.`; n/a`qn;

where s.`; n/ denotes the number of partitions of n into exactly ` parts where

no even part appears more than twice nor is divisible by 4. Note also that the

right-hand side of (3.8) generates partitions where parts may appear as in Schur’s

1926 partition theorem [15] (i.e. partitions into parts congruent to ˙1 .mod 6/),

dilated by a factor of 2, along with unrestricted appearances of odd parts. It is

a fairly common phenomenon for a Rogers–Ramanujan type identity to generate

partitions whose parts are restricted according to a well-known partition theorem,

dilated by a factor ofm, and where nonmultiples ofmmay appear without restric-

tion. See, e.g., Connor [8] and Sills [16].

A partner to (3.8) was found by Slater [17, p. 164, Eq. (110), corrected], cf. [11,

Eq. (1.19)]:

1X

j D0

qj 2C2j .q3I q6/j

.qI q2/j .qI q2/j C1.q4I q4/jD

Y

j =1

j �1.mod 2/ or j �˙4.mod 12/

1

1 � qj: (3.10)

An a-generalization of (3.10) is

1X

j D0

aj qj 2C2j .q3I q6/j

.qI q2/j .aqI q2/j C1.q4I q4/jD

Y

j =1

1C aq4j C a2q8j

1 � aq2j �1

DX

n;`=0

t .`; n/a`qn;

(3.11)


where t .`; n/ denotes the number of partitions of n into ` parts where even parts

appear at most twice and are divisible by 4.

Remark 3.5. An explanation as to why (3.8) and (3.10) admit a-generalizations

which include infinite products and (1.1) and (1.2) do not, may be found in the

theory of basic hypergeometric series. The Rogers–Ramanujan identities (1.1)

and (1.2) arise as limiting cases of Watson’s q-analog of Whipple’s theorem [18],

[10, p. 43, Eq. (2.5.1)]; see [10, pp. 44–45, §2.7]. In contrast, (3.9) and (3.11) are

special cases of Andrews’s q-analog of Bailey’s 2F1.12/ sum [1, p. 526, Eq. (1.9)],

[10, p. 354, Eq. (II.10)].

Remark 3.6. S. Corteel and J. Lovejoy interpreted (3.8) and (3.10) combinatori-

ally using overpartitions in [9].

4 A Family of Ramanujan and Slater

4.1 A Long-lost Relative

Let us define Q.w; x/ WD .�wx�1;�x;wIw/1.wx�2; wx2Iw2/1; where

.a1; a2; : : : ; ar Iw/1 WD

rY

kD1

.akIw/1:

Then it is clear that an identity is missing from the family

1X

j D0

q2j.j C2/.q3I q6/j

.q2I q2/2j C1.qI q2/jDQ.q18; q7/

.q2I q2/1(Slater [17, Eq. (125)]); (4.1)

1X

j D0

q2j.j C1/.q3I q6/j

.q2I q2/2j C1.qI q2/jDQ.q18; q5/

.q2I q2/1(Slater [17, Eq. (124)]); (4.2)

1X

j D0

q2j 2

.q3I q6/j

.q2I q2/2j .qI q2/jDQ.q18; q3/

.q2I q2/1(Ramanujan [6, Entry 5.3.4]): (4.3)

The following identity completes the above family:

1X

j D0

q2j.j C1/.q3I q6/j

.q2I q2/2j .qI q2/j C1

DQ.q18; q/

.q2I q2/1: (4.4)

Theorem 4.1. Identity (4.4) is valid.


Proof. We show that (4.2)Cq�(4.1) D (4.4). For the series side,

1X

j D0

q2j.j C1/.q3I q6/j

.q2I q2/2j C1.qI q2/jC q

1X

j D0

q2j.j C2/.q3I q6/j

.q2I q2/2j C1.qI q2/j

D

1X

j D0

q2j.j C1/.q3I q6/j .1C q2j C1/

.q2I q2/2j C1.qI q2/j

D

1X

j D0

q2j.j C1/.q3I q6/j


:

For the product side, we make use of the quintuple product identity:

Q.w; x/ D .wx3; w2x�3; w3Iw3/1 C x.wx�3; w2x3; w3Iw3/1:

Hence

Q.q18; q5/C qQ.q18; q7/

D .q33; q21; q54I q54/1 C q5.q3; q51; q54I q54/1

C q..q39; q15; q54I q54/1 C q7.q�3; q57; q54I q54/1/

D .q33; q21; q54I q54/1 C q5.q3; q51; q54I q54/1

C q..q39; q15; q54I q54/1 � q4.q51; q3; q54I q54/1/

D .q21; q33; q54I q54/1 C q.q15; q39; q54I q54/1

D Q.q18; q/:

The result now follows.

4.2 Combinatorial Interpretations

We interpret (4.3) combinatorially.

Theorem 4.2. The number of signed partitions � D .�; �/ of n, where

� `.�/ is even, and each positive part is even and = `.�/, and

� the negative parts are odd, less than `.�/, and may appear at most twice

equals the number of (ordinary) partitions of n into parts congruent to ˙2, ˙3,

˙4,˙8 .mod 18/.


Proof. Starting with the left-hand side of (4.3), we find

1X

j D0

q2j 2

.q3I q6/j

.q2I q2/2j .qI q2/jD

1X

j D0

q2j 2 Qj

kD1.1C q2k�1 C q4k�2/

.q2I q2/2j

D

1X

j D0

q2j 2 Qj

kD1q4k�2.1C q�.2k�1/ C q�.4k�2//

.q2I q2/2j

D

1X

j D0

q2j 2C4.1C2C��Cj /�2jQj

kD1.1C q�.2k�1/ C q�.4k�2//

.q2I q2/2j

D

1X

j D0

q4j 2

.q2I q2/2j�

jY

kD1

.1C q�.2k�1/ C q�.4k�2//:

Notice that1

.qI q/2j

is the generating function for partitions into at most 2j parts, thus

q2j 2 1

.qI q/2jDq

2j terms‚ …„ ƒ

j C j C j C � � � C j

.qI q/2j

is the generating function for partitions into exactly 2j parts, where each part is

at least j . Thus

q4j 2 1

.q2I q2/2j

is the generating function for partitions into exactly 2j parts, each of which is

even and at least 2j . Also,Qj

kD1.1 C q�.2k�1/ C q�.4k�2// is the generating

function for signed partitions into odd negative parts < 2j and appearing at most

twice each. Summing over all j , we find that the left-hand side of (4.3) is the

generating function for signed partitions � D .�; �/ of n, where `.�/ is even, and

each positive part is even and = `.�/, and the negative parts are odd, less than

`.�/, and may appear at most twice.

Now the right-hand side of (4.3) is

Q.q18; q3/

.q2I q2/1D.�q3;�q15I q18I q18/1.q

12; q24I q36/1

.q2I q2/1

DY

i=1

i�˙2;˙3;˙4;˙8.mod 18/

1

1 � qi;


which is clearly the generating function for partitions into parts congruent to

˙2;˙3;˙4;˙8 .mod 18/.

Remark 4.3. Andrews provided a different combinatorial interpretation of (4.3)

in [3, p. 175, Theorem 2].

Following the ideas in the proof of Theorem 4.2, the analogous combinatorial

interpretation of Identity (4.2) is as follows.


� `.�/ is odd, and each positive part is even and = `.�/ � 1, and



˙5,˙6 .mod 18/.

We next interpret (4.1) combinatorially. Note that the theorem equates the

number in a certain class of signed partitions of nC1 with the number in a certain

class of regular partitions of n.

Theorem 4.5. The number of signed partitions � D .�; �/ of nC 1, where

� `.�/ is odd, and each positive part is odd and = `.�/, and



˙7,˙8 .mod 18/.

Proof. The proof is similar to that of Theorem 4.2, except that

1X

j D0

q2j.j C2/.q3I q6/j

.q2I q2/2j C1.qI q2/jD

1

q

1X

j D0

q4j 2C4j C1

.q2I q2/2j C1

�

jY

kD1

.1Cq�.2k�1/Cq�.4k�2//;

and since

4j 2 C 4j C 1 D .2j C 1/C .2j C 1/C � � � C .2j C 1/„ ƒ‚ …

2j C1 terms

;

it follows thatq4j 2C4j C1

.q2I q2/2j C1

is the generating function for partitions into exactly 2j C 1 parts, each of which

is odd and at least 2j C 1.


Lastly, we give a combinatorial interpretation of (4.4).

Theorem 4.6. The number of signed partitions � D .�; �/ of nC 1, where

� � contains an odd positive part m (which may be repeated), exactly m � 1

positive even parts, all = m � 1, and

� negative parts are all odd, < m, and appear at most twice,


˙6,˙8 .mod 18/.

Proof. This time

1X

j D0

q2j.j C1/.q3I q6/j


D1

q

1X

j D0

q4j 2

.q2I q2/2j

q2j C1

1 � q2j C1

jY

kD1

.1C q�.2k�1/ C q�.4k�2//;

so that, as before,

q4j 2

.q2I q2/2j

is the generating function for partitions into exactly 2j parts, each of which is

even and at least 2j , and

q2j C1

1 � q2j C1

generates partitions consisting of the part 2j C 1 and containing at least one such

part.

5 Combinatorial Interpretations of a Family of

Mod 18 Identities

In [11], we presented the following family of Rogers–Ramanujan–Slater type

identities related to the modulus 18:

1X

j D0

qj.j C1/.�1I q3/j

.�1I q/j .qI q/2jD.q; q8; q9I q9/1.q

7; q11I q18/1

.qI q/1; (5.1)

1X

j D0

qj 2

.�1I q3/j

.�1I q/j .qI q/2jD.q2; q7; q9I q9/1.q

5; q13I q18/1

.qI q/1; (5.2)


1X

j D0

qj.j C1/.�q3I q3/j

.�qI q/j .qI q/2j C1

D.q3; q6; q9I q9/1.q

3; q15I q18/1

.qI q/1; (5.3)

1X

j D0


.q2I q2/j .qj C2I q/j C1

D.q4; q5; q9I q9/1.q; q

17I q18/1

.qI q/1: (5.4)

We give a combinatorial interpretation of (5.4).

Theorem 5.1. The number of signed partitions � D .�; �/ of nC 2, wherein

� �1, the largest positive part, is even,

� the integers 1; 2; : : : ; �1

2� 1 all appear an even number of times and at least

twice,

� the integer �1

2does not appear,

� the integers �1

2C 1; �1

2C 2; : : : ; �1 all appear at least once, and

� there are exactly �1

2� 1 negative parts, each � 1 .mod 3/ and 5

3�1

2� 2,

with the parts greater than 1 occurring at most once


˙6,˙7,˙8 .mod 18/.

Proof. We consider the general term on the left side of (5.4),


.q2I q2/j .qj C2I q/j C1

Dqj 2C2j

.q2I q2/j

q.3j 2C3j /=2

Qj

kD0.1 � qj C2Ck/

jY

kD1

.1C q�3k/

D1

q2

qj 2Cj

.q2I q2/j

q.3j 2C7j C4/=2

Qj

kD0.1 � qj C2Ck/

q�j

jY

kD1

.1C q�3k/:

The factorsqj 2Cj

.q2I q2/jDq2C4C6C��C2j

.q2I q2/j

generates parts in ¹2; 4; 6; : : : ; 2j º where each part appears at least once. Then

by mapping each even part 2r to r C r , we have parts ¹1; 2; 3; : : : ; j º where each

part appears an even number of times and at least twice.

The factors

q.3j 2C7j C4/=2

Qj

kD0.1 � qj C2Ck/

Dq.j C2/C.j C3/C��C.2j C2/

Qj

kD0.1 � qj C2Ck/


generates partitions from the parts ¹j C 2; j C 3; : : : ; 2j C 1; 2j C 2º and where

each part appears at least once. Lastly,

q�j

jY

kD1

.1C q�3k/

is the generating function for signed partitions with negative parts that are congru-

ent to 1 modulo 3, � 3j C 1, the parts greater than 1 occur at most once, and the

total number of parts is j (the number of 1’s being j minus the number of other

parts).

Upon summing over j � 0, we get that

1X

j D0

qj 2Cj

.q2I q2/j

q.3j 2C7j C4/=2

Qj

kD0.1 � qj C2Ck/

q�j

jY

kD1

.1C q�3k/

is the generating function for signed partitions with the properties itemized in the

statement of the theorem.

The right side of (5.4) is

.q4; q5; q9I q9/1.q; q17I q18/1

.qI q/1D

Y

i=1

i�˙2;˙3;˙6;˙7;˙8.mod 18/

1

1 � qi;

which is the generating function for partitions into parts congruent to ˙2, ˙3,

˙6,˙7,˙8 .mod 18/.

The corresponding combinatorial interpretation of (5.2) is given by the follow-

ing theorem.



� the integers 1; 2; : : : ; �1


twice,


2; �1




3�1

2� 1,

with the parts greater than 2 occurring at most once,


˙4,˙6,˙8 .mod 18/.


Proof. The proof is similar to that of Theorem 5.1, except we rewrite the general

term on the left side of (5.2) as follows

qj 2

.�1I q3/j

.�1I q/j .qI q/2jD

qj 2

.q2I q2/j �1

q.3j 2�3j /=2

Qj

kD0.1 � qj Ck/

j �1Y

kD1

.1C q�3k/

Dqj 2�j

.q2I q2/j �1

q.3j 2C3j /=2

Qj

kD0.1 � qj Ck/

q�2j

j �1Y

kD1

.1C q�3k/:

The identity at (5.1) may be interpreted combinatorially as follows.



� the integers 1; 2; : : : ; �1


twice,


2; �1




3�1

2� 2,

with the parts greater than 1 occurring at most once,


˙4,˙5,˙6 .mod 18/.

Proof. The general term on the left side of (5.1) may be written as

qj 2Cj .�1I q3/j

.�1I q/j .qI q/2jD

qj 2Cj

.q2I q2/j �1

q.3j 2�3j /=2

Qj

kD0.1 � qj Ck/

j �1Y

kD1

.1C q�3k/

Dqj 2�j

.q2I q2/j �1

q.3j 2C3j /=2

Qj

kD0.1 � qj Ck/

q�j

j �1Y

kD1

.1C q�3k/:

Finally, we provide a combinatorial interpretation of (5.3).

Theorem 5.4. The number of signed partitions � D .�; �/ of nC 1, wherein

� �1, the largest positive part, is odd,

� the integers 1; 2; : : : ; �1�12

all appear an even number of times and at least

twice,

� the integers �1�12C 1; �1�1


� there are exactly �1�12

negative parts, each� 1 .mod 3/ and 53�1

2�2, with

the parts greater than 1 occurring at most once,

equals the number of (ordinary) partitions of n into parts congruent to 1, 2, 4, 5,


7 or 8 modulo 9, such that for any nonnegative integer j , 9j C 1 and 9j C 2 do

not both appear, and for any nonnegative integer k, 9k C 7 and 9k C 8 do not

both appear.

Proof. The general term on the left side of (5.3) may be written as


.�qI q/j .qI q/2j C1

Dqj 2Cj

.q2I q2/j

q.3j 2C3j /=2

.qj C1I q/j C1

jY

kD1

.1C q�3k/

D1

q

qj 2Cj

.q2I q2/j

q.3j 2C5j C2/=2

.qj C1I q/j C1

q�j

jY

kD1

.1C q�3k/I

thus, the interpretation of the left side is similar to that of the previous identities.

The right side of (5.3) provides a challenge because of the double occurrence

of the factors .q3I q18/1 and .q15I q18/1 in the numerator. Accordingly, we turn

to a partition enumeration technique introduced by Andrews and Lewis. In [7,

p. 79, Eq. (2.2) with k D 9], they show that

.qaCbI q18/1

.qa; qbI q9/1(5.5)

is the generating function for partitions of n into parts congruent to a or b modulo

9 such that for any k, 9k C a and 9k C b do not both appear as parts, where

0 < a < b < 9.

With this in mind, we immediately see that

.q3; q6; q9I q9/1.q3; q15I q18/1

.qI q/1D

1

.q4; q5I q9/1�.q3I q18/1

.q; q2I q9/1�.q15I q18/1

.q7; q8I q9/1

generates the partitions stated in our theorem.

Acknowledgments. We thank the referee for carefully reading the manuscript

and supplying helpful comments, and the conference organizers for the opportu-

nity to present at Integers 2007.

References

[1] G. E. Andrews, On the q-analog of Kummer’s theorem and applications, Duke Math.

J. 40 (1973), 525–528.

[2] G. E. Andrews, The Theory of Partitions, Encyclopedia of Mathematics and its Ap-

plications, vol. 2, Addison-Wesely, 1976; reissued Cambridge, 1998.

[3] G. E. Andrews, Ramanujan’s “lost” notebook II: #-function expansions, Adv. in

Math. 41 (1981), 173–185.


[4] G. E. Andrews, Euler’s “De Partitio Numerorum”, Bull. Amer. Math. Soc. 44 (2007),

561–574.

[5] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part I, Springer, 2005

[6] G. E. Andrews and B. C. Berndt, Ramanujan’s Lost Notebook. Part II, Springer,

2009.

[7] G. E. Andrews and R. Lewis, An algebraic identity of F. H. Jackson and its implica-

tions for partitions, Discrete Math. 232 (2001), 77–83.

[8] W. G. Connor, Partition theorems related to some identities of Rogers and Watson,

Trans. Amer. Math. Soc. 214 (1975), 95–111.

[9] S. Corteel and J. Lovejoy, Overpartitions and the q-Bailey identity, to appear in

Proc. Edinburgh Math. Soc.

[10] G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed., Cambridge,

2004.

[11] J. Mc Laughlin and A. V. Sills, Identities of the Ramanujan–Slater type related to

the moduli 18 and 24, J. Math. Anal. Appl. 344 (2008), no. 2, 765–777.

[12] P. A. MacMahon, Combinatory Analysis, two volumes, Cambridge, 1915–1916.

Reprinted: Chelsea, 1960.

[13] L. J. Rogers, Second memoir on the expansion of certain infinite products, Proc.

London Math. Soc. 25 (1894), 318–343.

[14] I. Schur, Ein Beitrag zur additiven Zahlentheorie und zur Theorie der Kettenbrüche,

Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl. (1917), 302–321.

[15] I. Schur, Zur additiven Zahlentheorie, Sitz. Preuss. Akad. Wiss. Phys.-Math. Kl.

(1926), 488–495.

[16] A. V. Sills, Identities of the Rogers–Ramanujan–Slater type, Int. J. Number Theory

3 (2007), 293–323.

[17] L. J. Slater, Further identities of the Rogers–Ramanujan type, Proc. London Math.

Soc. (2) 54 (1952), 147–167.

[18] G. N. Watson, A new proof of the Rogers–Ramanujan identities, J. London Math.

Soc. 4 (1929), 4–9.

Author information

James McLaughlin, Department of Mathematics, West Chester University,

West Chester, PA 19383, USA.


Andrew V. Sills, Department of Mathematical Sciences, Georgia Southern University,

Statesboro, GA 30460, USA.



A Mock Theta Function for the

Delta-function

Ken Ono

Abstract. We define a “mock theta” function

M�.z/ D

1X

nD�1

a�.n/qn D 11Š � q�1 �

2615348736000

691� � � �

for Ramanujan’s Delta-function. Although it is not a modular form, we are able to deter-

mine its images under the Hecke operators. These images are in terms of Ramanujan’s

tau-function and the modular polynomials jm.z/ which encode the denominator formula

for the infinite dimensional Monster Lie algebra. Using these results, we obtain a criterion

for Lehmer’s Conjecture on the nonvanishing of �.n/. We show that �.p/ D 0 if and only

if M�.z/ j T�10.p/ is a modular form. We also relate Lehmer’s Conjecture to integrality

properties of the a�.n/.

Keywords. Mock theta function, Maass form.

AMS classification. 11F03, 11F11.

– For George E. Andrews in celebration of his 70th birthday.

1 Introduction and Statement of Results

As usual, let

�.z/ D

1X

nD1

�.n/qn WD q

1Y

nD1

.1 � qn/24 D q � 24q2 C 252q3 � � � � (1.1)

(note q WD e2�iz throughout) be the unique normalized weight 12 cusp form

on SL2.Z/. Its coefficients, the so-called values of Ramanujan’s tau-function,

provide examples of some of the deepest phenomena in the theory of holomorphic

modular forms. Here we show how its values also arise in the theory of non-

holomorphic modular forms, the so-called Maass forms.

142 Ken Ono

To make this precise, we first recall some classical modular forms. As usual,

let

E4.z/ WD 1C 240

1X

nD1

�3.n/qn;

E6.z/ WD 1 � 504

1X

nD1

�5.n/qn

denote the normalized weight 4 and 6 Eisenstein series for the modular group

SL2.Z/. Throughout, we let ��.n/ WDP

d jn d� . Let j.z/ be the classical modular

function

j.z/ WDE4.z/

3

�.z/D

1X

nD�1

c.n/qn D q�1 C 744C 196884q C � � � : (1.2)

We now recall an important sequence of modular functions related to the Mon-

ster. Let j0.z/ WD 1, and for every positive integerm, let jm.z/ be the unique mod-

ular function which is holomorphic on H, the upper-half of the complex plane,

whose q-expansion is of the form

jm.z/ D q�m C

1X

nD1

cm.n/qn: (1.3)

It is a standard fact that each jm.z/ is a monic degree m polynomial in j.z/ with

integer coefficients. The first few jm are:

j0.z/ D 1;

j1.z/ D j.z/ � 744 D q�1 C 196884q C � � � ;

j2.z/ D j.z/2 � 1488j.z/C 159768 D q�2 C 42987520q C � � � ;

j3.z/ D j.z/3 � 2232j.z/2 C 1069956j.z/ � 36866976

D q�3 C 2592899910q C � � � :

These polynomials are easily described using Hecke operators. For primes p

and integral weights k, the Hecke operator Tk.p/ is defined by

�

X

a.n/qn�

j Tk.p/ WDX

�

a.np/C pk�1a.n=p/�

qn: (1.4)

These operators generate all of the Hecke operators, and it turns out that

jm.z/ D m.j1.z/ j T0.m//:

A Mock Theta Function for the Delta-function 143

Remark. The jm.z/ have an elegant generating function (see [2, 11, 12]). If

polynomials Jm.x/ are defined by

1X

mD0

Jm.x/qm WD

E4.z/2E6.z/

�.z/�

1

j.z/ � xD 1C .x � 744/q C � � � ; (1.5)

then we have that jm.z/ D Jm.j.z//.

The jm.z/ encode the identity

j.�/ � j.z/ D p�1 exp

�

1X

mD1

jm.z/ �pm

m

!

;

where p D e2�i� . It is equivalent, by a straightforward calculation, to the famous

denominator formula for the Monster Lie algebra

j.�/ � j.z/ D p�1Y

m>0 and n2Z

.1 � pmqn/c.mn/:

We show that these polynomials also arise in the Hecke theory of certain har-

monic weak Maass forms (For more on such Maass forms and their applications,

see [3, 4, 5, 6, 7, 8, 9]). We shall recall the definition of these forms in Sec-

tion 2. The mock theta functions of Ramanujan provide non-trivial examples of

such Maass forms (for example, see [3, 5, 17]). As an example, it turns out that

q�1f .q24/C 2ip

3 �Nf .z/ (1.6)

is a weight 1/2 harmonic weak Maass form, where

Nf .z/ WD

Z i1

�24z

P

1

nD�1

�

nC 16

�

e3�i.nC16/

2�

p

�i.� C 24z/d�

is a period integral of a theta function, and f .q/ is Ramanujan’s mock theta func-

tion

f .q/ WD 1C

1X

nD1

qn2

.1C q/2.1C q2/2 � � � .1C qn/2:

All of Ramanujan’s mock theta functions are parts of such Maass forms in this

way.

Since such period integrals of modular forms are non-holomorphic, we shall

refer to the holomorphic projection of a harmonic weak Maass form as a mock

144 Ken Ono

theta function. Using Poincaré series, in Section 2.1 we define such a mock theta

function M�.z/

M�.z/ D

1X

nD�1

a�.n/qn D 39916800q�1 �

2615348736000

691� � � �

D 11Š � q�1 C24 � 11Š

B12

� 73562460235:68364 : : : q

� 929026615019:11308 : : : q2 � 8982427958440:32917 : : : q3

� 71877619168847:70781 : : : q4 � � � � :

(1.7)

Here B12 D �691=2730 is the 12th Bernoulli number. This function enjoys the

property that

M�.z/CN�.z/

is a weight �10 harmonic weak Maass form on SL2.Z/, where N�.z/ is the

period integral of �.z/

N�.z/ D .2�/11 � 11i � ˇ�

Z i1

�z

�.��/

.�i.� C z//�10d�: (1.8)

We shall see that ˇ� � 2:840287 : : : is the first coefficient of a certain Poincaré

series.

For positive integers n, the coefficients a�.n/ appear to be real numbers with-

out nice arithmetic properties. Indeed, we do not have a simple description for

any of these coefficients. However, we shall show that these coefficients satisfy

deep arithmetic properties thanks to the rationality of a�.�1/ and a�.0/. In par-

ticular, we shall describe the behavior of the Hecke operators on M�.z/ in terms

of power series with rational coefficients, and we shall also be able to speak of

congruences.

Although M�.z/ is not a modular form, for every prime p we shall show that

p11 �M�.z/ j T�10.p/ � �.p/ �M�.z/

D

1X

nD�p

�

p11a�.pn/ � �.p/a�.n/C a�.n=p/�

qn(1.9)

is a weight �10 weakly holomorphic modular form. It will turn out to be a mod-

ular form with integer coefficients. We explicitly describe these forms in terms of

Ramanujan’s tau-function and the modular polynomials jm.z/. For convenience,


if p is prime, then define the modular functions Ap.z/ and Bp.z/ by

Ap.z/ WD24

B12

.1C p11/C jp.z/ � 264

pX

mD1

�9.m/jp�m.z/; (1.10)

Bp.z/ WD ��.p/

�

�264C24

B12

C j1.z/

�

: (1.11)

Theorem 1.1. If p is prime, then

1X

nD�p

�

p11a�.pn/ � �.p/a�.n/C a�.n=p/�

qn

D11Š

E4.z/E6.z/��

Ap.z/C Bp.z/�

:

Theorem 1.1 gives many congruences for the mock theta functionM�.z/ such

as those given by the following corollary.

Corollary 1.2. The following congruences are true:

(1) If p is prime, then

1X

nD�p

�

p11a�.pn/ � �.p/a�.n/C a�.n=p/�

qn � 0 .mod 11Š/:

(2) If p is prime, then

1

11Š

1X

nD�p

�

p11a�.pn/ � �.p/a�.n/C a�.n=p/�

qn

� jp.z/ � �.p/j1.z/ .mod 24/:

Remark. Since 24B12D � 65520

691and since the coefficients of each jm.z/ are inte-

gers, it follows that the coefficients in Corollary 1.2 (1) are in 11Š691

Z. The absence

of 691 in the denominators follows from the famous Ramanujan congruence

�.n/ � �11.n/ .mod 691/:

It is interesting to note that this congruence and the appearance of 691 in the

formula for a�.0/ both arise from the fact that the numerator of B12 is 691.

146 Ken Ono

Remark. One may simplify the right hand side of Corollary 1.2 (2) for primes

p � 5. For such primes p, classical congruences of Ramanujan imply that

�.p/ � 1C p .mod 24/:

A famous conjecture of Lehmer asserts that �.n/ ¤ 0 for every positive in-

teger n. It is well known that the truth of this conjecture would follow from the

nonvanishing of �.p/ for all primes p. With this in mind, we obtain the following

corollary which reinterpretes this conjecture in terms of the mock theta function

M�.z/.

Corollary 1.3. The following are true for all primes p.

(1) We have that �.p/ D 0 if and only if

1X

nD�p

�

p11a�.pn/C a�.n=p/�

qn

is a weight �10 modular form on SL2.Z/.

(2) We have that �.p/ D 0 if and only if

1X

nD�p

�

p11a�.pn/C a�.n=p/�

qn D11Š

E4.z/E6.z/� Ap.z/:

(3) If �.p/ D 0, then for every positive integer n coprime to p we have that

p11a�.pn/ is an integer for which

p11a�.pn/ � 0 .mod 11Š/:

Based on numerical experiments, we make the following conjecture which

implies Lehmer’s Conjecture.

Conjecture. The coefficients a�.n/ are irrational for every positive integer n.

Remark. After this paper was written, Bruinier, Rhoades and the author [10]

proved general theorems relating the algebraicity of coefficients of certain inte-

ger weight harmonic weak Maass forms to the vanishing Hecke eigenvalues.

Remark. This conjecture is closely related to results obtained by the author and

Bruinier on Heegner divisors and derivatives ofL-functions (see [9]). In that work

results were obtained on the transcendence on the coefficients of harmonic weak

Maass forms. These results, combined with a well-known conjecture of Goldfeld

on modular L-functions, implies that “almost all” coefficients of certain weight

1=2 harmonic weak Maass forms are transcendental, and hence irrational. This

work provides further evidence supporting the plausibility of our conjecture.


In Section 2.1 we construct the mock theta function M�.z/ using results in

earlier joint work with Bringmann [4]. In particular, we give complicated exact

formulas for the coefficients of M�.z/, and we give a description of the real

number ˇ� in (1.8). In Section 3 we prove Theorem 1.1 and Corollaries 1.2 and

1.3. In the last section we conclude with a detailed discussion of Theorem 1.1 and

Corollary 1.2 when p D 2.

2 Harmonic Weak Maass Forms and M�.z/

We recall the notion of a harmonic weak Maass form of integer weight k on

SL2.Z/ due to Bruinier and Funke [8]. If z D x C iy 2 H with x; y 2 R,

then the weight k hyperbolic Laplacian is given by

�k WD �y2

�

@2

@x2C

@2

@y2

�

C iky

�

@

@xC i

@

@y

�

: (2.1)

A weight k harmonic weak Maass form on SL2.Z/ is any smooth function M W

H! C satisfying the following:

(1) For all A D�

a bc d

�

2 SL2.Z/ and all z 2 H, we have

M.Az/ D .cz C d/kM.z/:

(2) We have that �kM D 0.

(3) The functionM.z/ has at most linear exponential growth at the cusp infinity.

2.1 Poincaré Series and the Definition of M�.z/

Here we recall two relevant Poincaré series. The Poincaré series H.z/ will turn

out to equal ˇ��.z/, and the second Poincaré series R.z/ shall be of the form

R.z/ DM�.z/CN�.z/:

Here we recall the constructions of these Poincaré series following [4]. We

rely on classical special functions whose properties and definitions may be found

in [1]. Suppose that k 2 Z. For A D�

a bc d

�

2 SL2.Z/, define j.A; z/ by

j.A; z/ WD .cz C d/: (2.2)

As usual, for such A and functions f W H! C, we let

.f jk A/.z/ WD j.A; z/�kf .Az/: (2.3)

148 Ken Ono

Let m be an integer, and let 'm W RC ! C be a function which satisfies

'm.y/ D O.y˛/, as y ! 0, for some ˛ 2 R. If e.˛/ WD e2�i˛ as usual, then let

'�

m.z/ WD 'm.y/e.mx/: (2.4)

Such functions are fixed by the translations �1 WD ¹˙�

1 n0 1

�

W n 2 Zº. Given

this data, we define the Poincaré series

P.m; k; 'mI z/ WDX

A2�1nSL2.Z/

.'�

m jk A/.z/: (2.5)

Now define the function H.z/ by

H.z/ WD P.1; 12; e.iy/I z/ D

1X

nD1

b.n/qn: (2.6)

It is well known that H.z/ is a weight 12 cusp form on SL2.Z/, and so it follows

that

H.z/ D ˇ��.z/;

where ˇ� D b.1/ � 2:840287 : : : . It is well known that (for example, see Chap-

ter 3 of [15]) that ˇ� may be described in terms of the Petersson norm of H.z/.

Now we construct the relevant weight �10 harmonic weak Maass form. Let

M�; �.z/ be the usual M -Whittaker function. For complex s, let

Ms.y/ WD jyj�

k

2 Mk

2sgn.y/; s�

12.jyj/;

and for positive m let '�m.z/ WDM1�

k

2.�4�my/. We now let

R.z/ WD P.�1;�10; '�1I z/: (2.7)

Now recall the period integral N�.z/ from (1.8), and define M�.z/ by

M�.z/ WD R.z/ �N�.z/: (2.8)

We shall give an exact formula for the Fourier expansion of M�.z/ in terms of

the classical Kloosterman sums

K.m; n; c/ WDX

v.c/�

e

�

mv C nv

c

�

: (2.9)

In the sums above, v runs through the primitive residue classes modulo c, and

v denotes the multiplicative inverse of v modulo c. Theorem 1.1 of [4] (also

see [13, 14] for related earlier works) then implies the following result relating

H.z/;R.z/ and M�.z/.


Theorem 2.1. The following are true:

(1) The function R.z/ is a weight �10 harmonic weak Maass form on SL2.Z/.

(2) The function M�.z/ is holomorphic on H, and it has a Fourier expansion of

the form

M�.z/ D

1X

nD�1

a�.n/qn D �.12/q�1 �

212�12

�.12/C � � � ;

where for positive integers n we have

a�.n/ D �2��.12/n�112 �

1X

cD1

K.�1; n; c/

c� I11

�

4�pn

c

�

:

Here I11.x/ is the usual I11-Bessel function.

Remark. We note that M�.z/ begins with the terms

M�.z/ D 39916800q�1 �2615348736000

691C � � � :

Obviously, we have that a�.�1/ D �.12/ D 11Š. Using classical formulas for

Riemann’s zeta function at positive even integers, we find that a�.0/ D24�11ŠB12

.

Strictly speaking, the constant term for M�.z/ in [4] is given in terms of the

infinite sum1X

cD1

K.�1; 0; c/

c12:

A straightforward calculation relates these Kloosterman sums to �.c/, the classi-

cal Möbius function, and consequently provides the connection to the reciprocal

of �.12/.

Remark. It would be very interesting to have a simpler description of the coeffi-

cients a�.n/ for positive n.

Remark. It turns out that there are infinitely many choices for the mock theta

function M�.z/. This ambiguity arises for two reasons. First of all, one may

choose other descriptions of�.z/ in terms of Poincaré series to obtain other mock

theta functions. We simply selected the simplest choice. Secondly, we point out

that if F.z/ is a weakly holomorphic modular form of weight �10 on SL2.Z/

with algebraic coefficients, then the sum

M�.z/CN�.z/C F.z/

is also a weight �10 harmonic weak Maass form, and so M�.z/C F.z/ is also a

mock theta function for �.z/. In both situations one can easily modify the argu-

ments here to obtain the corresponding versions of Theorem 1.1 and Corollary 1.2.

150 Ken Ono

3 Proofs

Here we prove Theorem 1.1 and Corollary 1.2. The proof of Theorem 1.1 re-

lies on facts about harmonic weak Maass forms and their behavior under Hecke

operators. The proofs of Corollaries 1.2 and 1.3 are elementary.

3.1 Proof of Theorem 1.1

The Hecke operators T�10.p/ act independently on the holomorphic and non-

holomorphic parts of R.z/. Since �.z/ is a Hecke eigenform, it follows that

N�.z/ is an eigenform of T�10.p/ (for example, see Section 7 of [9]). In particu-

lar, since

�.z/ j T12.p/ D �.p/�.z/;

it then follows that

Lp.z/ WD R.z/ j T�10.p/ � �.p/p�11 �R.z/

is holomorphic on H, and so it is a weight�10 weakly holomorphic modular form

on SL2.Z/. Recall that a weakly holomorphic modular form is a meromorphic

modular form whose poles (if any) are supported at cusps. Obviously, we then

have that

Lp.z/ DM�.z/ j T�10.p/ � �.p/p�11 �M�.z/: (3.1)

Consequently, it follows thatE4.z/E6.z/Lp.z/ is a weakly holomorphic modular

form of weight 0. All such forms are polynomials in j.z/.

Now we turn to the problem of determining the exact formula for the modular

form E4.z/E6.z/Lp.z/. We first note that

M�.z/ j T�10.p/ D 11Š � p�11 � q�p C a�.0/.1C p�11/CO.q/;

where a�.0/ D24�11ŠB12

as before. Therefore, since

E4.z/E6.z/ D E10.z/ D 1 � 264

1X

nD1

�9.n/qn;

it follows that

E4.z/E6.z/ � .M�.z/ j T�10.p//

D 11Š � p�11 � q�p � 264 � 11Š � p�11

p�1X

mD1

�9.m/q�pCm

� 264 � 11Š � p�11.1C p9/C a�.0/.1C p�11/CO.q/:

(3.2)


Similarly, we have that

��.p/p�11 �E4.z/E6.z/M�.z/

D ��.p/p�11 ��

11Š � q�1 � 264 � 11ŠC a�.0/�

CO.q/:

Combining these observations, we find that

E4.z/E6.z/Lp.z/ D 11Š � p�11 � q�p � 264 � 11Š � p�11

p�1X

mD1

�9.m/q�pCm

� 264 � 11Š � p�11.1C p9/C a�.0/.1C p�11/

� �.p/p�11 ��

11Š � q�1 � 264 � 11ŠC a�.0/�

CO.q/

D 11Š � p�11

�

q�p � 264

p�1X

mD1

�9.m/q�pCm � 264.1C p9/

Ca�.0/p

11

11Š� .1C p�11/

�

� �.p/ � 11Š � p�11 �

�

q�1 � 264Ca�.0/

11Š

�

CO.q/:

Since every polynomial in j.z/ is uniquely determined by its “principal part” (i.e.,

the coefficients corresponding to non-positive exponents), it follows that

E4.z/E6.z/Lp.z/ D11Š

p11��

Ap.z/C Bp.z/�

:

Here we use the fact that each jm.z/ satisfies

jm.z/ D q�m CO.q/:

This completes the proof.

3.2 Proof of Corollary 1.2

Here we prove the two claimed congruences.

(1) With exception of the constant terms, by (1.10) and (1.11) we have that all of

the coefficients of Ap.z/ and Bp.z/ are integers. Since 24B12D � 65520

691, and since

E4.z/E6.z/ has integer coefficients, the first claim follows immediately from The-

orem 1.1 and the Ramanujan congruence

�.n/ � �11.n/ .mod 691/:

152 Ken Ono

(2) By Theorem 1.1, we have that

1

11Š

1X

nD�p

�

p11a�.pn/ � �.p/a�.n/C a�.n=p/�

qn

D1

E4.z/E6.z/� .Ap.z/C Bp.z//:

Moreover, its coefficients are in Z by part (1). Therefore, we may reduce this mod-

ulo 24. Since E4.z/E6.z/ � 1 .mod 24/, by (1.10) and (1.11) we then find that

1

11Š

1X

nD�p

.a�.pn/ � �.p/a�.n/C a�.n=p// qn

� Ap.z/C Bp.z/ .mod 24/

� jp.z/ � �.p/j1.z/ .mod 24/:

This gives the second claim.

3.3 Proof of Corollary 1.3

Theorem 1.1, combined with the fact that M�.z/ is not modular, gives the first

claim. Theorem 1.1 and (1.11) immediately implies the second claim. The third

claim now follows from Corollary 1.2 (1).

4 Examples

Here we give a detailed discussion of Theorem 1.1 and Corollary 1.2 when p D 2.

In particular, we use the exact formulas for the coefficients of M�.z/ to nu-

merically approximate the theoretical exact formulas for M�.z/ j T�10.2/ and

compare them with the exact formulas we derive using just the first two coeffi-

cients of M�.z/.

We begin by giving the numerical approximations for the first few terms of

M�.z/. Using Theorem 2.1, we find that

M�.z/ D

1X

nD�1

a�.n/qn

D 39916800q�1 �2615348736000

691� 73562460235:68364 : : : q

� 929026615019:11308 : : : q2 � 8982427958440:32917 : : : q3

� 71877619168847:70781 : : : q4 � 497966668914961:54321 : : : q5

� 3074946857439412:02739 : : : q6 � � � � :

Again we stress that a�.�1/ and a�.0/ are exact.


4.1 Theorem 1.1 when p D 2

Using Theorem 2.1, one numerically finds that

M�.z/ j T�10.2/ � �.2/2�11 �M�.z/

D155925

8� q�2 C 467775q�1 � 3831077250

� 929888675100:00000 : : : q

� 71888542118662:49999 : : : q2

� 3075052120267049:99999 : : : q3 � � � � :

(4.1)

The first three terms are exact.

The modular functions A2.z/ and B2.z/ are:

A2.z/ D j2.z/ � 264j1.z/ �227833992

691;

B2.z/ D 24j1.z/ �5950656

691:

Therefore, Theorem 1.1 implies that

M�.z/ j T�10.2/ � �.2/2�11 �M�.z/

D11Š

211E4.z/E6.z/� .A2.z/C B2.z//

D155925.j2.z/ � 240j1.z/ � 338328/

8E4.z/E6.z/

D155925

8� q�2 C 467775q�1 � 3831077250 � 929888675100q

�143777084237325

2� q2 � 3075052120267050q3 � � � � :

This agrees with the numerics in (4.1).

4.2 Corollary 1.2 when p D 2

Using the calculations from the previous subsection, we have that

M�.z/ j T�10.2/ � �.2/2�11M�.z/ D

155925.j2.z/ � 240j1.z/ � 338328/

8E4.z/E6.z/:

The coefficients are easily seen to lie in 18Z. This illustrates Corollary 1.2 (1).

154 Ken Ono

To illustrate Corollary 1.2 (2), simply multiply both sides of this expression by

211=11Š to obtain

211�

M�.z/ j � �.2/2�11M�.z/

�

11ŠDj2.z/ � 240j1.z/ � 338328

E4.z/E6.z/

� j2.z/ � �.2/j1.z/

� j2.z/ .mod 24/:

Here we used �.2/ D �24 and E4.z/E6.z/ � 1 .mod 24/.

Acknowledgments. The author thanks Matt Boylan, Kathrin Bringmann,

Amanda Folsom, and Rob Rhoades for their comments on early drafts of this

paper. The author also thanks the National Science Foundation for its generous

support, he thanks the support of the Manasse family, and the support of the Hill-

dale Foundation.

References

[1] G. E. Andrews, R. Askey and R. Roy, Special Functions, Cambridge Univ. Press,

Cambridge, 1999.

[2] T. Asai, M. Kaneko and H. Ninomiya, Zeros of certain modular functions and an

application, Comm. Math. Univ. Sancti Pauli 46 (1997), 93–101.

[3] K. Bringmann and K. Ono, The f .q/ mock theta function conjecture and partition

ranks, Invent. Math. 165 (2006), 243–266.

[4] K. Bringmann and K. Ono, Lifting cusp forms to Maass forms with an application

to partitions, Proc. Natl. Acad. Sci. USA 104 (2007), no. 10, 3725–3731.

[5] K. Bringmann and K. Ono, Dyson’s ranks and Maass forms, to appear in Ann. of

Math.

[6] K. Bringmann, K. Ono and R. Rhoades, Eulerian series as modular forms, J. Amer.

Math. Soc. 21 (2008), 1085–1104.

[7] J. H. Bruinier, Borcherds products onO.2; l/ and Chern classes of Heegner divisors,

Springer Lecture Notes in Mathematics 1780, Springer-Verlag, 2002.

[8] J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 125 (2004),

45–90.

[9] J. H. Bruinier and K. Ono, Heegner divisors,L-functions, and harmonic weak Maass

forms, to appear in Ann. Math.

[10] J. H. Bruinier, K. Ono and R. Rhoades, Differential operators for harmonic weak

Maass forms and the vanishing of Hecke eigenvalues, Math. Ann. 342 (2008), 673–

693.


[11] G. Faber, Über polynomische Entwicklungen, Math. Ann. 57 (1903), 389–408.

[12] G. Faber, Über polynomische Entwicklungen. II, Math. Ann. 64 (1907), 116–135.

[13] J. D. Fay, Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew.

Math. 293/294 (1977), 143–203.

[14] D. A. Hejhal, The Selberg Trace Formula for PSL.2; R/. Vol. 2, Springer Lect.

Notes in Math. 1001, Springer-Verlag, Berlin, 1983.

[15] H. Iwaniec, Topics in the Classical Theory of Automorphic Forms, Grad. Studies in

Math. 17, Amer. Math. Soc., Providence, R.I., 1997.

[16] K. Ono, The Web of Modularity: Arithmetic of the Coefficients of Modular Forms

and q-Series, Conference Board of the Mathematical Sciences 102, Amer. Math.

Soc., 2004.

[17] S. P. Zwegers, Mock Theta Functions, Ph.D. Thesis, Universiteit Utrecht, 2002.

Author information

Ken Ono, Department of Mathematics, University of Wisconsin,

Madison, Wisconsin 53706, USA.



On the Density of Integral Sets with

Missing Differences

Ram Krishna Pandey and Amitabha Tripathi

Abstract. For a given set M of positive integers, a well-known problem of Motzkin

asks for determining the maximal density �.M/ among sets of nonnegative integers in

which no two elements differ by M . The problem is completely settled when jM j � 2,

and some partial results are known for several families of M for jM j � 3. In this paper,

we consider the case M D ¹a; b; cº, with c a multiple of a or b. In most cases, we obtain

lower bounds for �.M/, which are conjecturally the exact values of �.M/, while in some

we obtain the exact value of �.M/.

Keywords. Density, M -set.

AMS classification. 11B05.

1 Introduction

For x 2 R and a set S of nonnegative integers, let S.x/ denote the number of

elements n 2 S such that n � x. The upper and lower densities of S , denoted by

ı.S/ and ı.S/ respectively, are given by

ı.S/ WD lim supx!1

S.x/

x; ı.S/ WD lim inf

x!1

S.x/

x:

If ı.S/ D ı.S/, we denote the common value by ı.S/, and say that S has density

ı.S/. Given a set of positive integersM , S is said to be anM -set if a 2 S , b 2 S

imply a � b …M . Motzkin in [3] asked to determine �.M/ given by

�.M/ WD supS

ı.S/

where S varies over the class of all M -sets. Cantor & Gordon in [1] showed the

existence of �.M/ for any M , determined �.M/ when jM j � 2:

�.¹m1º/ D1

2;

�.¹m1; m2º/ Db.m1 Cm2/=2c

m1 Cm2

for gcd.m1; m2/ D 1;

158 Ram Krishna Pandey and Amitabha Tripathi

and gave the following lower bound for �.M/:

�.M/ � supgcd.k;m/D1

1

mmin

ijkmi jm;

where mi are the elements of M and jxjm denotes the absolute value of the abso-

lutely least remainder of x mod m. Haralambis in [2] gave the equivalent expres-

sions for the right-hand side expression of the above inequality:

d1.M/ D supx2.0;1/

minikxmik;

d2.M/ D supgcd.k;m/D1

1

mmin

ijkmi jm;

d3.M/ D maxmDm

jCm

`

1�k�m

2

1

mmin

ijkmi jm

where kxk denotes the distance from the nearest integer. Thus d1.M/D d2.M/D

d3.M/, and we denote this common value by d.M/. Hence d.M/ serves as a

lower bound for �.M/. A useful upper bound for �.M/ is due to Haralambis

in [2]:

�.M/ � ˛ provided there exists a positive integer k such that S.k/ � .k C 1/˛

for every M -set S with 0 2 S .

In fact, Haralambis in [2] conjectured that �.M/ D d.M/ for jM j D 3, so that,

conjecturally, determining d.M/ D d3.M/ gives the value of �.M/.

We consider the problem for the families M D ¹a; b; cº, where c is a multiple

of a or b. By a result of Cantor & Gordon in [1], we know that �.kM/ D �.M/.

Thus, it is no loss of generality to assume that gcd.a; b/ D 1, and that a < b. In

most cases, we determine the value of d.M/, which is the lower bound for �.M/

and conjecturally equal to it, and in some cases we determine the value of �.M/.

2 Exact Results

We begin by dealing with one special case where we determine �.M/. We use

the upper and lower bounds for �.M/ to achieve this.

Theorem 1. LetM D ¹a; b; cº, where aC b is odd, gcd.a; b/ D 1, c 2 ¹na; nbº,

and n � ˙1.mod aC b/. Then

�.M/ DaC b � 1

2.aC b/:

On the Density of Integral Sets with Missing Differences 159

Proof. By Cantor & Gordon’s result we have �.M/ � �.¹a; bº/ D aCb�12.aCb/

.

For the reverse inequality, choose x such that ax � aCb�12

.mod a C b/. Since

cx � ˙ax � �bx .mod aC b/, �.M/ � aCb�12.aCb/

. Hence the result.

3 The Case M D ¹a; b; nbº

In this section, we deal with the family M D ¹a; b; nbº, with a < b, gcd.a; b/ D

1, and n � 2. We compute d3.M/ by comparing the rational numbers in the three

cases, as mentioned in Section 1.

Theorem 2. Let M D ¹a; b; nbº, where a < b, gcd.a; b/ D 1, and a, b are odd

integers. Then

�.M/ D

8

<

:

n2.nC1/

if n is evenI

12

if n is odd:

Proof. By Cantor and Gordon’s result we have �.M/ � �.¹a; bº/ D 12. If n is

odd, then ¹1; 3; 5; : : : º is an M -set. Hence �.M/ D 12

in this case. If n is even,

then �.M/ � �.¹b; nbº/ D �.¹1; nº/ D n2.nC1/

. To show the reverse inequality,

let m D .nC 1/b. Observe that m is odd. Choose x � m�12.mod m/. Then

ax � m�a2

.mod m/; �nbx � bx � m�b2

.mod m/:

Since 12.m � a/ > 1

2.m � b/ D 1

2nb, we have �.M/ � d.M/ � nb

2mD n

2.nC1/.


Lemma 1. For r; s � 0, let

Ar WD ¹2r.aC b/C 2t � 1 W 1 � t � bº;

Bs WD ¹2s.aC b/C 2b C 2t � 1 W 1 � t � aº:

Then ¹A0; A1; : : : ; B0; B1; : : : º partitions the set of positive odd integers 2N � 1.

Proof. Observe that jAr j D b and jBsj D a for each r; s � 0, and that ArC1 D

Ar C 2.a C b/ and BsC1 D Bs C 2.a C b/. The lemma now follows from

the observation that ¹A0; B0º partitions the odd integers in the interval Œ1; 2.a C

b/ � 1�.


Theorem 3. Let M D ¹a; b; nbº, where a < b, gcd.a; b/ D 1, a C b and n

are odd integers. Let the family of sets ¹Arºr�0 and ¹Bsºs�0 be defined as in

Lemma 1. If n 6� ˙1.mod aC b/, then

d.M/ D

8

<

:

m�.2rbC2t�1/2m

if n 2 Ar and where m D aC nbI

m�2.sC1/b2m

if n 2 Bs and where m D .nC 1/b:

Proof. We compute d.M/ by using the expression for d3.M/ in Section 1. Thus

there are three choices for m, and we determine d3.M/ by comparing the three

rational numbers corresponding to these case. By Lemma 1, n belongs to a unique

set among the two families ¹Arºr�0 and ¹Bsºs�0.

CASE I: (m D aC nb) Observe that m is odd, and that gcd.b;m/ D 1.

Subcase (i): (n 2 Ar ) Choose x such that

bx � m�.2rbC1/2

.mod m/:

Thus we have 2ax � �2nbx � n.2rb C 1/ D 2r.m � a/ C n � n � 2ra D

2rb C 2t � 1.mod m/, and

ax � �m�.2rbC2t�1/2

.mod m/:

Since nbx � �ax .mod m/,

min®

jaxjm; jbxjm; jnbxjm¯

D m�.2rbC2t�1/2

: (1)

We now show that

min®

jayjm; jbyjm; jnbyjm¯

� m�.2rbC2t�1/2

D m�12� `;

for each y, 1 � y � 12.m�1/, where ` D rbCt�1. Let I WD

�

m�12�`; mC1

2C`�

and J WD�

m�12� `; mC1

2C `

�

. We show that, for 1 � y � 12.m � 1/, if

by .mod m/ 2 I, then ay .mod m/ … J. Accordingly, write

by � m�12� `C i .mod m/:

Then by .mod m/ 2 I if and only if 0 � i � 2` C 1, and 2ay � �2nby �

n�

1C 2.` � i/�

.mod m/. Since n�

1C 2.` � i/�

is odd, we get

ay � m�12C nC1

2C n.` � i/ .mod m/:

To show that ay .mod m/ … J, we consider the two cases 0 � i � ` and `C 1 �

i � 2`C 1.


First consider the case 0 � i � `. For each k, 0 � k � r , define

Ik WD�

` � 1n

�

.k C 1/m � ` � nC12

�

; ` � 1n

�

km � ` � nC12

�

� 1�

;

J WD ¹` � kb W 1 � k � rº[ ¹`º:

Then it can be shown that I0 [ I1 [ : : : [ Ir [ J contains the set ¹0; 1; 2; : : : ; `º.

A simple computation shows that

ay 2

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

�

mC12C `;m

�

if i D `I

�

0; m�12� `

�

if i 2 J; i ¤ `I

�

kmC mC12C `; .k C 1/mC m�1

2� `

�

if i 2 Ik with 0 � k � r:

For the cases `C 1 � i � 2`C 1, define for 0 � k � r ,

I 0

k WD�

`C 1n

�

km � `C n�12

�

C 1; `C 1n

�

.k C 1/m � `C n�12

��

;

J 0 WD ¹`C 1C kb W 1 � k � rº:

Then it can be shown that I 0

0 [ I0

1 [ : : : [ I0r [ J

0 contains the set ¹` C 1; ` C

2; : : : ; 2` C 1º, and that ay .mod m/ … J. This completes the subcase when

n 2 Ar .

Subcase (ii): (n 2 Bs) Choose x such that

bx �m�

�

2.sC1/bC1�

2.mod m/:

The computation in subcase (i) can be employed to show that

ax � �m�

�

2.sC1/b�2.a�t/�1�

2.mod m/;

so that

min®

jaxjm; jbxjm; jnbxjm¯

Dm�

�

2.sC1/bC1�

2: (2)

To show that

min®

jayjm; jbyjm; jnbyjm¯

�m�

�

2.sC1/bC1�

2D m�1

2� `;

for each y, 1 � y � 12.m�1/, where ` D .sC1/b, we mimic the proof in subcase

(i), the only change being in the value of `. All notations and congruences carry

through, so the derivations are omitted. This completes the subcase when n 2 Bs ,

and Case I.


CASE II: (m D .nC 1/b) Observe that m is even. As in Case I, we consider two

subcases.

Subcase (i): (n 2 Ar ) For each x, 1 � x � 12m, we show that

min¹jaxjm; jbxjm; jnbxjmº�m2� .rb C t /: (3)

Suppose jbxjm > m2� .rb C t /. Then

�mC m2� rb � t < bx < �mC m

2C rb C t

for some integer �. Thus x 2�

�.nC 1/C nC12� r; �.nC 1/C nC1

2C r

�

. Write

x � nC12� r C i .mod nC 1/, with 0 � i � 2r . Since nC1

2D r.aC b/C t , it is

easy to verify that

ax � �ar C ai � nC12C br C t C ai .mod nC 1/ when b is odd

and

ax � br C t C ai .mod nC 1/ when b is even:

Now 2ar D .n C 1/ � 2.br C t /. Hence 0 � ai � .n C 1/ � 2.rb C t / for

0 � i � 2r , and jaxjm �m2� .br C t /, as desired. This completes the subcase

when n 2 Ar .

Subcase (ii): (n 2 Bs) As in subcase (i), we can choose an integer � such that

with x D �.nC 1/C nC12� .s C 1/, we have

bx � m2� .s C 1/b .mod m/; ax � m

2C .s C 1/b C .t � a/ .mod m/:

Hence

min¹jaxjm; jbxjm; jnbxjmºDm2� .s C 1/b: (4)

Again, an argument similar to the one in subcase (i) shows that

min¹jayjm; jbyjm; jnbyjmº�m2� .s C 1/b

for each y, 1 � y � 12m. This completes the argument in Case II.

CASE III: (m D a C b) Observe that m is odd, and that gcd.a;m/ D 1 D

gcd.b;m/. Choose x such that ax � �bx � �aCb�12

.mod m/. Since n is odd,

it is easy to see that

nbx � m�n2� ˙m�1

2.mod m/

if and only if n � ˙1.mod 2m/. Since m�12

is the maximum absolute remainder

mod m and since n � ˙1.mod 2.aC b// is excluded by assumption,

min¹jaxjm; jbxjm; jnbxjmº�aCb�3

2

for each x, 1 � x � 12.m � 1/.


To determine d.M/, we consider the two cases n 2 Ar and n 2 Bs separately,

and compare the values given by the three cases. For n 2 Ar , n 6� ˙1.mod 2.aC

b//, observe that

.nC1/b�2.rbCt/2.nC1/b

D 12� rbCt

.nC1/b< 1

2� 2.rbCt/�1

2.aCnb/D .aCnb/�.2rbC2t�1/

2.aCnb/;

andaCb�32.aCb/

D 12� 3

2.aCb/< 1

2� 2.rbCt/�1


2.aCnb/:

Thus the upper bounds for d.M/ in Cases II and III are each less than the value

of d.M/ in Case I. Hence, in this case

d.M/ D 12� 2.rbCt/�1


2.aCnb/:

For n 2 Bs , n 6� ˙1.mod 2.aC b//, we have

.aCnb/�.2.sC1/bC1/2.aCnb/

D 12�

.sC1/bC12

aCnb< 1

2� sC1

nC1D .nC1/�2.sC1/

2.nC1/;

andaCb�32.aCb/

D 12� 3

2.aCb/< 1

2� sC1

nC1D .nC1/�2.sC1/

2.nC1/:

Thus the upper bounds for d.M/ in Cases I and III are each less than the value of

d.M/ in Case II, and

d.m/ D 12� sC1

nC1D .nC1/�2.sC1/

2.nC1/

in this case. This completes the comparison, and the proof of the theorem.


A0

r WD ¹.2r C 1/.aC b/C 2t � 1 W 1 � t � bº;

B 0

s WD ¹.2s � 1/.aC b/C 2b C 2t � 1 W 1 � t � aº:

Then ¹A0

0; A0

1; : : : ; B0

0; B0

1; : : : º partitions the set b � aC .2N � 1/.

Proof. Observe that A0r D ArC .aCb/ and B 0

s D Bs� .aCb/ for each r; s � 0.

The proof is similar to that of Lemma 1. We have jA0r j D b, jB 0

sj D a for each

r; s � 0, A0

rC1 D A0r C 2.a C b/ and B 0

sC1 D B 0s C 2.a C b/. The lemma

now follows from the observation that ¹A0

0; B0

0º partitions the even integers in the

interval Œb � aC 1; 3b C a � 1�.


Theorem 4. Let M D ¹a; b; nbº where a < b, gcd.a; b/ D 1, a C b is odd,

n � b�aC1 and even. Let the family of sets ¹A0rºr�0 and ¹B 0

sºs�0 be as defined

in Lemma 2. If n 6� ˙1.mod 2.aC b//, then

d.M/ D

8

<

:

m�¹.2rC1/bC2t�1º

2mwhere m D aC nb and n 2 A0

r I

m�.2sC1/b2m

where m D .nC 1/b and n 2 B 0s:

Proof. We use the method of proof given in Theorem 3, and place every even

integer n � b � a C 1 in a unique set among the two families ¹A0rºr�0 and

¹B 0sºs�0.

CASE I: (m D aC nb) Observe that gcd.b;m/ D 1.

Subcase (i): (n 2 A0r ) Choose x such that

bx � m�Œ.2rC1/bC1�2

.mod m/:

This is an analogue of the corresponding subcase in Theorem 3 with 2rC1 replac-

ing 2r . The argument of this subcase carries through if we make this replacement

throughout this subcase. We omit the details. This completes the subcase when

n 2 A0r .

Subcase (ii): (n 2 B 0s) Choose x such that

bx � m�Œ.2sC1/bC1�2

.mod m/:

This is an analogue of the corresponding subcase in Theorem 3 with 2s C 1 re-

placing 2.s C 1/. The argument of this subcase carries through if we make this

replacement throughout this subcase. We omit the details. This completes the

subcase when n 2 B 0s , and Case I.

CASE II: (m D aC nb)

Subcase (i): (n 2 A0r ) This is an analogue of the corresponding subcase in The-

orem 3 with 2r C 1 replacing 2r , obtaining only an upper bound. We omit the

details.

Subcase (ii): (n 2 B 0s) This is an analogue of the corresponding subcase in Theo-

rem 3 with 2s C 1 replacing 2.s C 1/. We again omit the details.

CASE III: (m D aC b)

We may use the exact same computation of Case III in Theorem 3 for this case

as well. The rest of the proof is the same as that given in Theorem 3, and is

omitted.


Theorem 5. Let M D ¹a; b; nbº; where a < b, gcd.a; b/ D 1, a C b is odd,

n � b � a � 1, and n is even. Then �.M/ D n2.nC1/

:

Proof. By Cantor and Gordon’s result we have �.M/ � �.¹b; nbº/ D �.¹1; nº/

D n2.nC1/

. For the reverse inequality, let m D .n C 1/b and choose x such that

x � n2.mod nC 1/. Then x D �.nC 1/C n

2for some integer �, and a simple

calculation shows that

ax � m�a�.nC1/2

.mod m/” .2�C 1/a � �1 .mod b/:

If b is even, then a must be odd, and so any solution of ay � ˙1.mod b/ is

necessarily odd. If b is odd, then a must be even, and we may choose y to be odd

by replacing y by b � y, if necessary. In either case, we may choose � such that

a.2� C 1/ � ˙1.mod b/, and hence satisfy ax � m�a�.nC1/2

.mod m/. Since

bx � nb2D m�b

2.mod m/ and b � aCnC1, it follows that �.M/ � nb

2.nC1/bD

n2.nC1/

. This completes the proof.

4 The Case M D ¹a; b; naº

We deal with the family M D ¹a; b; nbº, with a < b, gcd.a; b/ D 1, and n � 2.

The results are analogous to those obtained in Section 3, and proofs similar. We

begin by considering the case where a; b are both odd. However, unlike the anal-

ogous case in Theorem 2, we are able to determine d.M/ only for all sufficiently

large n.

Theorem 6. Let M D ¹a; b; naº, where a < b, gcd.a; b/ D 1, a, b are odd

integers, and n � b.aCb�2/2a

and even. Then d.M/ D na2.naCb/

:

Proof. We compute d.M/ by using the expression for d3.M/ in Section 1.

CASE I: (m D na C b) Observe that m is odd. Choose x such that x � m�12

.mod m/. Then

ax � m�a2

.mod m/ bx � m�b2

.mod m/:

Thus

min¹jaxjm; jbxjm; jnaxjmºDm�b

2D na

2: (5)

We now show that

min¹jayjm; jbyjm; jnayjmº�m�b

2


for each y, 1 � y � 12.m � 1/, by an argument similar to the one in Theorem 2.

Let I WD�

m�b2; mCb

2

�

. We show that, for 1 � y � 12.m � 1/, by .mod m/ 2 I

and ay .mod m/ 2 I only when y � m�12.mod m/. With y � m�1

2Ci .mod m/,

a simple calculation shows that

by .mod m/ 2 I” i 2�

kmb; km

bC 1

�

for some integer k, with 0 � k � b � 1.

If k D 0, i D 0 gives y � m�12� x .mod m/while i D 1 gives y � �m�1

2�

�x .mod m/. For 1 � k � b � 1, let ka D qb C r , where 0 � r � b � 1. In

fact, r ¤ 0 since b j ka otherwise, and this is impossible since gcd.a; b/ D 1

and 1 � k � b � 1. A routine calculation shows that ay .mod m/ lies betweenm�a

2C mr

b.mod m/ and mCa

2C mr

b.mod m/, and another shows

mCb2� m�a

2C mr

b< mCa

2C mr

b� mC m�b

2;

the first and third inequalities being valid since 2na � b.aC b � 2/. This proves

our claim that ay .mod m/ … I for 1 � y < m�12

, and completes the proof in this

case.

CASE II: (m D .nC 1/a) As in Case I, m is odd. The same choice of x gives

min¹jaxjm; jbxjm; jnaxjmºDm�b

2D .nC1/a�b

2: (6)

The proof of

min¹jayjm; jbyjm; jnayjmº�m�b

2

for each y, 1 � y � 12.m � 1/ is similar to the one in Case I, and omitted.

CASE III: (m D a C b) In this case m is even, and gcd.a;m/ D gcd.b;m/ D 1.

Observe that

ax � �bx � m2.mod m/

implies nax � 0.mod m/ since n is even. Therefore

min¹jaxjm; jbxjm; jnaxjmº�m2� 1

for each x, 1 � x � 12m.

It is easy to check that na2.bCna/

> .nC1/a�b2.nC1/a

, and that na2.bCna/

� aCb�22.aCb/

if and

only if n � b.aCb�2/2a

. Hence the result.



Cr WD ¹2r.aC b/C 2t � 1 W 1 � t � aº;

Ds WD ¹2s.aC b/C 2aC 2t � 1 W 1 � t � bº:

Then ¹C1; C2; : : : ;D1;D2; : : : º partitions the set 2.aC b/C .2N � 1/.

Proof. Observe that the families ¹Crºr�0 and ¹Dsºs�0 are obtained from the fam-

ilies ¹Arºr�0 and ¹Bsºs�0 by interchanging a and b. Observe also that jCr j D a

and jDsj D b for each r; s � 0, and that CrC1 D Cr C 2.a C b/ and DsC1 D

Ds C 2.a C b/. The lemma now follows from the observation that ¹C1;D1º

partitions the odd integers in the interval Œ2.aC b/C 1; 4.aC b/ � 1�.

Theorem 7. Let M D ¹a; b; naº where a < b, gcd.a; b/ D 1, a C b is odd,

n � 2.aCb/C1 and odd. Let the family of sets ¹Crºr�1 and ¹Dsºs�1 be defined

as in Lemma 3. If n 6� ˙1.mod aC b/, then

d.M/ D

8

ˆ

ˆ

<

ˆ

ˆ

:

m�.2raC2t�1/2m

if n 2 Cr and where m D naC bI

m�2.sC1/a2m

if n 2 Bs , b � 2.s C 1/aC t ,

and where m D .nC 1/a:

Proof. The argument in Theorem 3 carries over with the roles of a and b inter-

changed. We omit the proof.

Remark 1. We remark that just as the families ¹Crºr�1 and ¹Dsºs�1 are obtained

from the families ¹Arºr�0 and ¹Bsºs�0 by interchanging a and b, so are the cor-

responding results from Theorem 3. However, these formulae do not hold for

n 2 C0 [D0.


C 0

r WD ¹.2r C 1/.aC b/C 2t � 1 W 1 � t � aº;

D0

s WD ¹.2s C 1/.aC b/C 2aC 2t � 1 W 1 � t � bº:

Then ¹C 0

0; C0

1; : : : ;D0

0;D0

1; : : : º partitions the set .aC b/C .2N � 1/.

Proof. Observe that C 0r D CrC.aCb/ andD0

s D DsC.aCb/ for each r; s � 0.

The lemma now follows from Lemma 3.

Theorem 8. Let M D ¹a; b; naº where a < b, gcd.a; b/ D 1, a C b is odd,

n � a C b C 1 and even. Let the family of sets ¹C 0rºr�0 and ¹D0

sºs�0 be defined


as in Lemma 4. If n 6� ˙1.mod aC b/, then

d.M/ D

8

ˆ

ˆ

ˆ

<

ˆ

ˆ

ˆ

:

m�

�

.2rC1/aC2t�1�

2mif n 2 C 0

r and where m D naC bI

m�.2sC3/a2m

if n 2 D0s , b � .2s C 3/aC t ,

and where m D .nC 1/a:

Proof. The argument in Theorem 4 carries over with the roles of a and b inter-

changed, and in addition, replacing s by s C 1 in the case n 2 B 0s . We omit the

proof.

5 Concluding Remarks

In the previous sections, we have been able to obtain exact values of d.M/ in

many cases, and in some special cases, even the value of �.M/. We expect that

the values of d.M/ are in fact equal to �.M/, although we have not been able to

show this. There are, however, a few cases where the exact value of d.M/ has

eluded us. We state this as the following concluding remark.

Remark 2. Let M D ¹a; b; naº, with a < b and gcd.a; b/ D 1. We have been

unable to determine d.M/ in the following cases:

(i) aC b is odd, n � aC b � 1 and even, and satisfies b � .2s C 3/aC t ;

(ii) aC b is odd, n � 2.aC b/ � 1 and odd, and satisfies b � 2.s C 1/aC t ;

(iii) a, b odd, n < b.aCb�2/2a

and even.

Acknowledgments. The authors are grateful to the referee for his or her careful

reading and comments.

References

[1] D. G. Cantor and B. Gordon, Sequences of integers with missing differences, J. Com-

bin. Theory Ser. A 14 (1972), 281–287.

[2] N. M. Haralambis, Sets of Integers with Missing Differences, J. Combin. Theory Ser.

A 23 (1977), 22–33.

[3] T. S. Motzkin, Unpublished problem collection.


Author information

Ram Krishna Pandey, Department of Mathematics, Indian Institute of Technology,

Hauz Khas, New Delhi – 110 016, India.


Amitabha Tripathi, Department of Mathematics, Indian Institute of Technology,

Hauz Khas, New Delhi – 110 016, India.



On Pseudosquares and Pseudopowers

Carl Pomerance and Igor E. Shparlinski

Abstract. Introduced by Kraitchik and Lehmer, an x-pseudosquare is a positive integer

n � 1 .mod 8/ that is a quadratic residue for each odd prime p � x, yet is not a square.

We use bounds of character sums to prove that pseudosquares are equidistributed in fairly

short intervals. An x-pseudopower to base g is a positive integer which is not a power of

g yet is so modulo p for all primes p � x. It is conjectured by Bach, Lukes, Shallit, and

Williams that the least such number is at most exp.agx= log x/ for a suitable constant ag .

A bound of exp.agx log log x= log x/ is proved conditionally on the Riemann Hypothesis

for Dedekind zeta functions, thus improving on a recent conditional exponential bound

of Konyagin and the present authors. We also give a GRH-conditional equidistribution

result for pseudopowers that is analogous to our unconditional result for pseudosquares.

Keywords. Pseudosquare, pseudopower, character sums.

AMS classification. 11L40.

1 Introduction

1.1 Pseudosquares

An x-pseudosquare is a nonsquare positive integer n such that n � 1 .mod 8/

and .n=p/ D 1 for each odd prime p � x. The subject of pseudosquares was

initiated by Kraitchik and more formally by Lehmer in [14]. It was later shown by

Weinberger (see [19]) that if the Generalized Riemann Hypothesis (GRH) holds,

then the least x-pseudosquare, call it Nx , satisfies Nx � exp.cx1=2/ for a positive

constant c. The interest in this inequality is that there is a primality test, due

to Selfridge and Weinberger, for integers n < Nx that requires the verification of

some simple Fermat-type congruences for prime bases p � x. Thus, a large lower

bound for Nx leads to a fast primality test, and in particular this result gives an

alternate and somewhat simpler form of Miller’s GRH-conditional polynomial-

time deterministic primality test. See [19] for details.

The first author was supported in part by NSF grant DMS-0703850. The second author was

supported in part by ARC grant DP0556431.

172 Carl Pomerance and Igor E. Shparlinski

By the GRH, we mean the Riemann Hypothesis for Dedekind zeta functions,

that is, for algebraic number fields. Note that this conjecture subsumes the Ex-

tended Riemann Hypothesis (ERH), which is the Riemann Hypothesis for rational

Dirichlet L-functions. The Weinberger lower bound for Nx in fact only requires

the ERH.

As the concept of an x-pseudosquare is a natural one, it is also of interest

to find a reasonable upper bound for Nx and also to study the distribution of x-

pseudosquares. Let

M.x/ DY

p�x

p

denote the product of the primes up to x. For nonsquare integers n coprime to

M.x/, the “probability” that n satisfy n � 1 .mod 8/ and .n=p/ D 1 for all odd

primes p � x is 2��.x/�1. Thus, it is reasonable perhaps to conjecture that

Nx � 2.1Co.1//�.x/;

for example, see Bach and Huelsbergen [1]. In [18], Schinzel proves conditionally

on the GRH that

Nx � 4.1Co.1//�.x/; (1)

and in particular, he conditionally shows that this inequality holds as well for the

smallest prime x-pseudosquare. Unconditionally, he uses the Burgess bound [5]

(see also [11, Theorem 12.6]) to show that

Nx � exp..1=4C o.1//x/: (2)

We start with an observation, communicated to us by K. Soundararajan, that

the pigeonhole principle (used in the same fashion as in [7, Lemma 10.1]) gives

an unconditional proof of (1), though not for prime pseudosquares. Indeed, let us

put X D 2�.x/x and consider the

.1C o.1//X

logXD .1C o.1//

2�.x/ log x

log 2

vectors of Legendre symbols

��

`

3

�

;

�

`

5

�

; : : : ;

�

`

p

��

for all primes ` 2 .x;X�, where p is the largest prime with p � x. Clearly

there are at most 2�.x/�1 possibilities for such vectors, so for large x there are

five distinct primes `1; : : : ; `5 2 .x;X� for which they coincide. Thus, at least

two of them, say `1; `2, have the property that `1 � `2 .mod 8/. Then `1`2 is an

On Pseudosquares and Pseudopowers 173

x-pseudosquare and we have Nx � `1`2 � X2, implying (1). (Note that it is not

necessary that the numbers ` be prime in this proof, just coprime to M.x/.)

Our contribution to the subject of pseudosquares is on their equidistribution.

For this we follow Schinzel’s proof of (2), but use a character sum estimate given

in [11, Corollary 12.14], which dates back to work of Graham and Ringrose [6],

to prove the following result. Let Sx be the set of x-pseudosquares.

Theorem 1. Uniformly for A > 0 and N � exp.3x= log log x/, we have

# .Sx \ .A;ACN�/ D .1C o.1//N

M.x/# .Sx \ .0;M.x/�/ ; x !1:

We also show

# .Sx \ .0;M.x/�/ D .1C o.1//M.x/

2�.x/C1e log x; x !1; (3)

so that one can rewrite Theorem 1 in a more explicit form.

We remark that, by the prime number theorem, M.x/ D exp ..1C o.1//x/

thus our result is nontrivial starting with very small values of N compared to

M.x/.

Note that Granville and Soundararajan [7] also discuss the equidistribution

of x-pseudosquares via the Graham–Ringrose result on character sums, but their

context is different and it is not clear that Theorem 1 follows directly from their

paper.

1.2 Pseudopowers

Let g be a fixed integer with jgj � 2. Following Bach, Lukes, Shallit, and

Williams [2], we say that an integer n > 0 is an x-pseudopower to base g if

n is not a power of g over the integers but is a power of g modulo all primes

p � x. Denote by qg.x/ the least x-pseudopower to base g.

In [2] it is conjectured that for each fixed g, there is a number ag such that for

x � 2,

qg.x/ � exp.agx= log x/: (4)

In addition, a heuristic argument is given for (4), with numerical evidence pre-

sented in the case of g D 2. For any g, we have (see [12]) the trivial bound

qg.x/ � 2M.x/C 1, where M.x/ is the product of the primes p � x. Thus,

qg.x/ � exp..1C o.1//x/:


Using an estimate for exponential sums due to Heath-Brown and Konyagin [9] and

results of Baker and Harman [3, 4] on the Brun–Titchmarsh theorem on average,

Konyagin, Pomerance, and Shparlinski [12] proved that

qg.x/ � exp.0:88715x/

for all sufficiently large x and all integers g with 2 � jgj � x. Further, it was

noted in [12] that the method implied that for fixed g,

qg.x/ � exp..1=2C o.1//x/;

assuming the GRH. In this paper we make further progress towards (4), again

assuming the GRH. Our proof makes use of the approach in Schinzel [18] for

pseudosquares.

Theorem 2. Assume the GRH. Then for each fixed integer g with jgj � 2 there is

a number ag such that for x � 3,

qg.x/ � exp.agx log log x= log x/:

We are also able to prove an equidistribution result conditional on the GRH

that is similar in strength to Theorem 1. Let Px be the set of x-pseudopowers

base g.

Theorem 3. Assume the GRH. Let g be a fixed integer with jgj > 1. There is a

positive number bg such that for any interval .A;A C N� � .0;1/, uniformly

over N � exp.bgx= log log x/, we have

# .Px \ .A;ACN�/ D .1C o.1//N

M.x/# .Px \ .0;M.x/�/ ; x !1:

We derive an asymptotic formula for # .Px \ .0;M.x/�/ in Section 3.2, see

(22), so that one can get a more explicit form of Theorem 3.

We note that in [12] an unconditional version of Theorem 3 is given which

however holds only for N � exp.0:88715x/. Under the GRH, the method of [12]

gives a somewhat stronger result but still requires N to be rather large, namely it

applies only to N � exp ..0:5C "/x/ for an arbitrary " > 0.

As for lower bounds for qg.x/, it follows from Schinzel [16, 17] that

qg.x/!1; x !1:

In [2] it is shown that assuming the GRH there is a number cg > 0 such that

qg.x/ � exp.cg

px.log log x/3=.log x/2/:


1.3 Notation

We recall that the notation U D O.V / and U � V are equivalent to the assertion

that the inequality jU j � c V holds for some constant c > 0.

2 Distribution of Pseudosquares

In this section we prove Theorem 1 by making use of the following character sum

estimate, which is [11, Corollary 12.14].

Lemma 4. Let � be a primitive character to the squarefree modulus q > 1. Sup-

pose all prime factors of q are at mostN 1=9 and let r be an integer withN r � q3.

Then for any number A,

ˇ

ˇ

ˇ

ˇ

X

A<n�ACN

�.n/

ˇ

ˇ

ˇ

ˇ

� 4N�.q/r=2r

q�1=r2r

;

where �.q/ is the number of positive divisors of q.

Recall thatM.x/ is the product of the primes in Œ1; x�. Let x be a large number

and let Sx denote the set of positive integers n � 1 .mod 8/ with .n=p/ D 1 for

each odd prime p � x. That is, Sx consists of the x-pseudosquares and actual

squares coprime to M.x/. In particular, Sx � Sx . We let M2.x/ D M.x/=2, the

product of the odd primes up to x.

Theorem 1 is routine once N is large compared withM.x/, so we assume that

N � M.x/2. Note that for a positive integer n with .8n C 1;M2.x// D 1, we

have that

Y

pjM2.x/

�

1C

�

8nC 1

p

��

D

´

2�.x/�1; if 8nC 1 2 Sx;

0; else.(5)

Thus, if A;N are positive numbers, then the sum

SA;N WDX

A<8nC1�ACN.8nC1;M2.x//D1

Y

pjM2.x/

�

1C

�

8nC 1

p

��

satisfies

SA;N D 2�.x/�1#.Sx \ .A;ACN�/: (6)


The product in (5) can be expanded, so that we have

SA;N DX


X

f jM2.x/

�

8nC 1

f

�

DX

f jM2.x/

X


�

8nC 1

f

�

:

(7)

The contribution to SA;N from f D 1 is

X


1 �N

4e log x(8)

uniformly forA;N withN � exp.x1=2/. This estimate follows immediately from

the fundamental lemma of the sieve; for example, see [8, Theorem 2.5].

Our goal now is to show that the contributions to SA;N coming from values of

f > 1 is small in comparison to (8). Suppose that f jM2.x/, f > 1 is fixed. We

can rewrite the contribution in (7) corresponding to f as

Rf DX

d jM2.x/=f

�.d/X

A<8nC1�ACNd j8nC1

�

8nC 1

f

�

; (9)

where �.d/ is the Möbius function.

The Pólya–Vinogradov inequality (see [11, Theorem 12.5]) immediately im-

plies that

jRf j � 3 � 2�.x/�2p

f logf < 2�.x/p

f log f (10)

for any choice of f > 1. We use (10) when f is not much larger than N , namely

we use it when

f � N r2r =.r2r�1C1/;

where r shall be chosen later. In this case it gives

jRf j � 2�.x/N 1�2=.r2r C2/ log.N 2/ � 2.1Co.1//�.x/N 1�2=.r2r C2/: (11)

For large values of f , that is, when

f > N r2r =.r2r�1C1/; (12)

we use a different approach which relies on Lemma 4.


Let rf D .1 � f2/=8, so that rf is an integer and 8rf � 1 .mod f /. Then

Rf DX

d jM2.x/=f

�.d/

�

d

f

�

X

A<dk�ACNk�d .mod 8/

�

k

f

�

DX

d jM2.x/=f

�.d/

�

8d

f

�

X

A<d.8lCd/�ACN

�

l C drf

f

�

DX

d jM2.x/=f

�.d/

�

8d

f

�

X

m2Id;f

�

m

f

�

;

where Id;f D ŒBd;f C 1; Bd;f CNd;f �, an interval of length

Nd;f DN

8dCO.1/:

Thus,

jRf j �X

d jM2.x/=f

ˇ

ˇ

ˇ

ˇ

X

m2Id;f

�

m

f

�ˇ

ˇ

ˇ

ˇ

: (13)

The character sums in (13) where 8d > N 0:1 are trivially bounded byNd;f D

O.N 0:9/ in absolute value, so their total contribution to Rf is

X

d jM2.x/=f

8d>N 0:1

ˇ

ˇ

ˇ

ˇ

X

m2Id;f

�

m

f

�ˇ

ˇ

ˇ

ˇ

� 2�.x/N 0:9: (14)

We now assume that 8d � N 0:1. Note that the conductors f of the characters

which appear in (13) are squarefree. We choose r as the largest integer with

r2r C 2 �log x

log log x

and apply Lemma 4 to the inner sum in (13) with this value of r . To do this we

need

.N=.8d//r � f 3 and x � .N=.8d//1=9:

These inequalities hold since

r D

�

1

log 2C o.1/

�

log log x and N � exp.3x= log log x/; (15)


so that�

N

8d

�r

� N 0:9r � exp.2:7rx= log log x/ �M.x/3 � f 3 (16)

and�

N

8d

�1=9

� N 0:1 � exp.0:3x= log log x/ � x (17)

for all large x.

Thus, by Lemma 4,

X

d jM2.x/=f

8d�N 0:1

ˇ

ˇ

ˇ

ˇ

X

m2Id;f

�

m

f

�ˇ

ˇ

ˇ

ˇ

� 4 � 2�.x/�1N2.�.x/�1/r=2r

f �1=r2r

: (18)

We now derive from (14), (18), and (12) that

jRf j � 2.1Co.1//�.x/N 0:9C2.1Co.1//�.x/Nf �1=r2r

� 2.1Co.1//�.x/N 1�2=.r2r C2/

which is also the bound in (11) for those f > 1 not satisfying (12).

Now summing over f we see that the contribution to SA;N from values of

f > 1 is at mostX

f jM2.x/f >1

jRf j � 4.1Co.1//�.x/N 1�2=.r2r C2/

D N 1�2=.r2r C2/ exp..log 4C o.1//x= log x/:

Note that by our choice of r we have

N 2=.r2r C2/ � N 2 log log x= log x � exp.6x= log x/:

Since 6 > log 4, we have that the contribution to (7) from terms with f > 1 is

small compared to the main term given by (8), so that

SA;N D .1C o.1//N

4e log x:

Together with (6), we now have

#.Sx \ .A;ACN�/ D .1C o.1//N

2�.x/C1e log x:

Since the number of squares in the interval .A;ACN� is at mostN 1=2, we obtain

# .Sx \ .A;ACN�/ D .1C o.1//N

2�.x/C1e log x:

TakingA D 0 andN DM.x/we derive (3) and conclude the proof of Theorem 1.

We remark that by being a little more careful with the estimates, we can prove

the theorem with “3” replaced with any fixed number larger than log 8.


3 Distribution of Pseudopowers

3.1 Proof of Theorem 2

Let g be a given integer with jgj � 2 which we assume to be fixed. Let pg.x/

be the least positive integer which is not a power of g yet is a power of g modulo

every prime p � x with p − g. It is easy to see that

qg.x/ � gpg.x/: (19)

Indeed, the integer gpg.x/ is not a power of g, it is a power of g modulo every

prime p � x with p − g, and it is zero modulo p for every prime p j g, and so is

a power of g modulo these primes too.

For every prime p − g let lg.p/ be the multiplicative order of g modulo p,

and let ig.p/ D .p � 1/=lg.p/, the index of the subgroup of powers of g in the

multiplicative group modulo p. Let

Mg.x/ DY

p�xp−g

p; Ig.x/ DY

pjMg.x/

ig.p/:

It follows from [12, Theorem 1] that Ig.x/ � exp.0:42x/ for all sufficiently large

x. We conditionally improve this result.

Lemma 5. Assume the GRH. There is a number cg such that for x � 3,

Ig.x/ � exp.cgx log log x= log x/:

Proof. In Kurlberg and Pomerance [13, Theorem 23], following ideas of Hoo-

ley [10] and Pappalardi [15], it is shown conditionally on the GRH that

X

pjMg.x/lg.p/�p=y

1�g�.x/

yCx log log x

.log x/2

for 1 � y � log x. Applying this result with y D log x, we have

X

pjMg.x/ig.p/�log x

1�gx log log x

.log x/2:

Indeed, ig.p/ � y implies that lg.p/ � .p � 1/=y < p=y. Since we trivially

have ig.p/ � x for each prime p jMg.x/ with p − g, we thus have

Ig.x/ DY

pjMg.x/ig.p/<log x

ig.p/Y

pjMg.x/ig.p/�log x

ig.p/ � .log x/�.x/xOg.x log log x=.log x/2/:

The lemma follows. ut


For each prime p − g, let �p be a character modulo p of order ig.p/. Then

ig.p/X

j D1

�jp.n/ D

´

ig.p/; if n is a power of g modulo p;

0; else:

Thus,

Y

pjMg.x/

ig.p/X

j D1

�jp.n/

D

´

Ig.x/; if n is a power of g modulo every p jMg.x/;

0; else:

(20)

Let ƒ.n/ denote the von Mangoldt function. From the definition of pg.x/ we

deduce that

Sg WDX

n<pg.x/

ƒ.n/Y

pjMg.x/

ig.p/X

j D1

�jp.n/ D Ig.x/

X

n<pg.x/n is a power of g

ƒ.n/:

The last sum is 0 if g is not a prime or prime power, and in any event is always at

most logpg.x/� x. Thus,

Sg � Ig.x/x: (21)

We now multiply out the product in (20); it is seen as a sum of Ig.x/ characters

modulo Mg.x/. The contribution to Sg from the principal characterQ

p �ig.p/p

is .pg.x// C O.x/. We may assume that pg.x/ � e�.x/ since otherwise the

theorem follows immediately from (19), so that the contribution to Sg from the

principal character is .1C o.1//pg.x/, by the prime number theorem.

The contribution to Sg from each nonprincipal character � is

X

n<pg.x/

�.n/ƒ.n/� pg.x/1=2�

.logMg.x//2 C .logpg.x//

2�

� pg.x/1=2x2

assuming the GRH. Hence, the contribution to Sg from nonprincipal characters is

O.Ig.x/pg.x/1=2x2/. Thus,

Sg D .1C o.1//pg.x/CO.Ig.x/pg.x/1=2x2/;

so that from (21) we deduce that

pg.x/� Ig.x/2x4:

Theorem 2 now follows immediately from (19) and Lemma 5.


We remark that an alternate way to handle primes dividing g is to eschew (19)

and instead multiply the product in (20) byP

� mod g �.n/. This sum is '.g/when

n � 1 .mod g/ and is 0 otherwise. Note that a number that is 1 mod p is always a

power of g modulo p. Although this is somewhat more complicated, it does lead

to a proof that there is a prime number below the bound exp.agx log log x= log x/

that is an x-pseudopower base g.

3.2 Proof of Theorem 3

We use the method of proof of Theorem 1. Accordingly we only outline some

new elements and suppress the details.

For a nonzero integer n, let rad.n/, the radical of n, be the largest squarefree

divisor of n. That is, rad.n/ is the product of the distinct prime factors of n. Also,

let !.n/ denote the number of distinct prime factors of n. Let P x denote the set

of positive integers which are either an x-pseudopower base g or a true power of

g. Then a positive integer n 2 P x if and only if both

(i) n is in the subgroup hgi of .Z=pZ/� when p � x and p − g;

(ii) n � 0 or 1 .mod p/ when p � x and p j g.

Assuming then that x � jgj, the cardinality of P x \ .0;M.x/� is

2!.g/Y

pjMg.x/

lg.p/ D 2!.g/Y

pjMg.x/

p � 1

ig.p/D

2!.g/'.Mg.x//

Ig.x/

�2!.g/M.x/

e '.rad.g//Ig.x/ log x;

by the formula of Mertens.

It is easy to see that this expression is exponentially large, either from the

observation that lg.p/ � 2 whenever�

gp

�

D �1, so that #.P x \ .0;M.x/�// �

2.1=2Co.1//�.x/, or using Ig.x/ � e0:42x from [12]. Further, the number of true

powers of g in .0;M.x/� is small; it is O.x/. Thus,

# .Px \ .0;M.x/�/ D .1C o.1//2!.g/M.x/

e '.rad.g//Ig.x/ log x; x !1: (22)

To prove Theorem 3 we again use Lemma 5, which is GRH-conditional. But

the framework of the proof follows the argument of Theorem 1, and in particular

it uses the unconditional Lemma 4. Notice that the proof of Theorem 2 used

Riemann Hypotheses a second time, namely in the estimation of the weighted


character sums. Now we use unweighted character sums and so are able to use

Lemma 4. The set-up is as follows. Let

PA;N DX

abDrad.g/

X

A<n�ACNn�0 .mod a/n�1 .mod b/

Y

pjMg.x/

ig.p/X

j D1

�jp.n/: (23)

This expression counts integers n 2 .A;ACN� that are 0 or 1 .mod p/ for each

prime p j g and in the subgroup hgi of .Z=pZ/� for each prime p � x with

p − g. Namely, it counts members of P x , and does so with the weight Ig.x/.

To prove Theorem 3 one then expands the product in (23). As usual, the contri-

bution from the principal character is easily estimated: it is the number of integers

n 2 .A;ACN� which are 0 or 1 .mod p/ for each prime p j g and are coprime

to Mg.x/. Thus, the principal character gives the contribution

.1C o.1//2!.g/

rad.g/�'.Mg.x//

Mg.x/N D .1C o.1//

2!.g/N

e '.rad.g// log x;

which when divided by the weight Ig.x/ gives the main term for our count.

The nonprincipal characters have conductors corresponding to those divisors

f of Mg.x/ with f > 1 and ig.p/ > 1 for each prime p j f . For such integers

f , the characters that occur with conductor f are induced by characters in the set

Xf D°

Y

pjf

�jp

p W 1 � jp � ig.p/ � 1 for p j f±

:

Thus, the contribution of the nonprincipal characters to PA;N is

P �

A;N WDX

abDrad.g/

X

f jMg.x/f >1

X

�2Xf

X

A<n�ACNn�0 .mod a/n�1 .mod b/

.n;Mg.x/=f /D1

�.n/;

where Xf is empty if ig.p/ D 1 for some prime p j f . The inner sum is

X

d jMg.x/=f

�.d/X

A<n�ACNn�0 .mod ad/n�1 .mod b/

�.n/:

As before we estimate the character sum here trivially if d is large, we use the

Pólya–Vinogradov inequality if f is small, and we use Lemma 4 in the remaining


cases. However, we modify slightly the choice of r in the proof of Theorem 1 and

take it now as the largest integer with

r2r C 2 �log x

.log log x/2:

Then we still have (15) and thus the conditions (16) and (17) are still satisfied so

that Lemma 4 may be used. Since each jXf j < Ig.x/, we obtain

jP �

A;N j � 4.1Co.1//�.x/Ig.x/N1�2=.r2r C2/:

This leads us to the asymptotic formula

PA;N D .1C o.1//2!.g/N

e '.rad.g// log xCO

�

Ig.x/4.1Co.1//�.x/N 1�2=.r2r C2/

�

:

For N � exp.bgx= log log x/ we have

N 2=.r2r C2/ � N 2.log log x/2= log x > exp�

2bgx log log x= log x�

:

Using Lemma 5 and taking bg D cg , we conclude the proof of Theorem 3.

Acknowledgments. The authors are grateful to K. Soundararajan for commu-

nicating the proof we presented in Section 1.1 and for his permission to give it

here.

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and Func. Anal. 13 (2003), 992–1028.

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185.

Author information

Carl Pomerance, Department of Mathematics, Dartmouth College,

Hanover, NH 03755-3551, USA.


Igor E. Shparlinski, Department of Computing, Macquarie University,

Sydney, NSW 2109, Australia.



Maier Matrices Beyond Z

Frank Thorne

Abstract. This paper is an expanded version of a talk given at the 2007 Integers Con-

ference, giving an overview of the Maier matrix method and surveying the author’s work

in extending it beyond the integers.

Keywords. Maier matrix method, distribution of primes, Shiu’s theorem, function fields.

AMS classification. 11N05, 11N13, 11T55.

1 Maier Matrices

Loosely speaking, a Maier matrix is a combinatorial device used to prove the ex-

istence of irregular or interesting patterns in the distribution of primes or related

sequences. We will illustrate the technique with two particularly interesting ex-

amples. The first is Maier’s 1985 proof [6] that “unexpected” irregularities exist

in the distribution of primes in short intervals. In particular, Maier proved that for

any A > 0 there exists a constant ıA > 0 such that

lim supn!1

�.nC logA n/ � �.n/

logA�1 n� 1C ıA;

lim infn!1

�.nC logA n/ � �.n/

logA�1 n� 1 � ıA:

(1.1)

These irregularities are unexpected in the sense that they contradict probabilistic

heuristics for A > 2.

The proof is as follows. For a variable y, let Q DQ

p<y p, let x1 D QD for

some fixed largeD, and let C be a parameter to be determined later. Consider the

following matrix of integers:

2

6

6

6

4

Qx1 C 1 Qx1 C 2 : : : Qx1 C yC

Q.x1 C 1/C 1 Q.x1 C 1/C 2 : : : Q.x1 C 1/C yC

::::::

::::::

Q.2x1/C 1 Q.2x1/C 2 : : : Q.2x1/C yC

3

7

7

7

5

:

The author is grateful for financial support from an NSF VIGRE fellowship.

186 Frank Thorne

The columns form arithmetic progressions modulo Q, and so the prime number

theorem for arithmetic progressions predicts1 that for each i 2 Œ1; yC � which is

coprime to Q, the corresponding column should contain � Q�.Q/

x1

log.Qx1/primes.

Therefore, the number of primes in the matrix, and thus in an average row, can

be asymptotically determined by counting the number of such i . In fact, the latter

quantity is

ˆ.yC ; y/ � yC �.Q/

Qe !.C/; (1.2)

for a function !.C/ which converges to e� , but oscillates above and below e�

as C ! 1.2 The short intervals occurring in the Maier matrix are of the sort

considered in (1.1), and Maier’s theorem soon follows.

In 1997, Shiu [8] similarly proved the remarkable result that if a; q, and k are

arbitrary integers with .a; q/ D 1, there exists a string of k consecutive primes

pnC1 � pnC2 � � � � � pnCk � a .mod q/:

(Here pn denotes the nth prime.) Furthermore, for k sufficiently large in terms of

q, these primes can be chosen to satisfy the bound

1

�.q/

�

log logpnC1 log log log logpnC1

.log log logpnC1/2

�1=�.q/

� k: (1.3)

To prove (1.3) Shiu constructed a similar Maier matrix; the primary difference is

in the choice of Q. For example, if a D 1, primes 6� 1 .mod m/ are excluded

from the product. This forces most primes in the matrix to be � 1 .mod m/, and

Shiu’s result easily follows.

The method has been similarly adapted to prove a host of interesting results

about the distribution of the primes and related integer sequences. For more

on this, we recommend the outstanding survey articles of Granville [4] and of

Soundararajan [9]. In this article we will consider the problem of adapting the

Maier matrix method to different settings. In particular we will describe exten-

sions of the method to the polynomial ring FqŒt � and to imaginary quadratic fields,

where we obtained analogous results.

2 Maier Matrices in FqŒt�

The polynomial ring FqŒt � (here Fq is a finite field) has long been studied as an

analogue to the integers. Like the integers, FqŒt � enjoys unique factorization, and

1This is not known to be true for allQ, except under GRH. However, a theorem of Gallagher [3]

implies the correct asymptotic for an infinite set of such Q, where the error term in the asymptotic

depends on D.2In general, ˆ.x; y/ denotes the number of n � x, all of whose prime factors are at least y.

Maier Matrices Beyond Z 187

has the additional property that the residue class rings are all finite, so that one

may naturally talk about the ‘size’ of elements.

Classical methods of analytic number theory have been extremely successful

in analyzing the distribution of primes (e.g., monic irreducible polynomials) in

FqŒt �. For example, one defines the zeta function as

�FqŒt�.s/ WDX

x2FqŒt�

1

jxjs; (2.1)

where the sum is over all monic polynomials x, and jxj WD #jFqŒt �=.x/j D qdeg x :

As there are exactly qn monics of degree n, one easily obtains the formula

�FqŒt�.s/ D1

1 � q1�s:

The Riemann hypothesis is then a triviality, and in fact one has an exact prime

number theorem

�.n/ D1

n

X

d jn

�.d/qn=d :

Thinking of n as logq.qn/, this closely mirrors the classical case.

It is therefore natural to ask whether the Maier matrix method can be adapted

to FqŒt �, and we have answered this question in the affirmative. To start with, we

have the following

Theorem 2.1 ([11, Theorem 1.1]). For any fixed A > 0, there exists a constant

ıA > 0 (depending also on q) such that

lim supk!1

supdeg f Dk

�.f; dA log ke/

qdA log keC1=k� 1C ıA;

lim infk!1

infdeg f Dk

�.f; dA log ke/

qdA log keC1=k� 1 � ıA:

Here �.f; i/ denotes the number of irreducible monic polynomials p with deg.f �

p/ � i .

The proof follows similar lines. We write Q DQ

deg p�n p, and consider the

following matrix:

2

6

6

6

4

Qg1 C h1 Qg1 C h2 : : : Qg1 C hj

Qg2 C h1 Qg2 C h2 : : : Qg2 C hj

::::::

::::::

Qgi C h1 Qgi C h2 : : : Qgi C hj

3

7

7

7

5

:

188 Frank Thorne

Here g1 through gi range through all monic polynomials of degree 2 degQ (or of

degree ˛ degQ for any fixed ˛ > 1), and h1 through hj run through all polynomi-

als of degree s (of arbitrary leading coefficient), for a parameter s to be determined

later. The number of primes in the whole matrix is � ij3 deg Q

e !.s=n/, and by

appropriately choosing s in terms of n, the matrix and thus some row can be made

to contain more or fewer primes than expected.

We also proved the following function field analogue of Shiu’s theorem:

Theorem 2.2 ([11, Theorem 1.2]). Suppose that k is a positive integer, and a and

m are polynomials with m monic and .a;m/ D 1. Then there exists a string of

consecutive primes

prC1 � prC2 � � � � � prCk � a .mod m/:

Furthermore, for sufficiently large k, these primes may be chosen so that their

common degree D satisfies

1

�.m/

�

logD

.log logD/2

�1=�.m/

� k: (2.2)

The implied constant depends only on q.

The observant reader will notice that it is not obvious what “consecutive”

means; the elements of FqŒt � are not naturally ordered in the same way as the

integers. We may in fact order our primes with respect to any ordering compatible

with our Maier matrix construction, and in particular our theorem applies with

respect to lexicographic order.

This theorem was extended in an interesting way by Tanner [10], who proved

the following:

Theorem (Tanner). Under the same hypotheses there exists an integer D0 (de-

pending on q, k, and m) such that for each D � D0 there exists a string of

consecutive primes

prC1 � prC2 � � � � � prCk � a .mod m/

of degree D. Furthermore, for sufficiently large k, D0 satisfies (2.2).

Tanner’s proof is an extension of the author’s proof; the point is that since

there are many polynomials of the same degree in FqŒt �, it is possible to construct

appropriate Maier matrices where all the polynomials in the matrix are of a given

degree.


3 ‘Prime Bubbles’ in Imaginary Quadratic Fields

We will now consider the problem of adapting Maier’s matrix method to number

fields. Let K be an imaginary quadratic field. In this setting a positive propor-

tion of ideals correspond to elements (although the unit group interferes), and the

prime elements of OK can be naturally visualized as lattice points in Z. Adapt-

ing the proof of Shiu’s theorem, we proved that there are clumps of primes, all of

which lie in an arbitrary fixed arithmetic progression, up to multiplication by units:

Theorem 3.1 ([12, Theorem 1.1]). Suppose K is an imaginary quadratic field, k

is a positive integer, and a and q are elements of OK with q ¤ 2 and .a; q/ D 1.

Then there exists a “bubble”

B.r; x0/ WD ¹x 2 C W jx � x0j < rº (3.1)

with at least k primes, all of which are congruent to ua modulo q for units u 2

OK . Furthermore, for k sufficiently large in terms of q (andK), x0 can be chosen

to satisfy

1

�K.q/

�

log log jx0j log log log log jx0j

.log log log jx0j/2

�!K=hK�K.q/

� k: (3.2)

The implied constant is absolute.

Here !K denotes the number of units in OK , hK is the class number ofK, and

�K.q/ WD j.OK=.q//�j: As an example of such a ‘prime bubble’ in ZŒi � (which

we found by computer search), the ball of radiusp

23:5 centered at 59 C 779i

contains six primes, all congruent to˙1 or˙i modulo 5C i .

To prove our result we construct the following Maier matrix:

2

6

6

6

4

Qi1 C b1 Qi1 C b2 : : : Qi1 C bJ

Qi2 C b1 Qi2 C b2 : : : Qi2 C bJ

::::::

::::::

QiI C b1 QiI C b2 : : : QiI C bJ

3

7

7

7

5

: (3.3)

Q is defined to be any generator of the ideal Q, given by

Q WD qY

p2P ;p¤p0

p; (3.4)

where P ranges over primes of norm � y with restrictions on the residue classes

modulo q (which depend on a). The need to exclude one prime p0 will be ex-

plained shortly.

The i range over all elements of OK with norm in .NQD; 2NQD/, and the

b range over all elements of norm less than either yz or 9yz (where z will be

190 Frank Thorne

chosen later in terms of y). In effect we are constructing two Maier matrices, a

“good” matrix (the smaller one, where Nb < yz) and a “bad” one. We then prove

that nearly all of the primes in the matrix are� a .mod q/, where “good” primes

� a .mod q/ are counted only in the good matrix, and “bad” primes 6� a .mod q/

are counted in the larger bad matrix.

There are two more important ingredients in the proof. The first is an appropri-

ate version of the prime number theorem for arithmetic progressions, valid when

the relative size of Q and i is as in (3.3). We cannot expect to prove such a re-

sult for all Q, but we can for a large class of moduli Q, as defined in (3.4). The

starting point is a zero-density estimate for Hecke L-functions proved by Fogels

[2], and we then follow techniques of Gallagher [3] and Shiu to obtain our result.

(We remove the prime p0 to ensure that the associated L-functions do not have

any Siegel zeroes.)

The last ingredient in the proof is a bit of combinatorial geometry. Using the

above techniques, we can prove that that some row of our Maier matrix is a pair

of concentric balls in the complex plane, such that the inner ball contains many

more good primes than the outer ball has bad. We must now prove that this bubble

contains a sub-bubble containing many good primes and no bad ones.

To do this we rely on the existence of a Delaunay triangulation. The Delaunay

triangulation of a set of points has the property that no point in the triangulation

is inside the circumcircle of any triangle. We take our set of points to be the set

all bad primes within the outer ball, as well as a regular 7-gon (of a certain radius)

outside the inner ball but inside the outer one. The circumcircles associated to

the Delaunay triangulation contain all of the good points and none of the bad,

and the number of such circumcircles is easily bounded from above. Moreover,

any circumcircle intersecting the inner ball can be proved to lie entirely within

the outer ball. The circumcircle containing the most good primes is therefore our

bubble of congruent primes.

4 Concluding Remarks

In the first place, we would like to discuss some additional results which we do not

have the space to fully describe here. In particular, Granville and Soundararajan

[5] recently generalized Maier’s theorem and proved that similar irregularities

occur in any arithmetic sequence. Here an “arithmetic sequence” is any sequence

A of integers, such that for all integers d coprime to some ‘bad’ modulus S , the

proportion of elements of A divisible by d is asymptotic to h.d/=d , where h.d/

is a multiplicative function h.d/ taking values in Œ0; 1�. It is also assumed that a

suitable weighted average of h.p/ is sufficiently smaller than 1.


Examples of such sequences include the primes and arbitrary subsets thereof,

almost primes, sums of two squares, norms of algebraic integers from extensions

of Q, and many other interesting sequences. Granville and Soundararajan’s main

result is then that any such sequence cannot be uniformly distributed in both short

intervals and arithmetic progressions to somewhat large moduli. To prove their

result they combine a generalized Maier matrix construction with a detailed anal-

ysis of oscillation in arithmetic functions (such as the function !.C/ occurring

in (1.2)).

In [13], the present author translated their mechanism to FqŒt �. In brief, the

method works. In particular we obtained several results on general arithmetic se-

quences in FqŒt �, exactly along the lines suggested by Granville and Soundarara-

jan’s work. Furthermore, in some cases we were be able to be quite precise about

where irregularities occur, proving (for example) that they occur among the poly-

nomials of every sufficiently large degree.

We conclude with a few remarks about some related work and some questions

that remain. Recently, Pollack [7] has proved an FqŒt � version of the quantita-

tive Bateman–Horn conjecture (which implies the Hardy–Littlewood prime tuple

conjecture as a special case), valid when q is coprime to 2n and large in relation

to n. Conversely, Conrad, Conrad, and Gross [1] have found a global obstruc-

tion to a somewhat different version of this conjecture. This obstruction is related

to a certain average of the Möbius function, and these authors propose a revised

conjecture based on geometric considerations as well as numerical calculations.

Finite extensions of FqŒt � are naturally associated to algebraic curves, and one

wonders whether the geometry of these curves may have additional consequences

for the distribution of primes.

In the number field case we have only scratched the surface, and one could

hope to prove all sorts of additional results. For example, one might ask whether

one could prove a result similar to Theorem 2.2 for any number field. The state-

ment of such a result might be somewhat more involved, but certainly we believe

that the proof should generalize.

One might also ask whether irregularities of the form (1.1) can be proved to

exist in number fields. The norms of primes are already known to be irregularly

distributed, as these form an arithmetic sequence in the sense of [5]. But nothing

has yet been proved about these sequences themselves. We are optimistic that

techniques similar to those discussed in this article should yield interesting results.

Acknowledgments. This work was completed at the University of Wisconsin,

where the author was a Ph.D. student. I would like to thank the very many people

who read my papers and who listened to my talk at Integers Conference 2007

and elsewhere, and who made many useful suggestions. I would also like to again

192 Frank Thorne

point out the excellent survey articles of Granville [4] and Soundararajan [9], from

which I learned much.

References

[1] B. Conrad, K. Conrad and R. Gross, Prime specialization in genus 0, Trans. Amer.

Math. Soc. 360 (2008), 2867–2908.

[2] E. Fogels, On the zeros of L-functions, Acta Arith. 11 (1965), 67–96.

[3] P. X. Gallagher, A large sieve density estimate near � D 1, Invent. Math. 11 (1970),

329–339.

[4] A. Granville, Unexpected irregularities in the distribution of prime numbers, in: Pro-

ceedings of the International Congress of Mathematicians (Zürich, 1994), pp. 388–

399, Birkhäuser, Basel, 1995.

[5] A. Granville and K. Soundararajan, An uncertainty principle for arithmetic se-

quences, Ann. of Math. 165 (2007), no. 2, 593–635.

[6] H. Maier, Primes in short intervals, Michigan Math. J. 32 (1985), 221–225.

[7] P. Pollack, Simultaneous prime specializations of polynomials over finite fields, ac-

cepted for publication in Proc. London Math. Soc.

[8] D. K. L. Shiu, Strings of congruent primes, J. London Math. Soc. 61 (2000), 359–

373.

[9] K. Soundararajan, The distribution of prime numbers, in: Equidistribution in Num-

ber Theory. An Introduction, pp. 59–83, NATO Sci. Ser. II Math. Phys. Chem. 237,

Springer, Dordrecht, 2007.

[10] N. Tanner, Strings of consecutive primes in function fields, accepted for publication

in Int. J. Number Theory.

[11] F. Thorne, Irregularities in the distribution of primes in function fields, J. Number

Theory 128 (2008), 1784–1794.

[12] F. Thorne, Bubbles of congruent primes, submitted.

[13] F. Thorne, An uncertainty principle for function fields, submitted.

Author information

Frank Thorne, Department of Mathematics, University of South Carolina,

1523 Greene Street, Columbia, SC 29208, USA.

Current address:

Department of Mathematics, Stanford University,

450 Serra Mall, Stanford, CA 94305, USA.



Linear Equations Involving Iterates

of �.N /

Tomohiro Yamada

Abstract. We study integers N satisfying the equation �.�.N // D A�.N/C BN .

Keywords. Perfect numbers, superperfect numbers, sum of divisors.

AMS classification. 11A05, 11A25.

1 Introduction

We denote by �.N / the sum of divisors of N . N is said to be perfect if �.N / D

2N and multiperfect if �.N / D kN for some integer k. It is not known whether or

not an odd perfect/multiperfect number exists. There are many known conditions

which must be satisfied by such a number. But these results are far from answering

whether or not an odd perfect/multiperfect number exists.

Suryanarayana [7] called N superperfect if �.�.N // D 2N and Pomerance

[5] called N super multiply perfect if �.�.N // D kN for some integer k. More

generally, Cohen and te Riele [2] defined N to be .m; k/-perfect if � .m/.N / D

kN , where � .m/.N / denotes �.� .m�1/.N // with � .0/.N / D N .

We introduce another analogous notion of perfect/multiperfect numbers. We

say that N is .nI a0 W : : : W an/-perfect ifPn

iD0 an�i�.i/.N / D 0 and .nI a1; : : : ;

an/-perfect if N is .nI �1 W a1 W : : : W an/-perfect. In particular, N is .2IA;B/-

perfect if �.�.N // D A�.N/C BN .

We begin by noting that almost all integers are .2IA;B/-perfect for some in-

tegers A;B . This fact follows from Katai and Subbarao [3, Theorem 1]. They

proved that for any fixedm � 1, we have .N; � .m/.N // DQ

p;pajjN pa, where p

runs over all primes below xm2 , for all integers N � x with o.x/ exceptions (Here

we use the notation x0 D x; xiC1 D max¹1; log xiº for all i � 0, where log x

is the natural logarithm of x, introduced in [4]). Hence, for almost all integers

N , .N; �.N // divides � .2/.N /, which implies that there exist some integers A;B

such that �.�.N // D A�.N/C BN .

On the other hand, we can show that for any fixed positive integers A;B , the

set of .2IA;B/-perfect numbers has density zero.

194 Tomohiro Yamada

Theorem 1.1. Let A;B;C be integers not all zero and satisfying AC � 0. Then

the number of .2IA WB WC/-perfect numbers below x is at most x exp.�.2�1=3 C

o.1//.log x/1=3.log log x/2=3/.

Our argument is similar to the argument of Pomerance [6] to study the distri-

bution of integers n satisfying �.n/ � a .mod n/. His argument rests on the fact

that almost all integers n can be written as mp, where p − m is prime, large and

uniquely determined by m except some special cases. Our argument adopts a fac-

torization of p C 1 into lq with q large and .l; q/ D 1, to show that q is uniquely

determined by m and l under some condition.

We note that this result does not seem to apply to the caseAC > 0. If p; 2p�1

are both prime, then n D 2p�1 satisfies 2�.�.n//�3�.n/C6n D 0 and therefore

n is .2I 2 W �3 W 6/-perfect. More generally, we can easily confirm the following

result.

Theorem 1.2. Let R be an arbitrary positive integer. If p and N D Rp � 1 are

both prime, then N is .2IR W �.RC 1/�.R/ W R�.R//-perfect.

Corollary 1.3. If R is a k-multiperfect number and p;N D Rp � 1 are both

prime, then N is .2I k.RC 1/ W �kR/-perfect.

A well-known conjecture states that, for any even integer R, the number of

primes p < x with Rp � 1 also prime is asymptotically equal to cx=.log x/2

for some constant c > 0 depending on R, contrary to the above given estimate

O.x exp.�.2�1=3 C o.1//.log x/1=3.log log x/2=3//.

2 Notations and Preliminary Lemmas

We denote by P.n/; p.n/ the largest and smallest prime factor of n respectively.

For the positive real number x, let us denote x0 D x; xiC1 D max¹1; log xiº

as mentioned in the previous section. We denote by c some positive constant

not necessarily same at every occurrence. Furthermore, we denote by x; y; z real

numbers and we put u D log x= logy and v D logy= log z.

Lemma 2.1. Denote by ‰.x; y/ the number of integers n � x divisible by no

prime > y. If y > x21 , then we have‰.x; y/ < x exp.�.1Co.1//u logu/ as x; u

tend to infinity.

Proof. This follows from a well-known theorem of de Bruijn [1]. For details on

the distribution of integers free from large prime factors, we refer the readers to

[8, Chapter III. 5], where a simple proof of the lemma is also given.

Linear Equations Involving Iterates of �.N / 195

Lemma 2.2. Let

s.x; k/ DX

p�x;p��1 .mod k/

1

p: (1)

Uniformly in k and x � e2, we have

s.x; k/�x2

'.k/: (2)

Proof. This inequality can be immediately obtained using partial summation and

the Brun–Titchmarsh inequality. The complete proof is given in [4, Lemma 2].

Lemma 2.3. Let S.x/ D ¹n j n � x; pa j n for some p; a with pa > y; a � 2º:

Then we have #S.x/� xy�1=2.

Proof. Let ….t/ be the number of perfect powers below t . It is clear that ….t/ <

t1=2C t1=3C� � �C t1=k < t1=2Cct1=3 log t � t1=2, where k D b.log x/=.log 2/c.

Let us denote by p the smallest integer for which p > y and > 1.

Clearly we have #S.x/ � xP

p�x p� p . Since p p > y, we have by partial

summation

X

p�x

1

p p

�….x/

x�….y/

yC

Z x

y

….t/

t2dt � y�1=2: (3)

This proves the lemma.

Lemma 2.4. If y > z > 2x21 and v > x2, then the number of integers � x

divisible by some prime p � y with P.p C 1/ < z is at most x exp.�.1 C

o.1//v log v/.

Proof. The number of such integers is at most

X

y�p�x;P.pC1/<z

x

p� x

X

y�m�x;P.mC1/<z

1

m; (4)

where p and m respectively run over primes and integers satisfying the described

conditions. By partial summation, we find that the last sum is at most

‰.y; z/

yC

Z 2x

y

‰.t; z/dt

t2: (5)

Since we have ‰.t; z/=t < exp.�.1C o.1//v log v/ uniformly for t 2 Œy; 2x�

by Lemma 2.1, the last sum in (4) can be bounded from above by

exp.�.1C o.1//v log v/

1C

Z 2x

y

dt

t

!

: (6)

196 Tomohiro Yamada

This integral is O.x1/ D O.exp v/ since, by assumption, x2 < v. Thus we obtain

xX

y<m�x;P.mC1/<z

1

mD x exp.�.1C o.1//v log v/: (7)


3 Proof of Theorem 1.1

Let y; z be real numbers and put u D log x= logy and v D logy= log z. We

choose y; z later so that x > y > z > x21 ; v > x2 and u; v; y; z tend to infinity as

x does so.

Let

S1 D ¹n j n � x; P.n/ � yº (8)

and

S2 D ¹n j n � x; pa j n for some p; a with pa � z; a � 2º: (9)

We immediately obtain #S1 D O.x exp.�.1C o.1//u logu// by Lemma 2.1

and #S2 D O.x=z1=2/ by Lemma 2.3.

Denote by S3 the set of integers n � x not in S1 [ S2 which can be written

in the form mp, where p is a prime � y, m is an integer not divisible by p and

.�.m/; p C 1/ is divisible by some prime q � z.

Let n be an integer in S3 and write n D mp in the above way. Then q j �.ra/

for some prime r dividing m and some integer a with ra jj m. Since q � z, we

have a D 1 by the assumption n 62 S2. Hence, m is divisible by some prime r

congruent to �1 .mod q/.

Sincem � x=p, the number of integers n satisfying n D mp; q j .�.m/; pC1/

for some q � z is at most

X

q�z

X

p��1 .mod q/

X

r��1 .mod q/

x

pr�X

q�z

cxx22

q2�cxx2

2

z; (10)

by Lemma 2.2. Hence, we have #S3 D O.xx22=z/.

Denote by S4 the set of integers n � x divisible by some prime p � y with

P.p C 1/ < z or q2 j .p C 1/ for some q � z. By Lemma 2.4, the num-

ber of integers n � x divisible by some prime p � y with P.p C 1/ < z is

O.x exp.�.1C o.1//v log v//. The number of integers n � x divisible by some

prime p with q2 j .p C 1/ for some q � z is at most

xX

q�z

X

p�1 .mod q2/

1

p� cxx2

X

q�z

1

q2�cxx2

zx1=31

�x

z; (11)

by the assumption that z > x21 .

Linear Equations Involving Iterates of �.N / 197

Combining these estimates yield #S4 D O.x.1=zCexp.�.1Co.1//v log v///.

We may assume that at least one of A and C is nonzero since the equation

A�.�.N //C B�.N/C CN D 0 does not hold if exactly one of A, B and C is

nonzero. Now let us denote by T the set of .2IA W B W C/-perfect numbers n � x

belonging to none of Si .i D 1; 2; 3; 4/. We assume that n 2 T . Since n 62 S1[S2,

we have P.n/ > y and P.n/2 − n. Thus n can be expressed as n D mp, p > y

and p − m. Now it follows from n 62 S3 that .�.m/; p C 1/ has no prime factor

� z. Let Tm denote the set of such integers. Moreover, we write p C 1 D N1N2

in the way P.N1/ < z � p.N2/ and divide each of Tm into sets Tm;N1according

to the value ofN1. Since n 62 S4, we have pC1 is divisible by some primeQ � z

exactly once. By the definition of N1; N2, we have N2 D N3Q and Tm;N1can be

covered by sets Tm;N1;N3according to the value of N3.

We shall show each Tm;N1;N3consists at most one element. We have �.n/ D

�.m/.pC1/ and �.�.n// D �.M1N1/�.M2/�.N2/, whereM1;M2 are uniquely

determined by �.m/ D M1M2, P.M1/ < z � p.M2/. Furthermore, noting

that Q does not divide .p C 1/=Q by the assumption that n 62 S4, we obtain

�.N2/ D �.N3/.Q C 1/. Since A�.�.n// C B�.m/.p C 1/ C Cmp D 0, we

have

A�.M1N1/�.M2/�.N3/.QC 1/C .B�.m/C Cm/N1N3Q � Cm D 0: (12)

Denoting

C1 D A�.M1N1/�.M2/�.N3/; C2 D .B�.m/C Cm/N1N3; C3 D Cm; (13)

this equation can be written

.C1 C C2/Q D .C3 � C1/: (14)

Since AC � 0, we have C1 ¤ C3 and therefore Q can be uniquely determined as

Q DC3 � C1

C1 C C2

: (15)

This is the crucial point where we use the assumption AC � 0. The uniqueness

of Q yields that #Tm;N1;N3� 1 for any m;N1; N3.

By the definition of Tm;N1;N3, each element of T must belong to Tm;N1;N3

for

at least one triple .m;N1; N3/. Since N1N3 D .p C 1/=Q � .x=m C 1/=z �

2x=.mz/, we have

#T �X

m�x=y

X

N1

X

N3

1 �X

m�x=y

2x

mz�cxx1

z�

cx

z1=2: (16)

198 Tomohiro Yamada

Now we conclude that the number of .2IA;B/-perfect numbers� x is at most

#S1 C #S2 C #S3 C #S4 C #T , which is

O.x.z�1=2 C exp.�.1C o.1//u logu/C exp.�.1C o.1//v log v/// (17)

by those estimates for #Si ’s and #T given above.

In order to search the optimal estimate, we put log z D c1x1=31 x

2=32 ; log y D

c2x2=31 x

1=32 and we have u logu D .1 C o.1//c�1

2 x1=31 x

2=32 and v log v D .1 C

o.1//.c2=c1/x1=31 x

2=32 . We see that max¹c1=2; c2=c1; 1=c2º� 2�1=3 with the equal-

ity attained when we choose c1 D 22=3; c2 D 21=3. This choice gives the desired

estimate. This completes the proof of Theorem 1.1.

We remark that .2IR W �.R C 1/�.R/ W R�.R//-perfect numbers given in

Theorem 1.2 correspond to the case m D N1 D 1; C D A�.R/ D �.B C C/R.

Acknowledgments. The author thanks Prof. Florian Luca for his advice on

exceptional triples .A;B; C /.

References

[1] N. G. de Bruijn, On the number of positive integers � x and free of prime factors

> y, Nederl. Akad. Wetensch. Proc. Ser. A 54 (1951), 50–60.

[2] G. L. Cohen and H. J. J. te Riele, Iterating the Sum-of-Divisors Function, Experiment.

Math. 5 (1996), 91–100.

[3] I. Kátai and M. V. Subbarao, Some further remarks on the ' and � functions, Annales

Univ. Sci. Budapest. Sect. Comput. 26 (2006), 51–63.

[4] I. Kátai and M. Wijsmuller, On the iterates of the sum of unitary divisors, Acta Math.

Hungar. 79 (1998), 149–167.

[5] C. Pomerance, On multiply perfect numbers with a special property, Pacific J. Math.

57 (1975), 511–517.

[6] C. Pomerance, On the congruences �.n/ � a .mod n/ and n � a .mod '.n//, Acta

Arith. 26 (1975), 265–272.

[7] D. Suryanarayana, Super perfect numbers, Elem. Math. 24 (1969), 16–17.

[8] G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Cam-

bridge University Press, 1995.

Author information

Tomohiro Yamada, Department of Mathematics, Kyoto University,

Kyoto, 606-8502, Japan.



Heron Sequences and Their Modifications�

Paul Yiu, K. R. S. Sastry and Shanzhen Gao

Abstract. We study sequences whose consecutive terms determine Heron triangles.

With a few exceptions, we show that given two positive integers u < v, there is an

infinite sequence every three consecutive terms of which determine a Heron triangle. We

also construct arbitrarily long sequences every three consecutive terms of which are the

sides of a Heron triangle.

Keywords. Heron triangles, Pell equations, Fibonacci numbers.

AMS classification. 11D09, 51M04, 11B99.

1 Heron Sequences

Heron triangles are triangles with integer sides and integer areas. They have been

studied since ancient times. For some recent studies, see [1, 2, 3, 5, 6, 7]. It is clear

that the angles of a Heron triangle all have rational sines and cosines. Such are

called Heron angles, see for example [4]. Conversely, if the angles of a triangle

are all Heron angles, then an appropriate magnification yields a Heron triangle.

In this paper we study natural number sequences associated with Heron triangles.

One most natural and interesting question is whether there is an infinite increasing

sequence in which every three consecutive terms are the sidelengths of a Heron

triangle. While we do not know the answer to this question, we can prove that

such a sequence can be arbitrarily long. The proof makes use of the Fibonacci

numbers Fn defined recursively by FnC1 D Fn C Fn�1, F1 D 1, F2 D 1.

Theorem 1. Given an integer n � 3, there is a sequence

a1; a2; : : : ; an

every three consecutive terms of which are the sides of a Heron triangle.

Proof. Let � be a Heron angle such that FnC1� <�2

. Consider the points

Pk D .cos 2FkC2�; sin 2FkC2�/;

�Sequences from The Encyclopedia of Integer Sequences discussed in this paper: A003500,

A103977, A103974, A104009, A103772, A104008.

200 Paul Yiu, K. R. S. Sastry and Shanzhen Gao

for k D 1; : : : ; n C 1. These are rational points on the unit circle. For k D

1; : : : ; n, the length of the chord PkPkC1 is

ak D 2 sin2FkC3 � 2FkC2

2� D 2 sinFkC1� 2 Q:

Since FnC1� < �2

, the sequence of rational numbers a1; a2; : : : ; an is strictly

increasing. We claim that every three consecutive terms are the sides of a triangle

with rational area. To see this, note that for k D 3; : : : ; n, two of the sides of

triangle Pk�2Pk�1Pk have lengths ak�2 and ak�1. The length of the third side

Pk�2Pk is

2 sin2FkC2 � 2Fk

2� D 2 sinFkC1� D ak :

Since the vertices of the triangle Pk�2Pk�1Pk are on the unit circle, its area

is ak�2ak�1ak

4, a rational number. By clearing denominators, we obtain an n-

term sequence of integers every three consecutive terms of which form a Heron

triangle.

The numbers realizing the Heron triangles in the above construction increase

very rapidly in n. By a routine computer search beginning with two positive

integers< 1000, we have found that the longest Heron sequence contains 9 terms.

There are three such sequences. One of these is

sides 60 275 325 500 525 697 746 1345 1797

area 4950 41250 78750 130872 175644 175644 452844

This sequence has been recorded in the Encyclopedia of Integer Sequences as

sequence A134587.

The other two 9-term sequences are obtained by multiplying the sides in this

sequence by 2 and 3 respectively. Again, beginning with two integers < 1000,

there are two 8-term modified Hereon sequences:

sides 445 485 850 1095 1435 2516 2691 3505

area 80100 197100 464100 165648 1782270 3369960

sides 825 975 1500 1575 2091 2238 4035 5391

area 371250 708750 1177848 1580796 1580796 4075596

Heron Sequences and Their Modifications 201

2 Modified Heron Sequences

A common way to construct Heron triangles is to choose three integers u, v, w

such that

uvw.uC v C w/ D 42 (1)

for an integer4. The triangle with sides

a D uC v; b D uC w; c D v C w

is a Heron triangle with area 4. By a modified Heron sequence we mean an

increasing sequence .un/ of positive integers such that every three consecutive

terms un�2, un�1 and un determine a Heron triangle with sides

an D un�2 C un�1; bn D un�2 C un; cn D un�1 C un;

and integer area 4n. While it is not a priori obvious that a Heron sequence may

be infinite, we show that, apart from a few exceptions, an infinite modified Heron

sequence always results if one begins with two distinct positive integers.

Theorem 2. Let u < v be positive integers such that

.u; v/ ¤ .1; 4/; .1; 9/; .2; 8/; .2; 18/; .4; 16/: (2)

There is an infinite modified Heron sequence .un/ satisfying

(i) u�1 D u, u0 D v;

(ii) un is the smallest positive integer > un�1 such that the triangle determined

by un�2, un�1, and un is a Heron triangle.

Proof. Given two positive integers u < v, to find an integer w which together

with u and v gives a Heron triangle we solve (1) by rewriting it in the form

x2 � uv � y2 D �uv.uC v/2 (3)

by putting x D 24 and y D 2w C uC v.

Case I. uv not equal to a square. It is readily seen that .x; y/ D .0; u C v/

is a solution of (3). If uv is not the square of an integer, then there is an infi-

nite sequence of integer solutions generated by the fundamental solution of the

Fermat–Pell equation

x2 � uv � y2 D 1: (4)


Namely, if .a; b/ is the smallest positive solution of (4), then there is a sequence

.xn; yn/ of solutions of (3) given by

xnC1

ynC1

!

D

a uvb

b a

!

xn

yn

!

;

x0

y0

!

D

0

uC v

!

: (5)

Note that a � 3 except when uv D 3. In particular, y1 D a.uCv/ � 3.uCv/.

We can determine an integerw from y1 D 2wCuCv which satisfiesw � uCv >

v. For .u; v/ D .1; 3/, the solution y2 would give w > v.

Case II. uv equal to a square. If uv is a square, equation (3) reduces to

.2w C uC v/2 D .uC v/2 C z2 (6)

for some integer z. This requires u C v to be a shorter side of a Pythagorean

triangle whose hypotenuse is 2w C u C v for some integer w > v. Now, it

is well known (and easy to verify) that every integer � 3 is a shorter side of a

Pythagorean triangle. However, to ensure a hypotenuse of sufficient length, we

examine the details. Note that we require the hypotenuse to be 2wCuCv, which

has the same parity as uC v.

(i) If uC v D 4k for some integer k, consider the Pythagorean triangle

.4k; 2.k2 � 1/; 2.k2 C 1//:

If we set the hypotenuse to be 2w C 4k, then w D .k � 1/2. If k � 6, w D

.k � 1/2 > 4k > v. The only pairs .u; v/ which do not satisfy this condition are

.u; v/ D .2; 18/ and .4; 16/.

(ii) If uC v D 4k C 2 for some integer k, consider the Pythagorean triangle

.2.2k C 1/; .2k C 1/2 � 1; .2k C 1/2 C 1/:

If we set 2w C 4k C 2 D .2k C 1/2 C 1, then w D 2k2. If k � 3, then

w D 2k2 > 4kC 2 > v. The only pairs .u; v/ which do not satisfy this condition

are .1; 9/ and .2; 8/.

(iii) If uC v D 4k C 1 for some integer k, consider the Pythagorean triangle

.4k C 1; 4k.2k C 1/; .2k C 1/2 C .2k/2/:

If we set 2wC4kC1 D .2kC1/2C .2k/2, then w D 4k2. If k � 2, w D 4k2 >

4k C 1 > v. The only pair .u; v/ which does not satisfy this condition is .1; 4/.

(iv) If uC v D 4k C 3 for some integer k, consider the Pythagorean triangle

.4k C 3; 4.k C 1/.2k C 1/; .2k C 1/2 C .2k C 2/2/:

Heron Sequences and Their Modifications 203

If we set 2w C 4k C 3 D .2k C 1/2 C .2k C 2/2, then w D .2k C 1/2, which

always exceeds 4k C 3 for k � 1.

It follows that if uv is a square, then apart from the pairs in the list (2), there

exists an integer w > v such that u, v, w determine a Heron triangle.

For .u; v/ given in the list (2), it is routine to check that Pythagorean triangle

in (6) is an integer multiple of .5; 12; 13/. In each case, the corresponding value

of w is smaller than v.

Therefore, beginning with .u�1; u0/ which is not one of the pairs in the list

(2) we can determine inductively for n � 1, un as the smallest integer > un�1

such that un�2, un�1, un determine a Heron triangle. The sequence so obtained

is infinite.

Remark. The above constructions may or not give the smallest possible w in

question. However, the existence of the smallest w > v is guaranteed.

We conclude with some examples of modified Heron sequences each deter-

mined by two initial entries. The first three have been recorded in the Encyclope-

dia of Integer Sequences

EIS number Modified Heron sequence

A134588 1; 2; 3; 10; 27; 98; 120; 327; : : :

A134589 1; 3; 12; 49; 108; 243; 624; : : :

A134590 2; 5; 63; 112; 140; 315; 364; : : :

6; 7; 8; 27; 70; 750; 972; : : :

The Heron triangles determined by the last sequence are

.13; 14; 15I 84/; .15; 34; 35I 252/; .35; 78; 97I 1260/;

.97; 777; 820I 34650/; .820; 1042; 1722I 302400/; : : : :

Acknowledgments. The authors thank the referee for instructive comments

leading to improvements on the presentation of this paper.

References

[1] A. Bremner, On Heron triangles, Ann. Math. Inform. 33 (2006), 15–21.

[2] R. H. Buchholz and R. L. Rathbun, An infinite set of Heron triangles with two rational

medians, Amer. Math. Monthly 104 (1997), 107–115.

[3] R. H. Buchholz and R. L. Rathbun, Heron triangles and elliptic curves, Bull. Austral.

Math. Soc. 58 (1998), 411–421.


[4] K. R. S. Sastry, Heron angles, Math. Comput. Ed. 35 (2001), 51–60.

[5] K. R. S. Sastry, Two Brahmagupta problems, Forum Geom. 6 (2006), 301–310.

[6] P. Yiu, Construction of indecomposable Heronian triangles, Rocky Mountain J. Math.

28 (1998), 1189–1202.

[7] P. Yiu, Heronian triangles are lattice triangles, Amer. Math. Monthly 108 (2001), 261–

263.

Author information

Paul Yiu, Department of Mathematical Sciences, Florida Atlantic University,

Boca Raton, Florida 33431-0991, USA.


K. R. S. Sastry, Jeevan Sandhya, Dodda Kalsandra Post,

Raghuvana Halli, Bangalore, 560 062, India.


Shanzhen Gao, Department of Mathematical Sciences, Florida Atlantic University,

Boca Raton, Florida 33431-0991, USA.


List of participants

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Andrews, George

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Siqueira

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Boumenir, Amin

Boylan, Matthew

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Corcino, Roberto B.

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Hopkins, Brian

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Keith, William J.

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