m.n. huxley-the distribution of prime numbers_ large sieves and zero-density theorems (oxford...

7/23/2019 M.N. Huxley-The Distribution of Prime Numbers_ Large Sieves and Zero-Density Theorems (Oxford Mathematical

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O X F O R D M A T H E M A T IC A L M O N O GR A PH S

M E R O M O E P H I C F U N C T IO N S

By w. k . h a x m a n . 1963

T H E T H E O R Y OF L A M I N A R B O U N D A R Y L A Y E R S

I N C O M P R E S S I B L E F L U I D S

By K. SIEWASTSON. 1964

C L A SS IC A L H A R M O N I C A N A L Y S I S A N D

L O C A L L Y C O MP AC T G R O U P S

By H. EBITER. 1968

Q U A N T U M - S T A T I ST I C A L F O U N D A T I O N S OF

C H E M I C A L K I N E T I C S

By s. g o l d e n . 1969

C O M P L E M E N T A R Y V A R I A T I O N A L P R I N C IP L E S

By a . m . a e t h u b s . 1970

V A R I A T I O N A L P R I N C I P L E S I N H E A T T R A N S F E R

By MAURICE A. BIOT. 1970

P A R T I A L W A V E A M P L I T U D E S A ND R E S O N A N C E PO L ES

By J. Ha m i l t o n and b . t r o m b o r g . 1972


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THE DISTRIBUTION OF

PRIME NUMBERS

L a r ge si eves an d zer o -d en si t y t h eo r em s

B Y

M. N. H U X L E Y

O X F O R D

AT T H E C L A R E N D O N P R E SS

1972


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Oxford University Press, Ely House, London W. 1

GLASGOW NEW YOR K TORONTO MELBOURNE WELLINGTON

CAPE TOWN IBADA N NAIRO BI DAK ES SALAAM LUSAKA ADDIS ABA llA

DELHI BOMBAY CALCUTTA MADRAS KARACHI LAHORE DACCAKUALA LUMPUR SINGAPORE HONG KONG TOKYO

Oxford University Press 1072

Printed in Great Britainat the University Press, Oxford

by Vivian MidlerPrinter to the University


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TO

T H E M E M O R Y O F

P R O F E S S O R H. D A V E N P O R T


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6/138

P R E F A C E

T h i s book has grown out of lectures given at Oxford in 1970 and at

University College, Cardiff, intended in each ease for graduate students

as an introduction to analytic number theory. The lectures were based

on Davenport sMultiplicative Number Theory,but incorporated simpli

fications in several proofs, recent work, and other extra material.

Analytic number theory, whilst containing a diversity o f results, has

one unifying method: that o f uniform distribution, mediated by certain

sums, which may be exponential sums, character sums, or Dirichlet

polynomials, according to the type of uniform distribution required.

The study o f prime numbers leads to all three. Hopes o f elegant asym

ptotic formulae are dashed by the existence of complex zeros of the

Riemann zeta function and o f the Dirichlet L-functions. The prime-

number theorem depends on the qualitative result that all zeros have

real parts less than one. A zero-density theorem is a quantitative result

asserting that not many zeros have real parts close to one. In recent

years many problems concerning prime numbers have been reduced to

that of obtaining a sufficiently strong zero-density theorem.The first part of this book is introductory in nature; it presents the

notions o f uniform distribution and o f large sieve inequalities. In the

second part the theory o f the zeta function and L-functions is developed

and the prime-number theorem proved. The third part deals with large

sieve results and mean-value theorems for L-functions, and these are

used in the fourth part to prove the main results. These are the theorem

of Bombieri and A. I. Vinogradov on primes in arithmetic progressions, a

result on gaps between prime numbers, and I. M. Vinogradovs theorem

that every large odd number is a sum of three primes. The treatment is

self-contained as far as possible; a few results are quoted from Hardy

and Wright (1960) and from Titchmarsh (1951).

Parts of prime-number theory n ot touched here, such as the problem

of the least prime in an arithmetical progression, are treated in Prachars

Primzahlverteilung (Springer 1957). Further work on zero-density

theorems is to be found in Montgomery (1971), who also gives a wide list

o f references covering the field.

M. N. H.Cardiff

1971


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C O N T E N T S

P A R T I. I N T R O D U C T O R Y R E S U L T S

1. Arithmetical functions 1

2. Some sum functions 6

3. Characters 10

4. Polya s theorem 14

5. Dirichlet series 18

6. Schinzels hypothesis 23

7. The large sieve 28

8. The upper-bound sieve 32

9. Franels theorem 36

P A R T I I . T HE P R I M E - N U M B E R T H E O R E M

10. A modular relation 4011. The functional equations 45

] 2. HadamarcPs product formula 50

13. Zeros of(s) 55

14. Zeros of{s, x) 58

15. The exceptional zero 61

16. The prime-number theorem 66

17. The prime-number theorem for an arithmetic progression 70

P A R T I II . T H E N E C E S S A R Y T O O LS

18. A survey of sieves 73

19. The hybrid sieve 79

20. An approximate functional equation (I) 84

21. An approximate functional equation (II) 89

22. Fourth powers of ^-functions 93


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P A E T I V . Z E R OS A N D P R I M E N U M B E R S

23. Inghams theorem 98

24. Bombieri s theorem 103

25. I. M. Vinogradovs estimate 107

26. I. M. Vinogradovs three-primes theorem 110

27. Halaszs method 114

28. Gaps between prime numbers 118

X C O N T E N T S

N O T A T I O N

B I B L I O G R A P H Y

I N D E X

123

124

127


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P A R T I

Introductory Results

1

A R I T H M E T I C A L F U N C T I O N S

An Expotition . . . means a long line of everybodyI. 110

T h i s chapter serves as a brief resume o f the elementary theory o f prime

numbers. A positive integer m can be written uniquely as a product

o f primes m _ (1,1)

where th e ^ are primes in increasing order o f size, and the aiare positive

integers. We shall reserve the letter p for prime numbers, and write a

sum over prime numbers as 2 ancl a product as JT- The proof ofp i>

unique factorization rests on Euclids algorithm that the highest com

mon factor (m, n)of two integers (not both zero) can be written as

(m,n) = mu+nv, (1.2)

where u, v are integers. We use (m, n) for the highest common factor

and \m, n\ for the lowest common multiple of two integers where these

are defined.

Let # be a positive integer. Then the statement that m is congruent

to n (mod#), written m = n(mod#), means that m nis a multiple of q.

Congruence m od qis an equivalence relation, dividing the integers into

q classes, called residue classes mod#. A convenient set o f representa

tives of the residue classes mod q is 0, 1, 2,..., qI. The residue classes

mod# form a cyclic group under addition, and the exponential maps

m->eQ(am),

where a is a fixed integer, and

(1.3)


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2 I N T R O D U C T O R Y R E S U L TS 1.1

e(a) = exp(27ria), eg(a) = exp(27ria/g), (1.4)

are homomorpliisms from this group to the group of complex numbers

o f unit modulus under multiplication. There are qdistinct maps, corre

sponding to a = 0, 1, 2,..., q 1. They too can be given a group structure,

forming a cyclic group of order q. They have the important property

(i-5>

where the summation is over a complete set of representatives of the

residue classes mod# (referred to briefly as a complete set of residues

mod#). I f on the left-hand side of eqn (1.5) we replace to by to + 1 , the

sum is still over a complete set of residues, but it has been multipliedby ea(a),which is not unity unless a = 0 (mod#). The sum is therefore

zero unless a = 0 (modg), when every term is unity. Interchange of a

and to leads to a corresponding identity for the sum of the images of m

under a complete set o f maps (a 0, 1,..., q1). These identities arise

because the images lie in a multiplicative not an additive group.

From Euclids algorithm comes the Chinese remainder theorem: if

to, nare positive integers and (m, n) = 1, then any pair o f residue classes

a (mod to) and b (mod?i) (which are themselves unions o f residue classesmodwm) intersect in exactly one class c (modtow), given by

c = bmu-\-anv (mod tow) (1,6)

in the notation o f eqn (1.2). Now let /(to) be the number of solutions

(ordered sets (x1,...,xr)of residue classes) of a set of congruences

gi{xi,.. .,xr) = 0 (modto), (1.7)

where thegiare polynomials in xrwith integer coefficients. When

(m,n) = 1, gi(x1>.. .,xr) is a multiple of mni f and only if it is a multiple

both of to and of n. Hence

f(m n) = f(m )f(r i) whenever (m ,n ) = 1. (1.8)

Equation (1.8) is the defining property of a multiplicative arithmetical

function. An arithmetical function is an enumerated subset of the

complex numbers, that is, a se q u e n ce /(l ), /(2),... of complex numbers.

The property f (m n )= f(m )f (n ) (1.9)

for all positive integers mand nseems more natural; if eqn (1.9) holds

as well as (1.8) then/(to) is said to be totally multiplicative, but (1.8) is

the property fundamental in the theory.


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1.1 A R I T H M E T I C A L F U N C T IO N S 3

The Chinese remainder theorem enables us to construct more compli

cated multiplicative functions. W e call a residue class a(mod#) reduced

if the highest com mon factor (a, q) is unity. A sum over reduced residue

classes is distinguished by an asterisk.With this notation we introduce Eulers function


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4 I N T R O D U C T O R Y R E S U L T S 1.1

a totally multiplicative periodic function, which is called a Dirichlet s

character mod#, or more briefly a character. Characters can be defined

as those totally multiplicative functions that are periodic. Since negative

integers also belong to well-defined residue classes mod q,we can speako f x(m) when is a negative integer; in particular, we shall refer to

X( !)It is possible to build new multiplicative functions from old. W e say

that d divides m, written d|to, when the integer to is a multiple o f the

positive integer d ; another paraphrase is dis a divisorof to . (Note that

the divisors o f 6 are 1, 2, 3, 6.) Nowlet/(TO) andgf(TO) be multiplicative.

Then so are the arithmetical functions

h(m) = f(m)g(m), (1.19)

H>) = 2 M , (1-20)d\m

and h(m) = '2,f(d)g(mjd). (1-21)d\m

We shall consider eqn (1.21), since (1.20) is a special case, and (1.19) is

evident. When (m,n) 1, the divisor do f mncan be written uniquely

as d = ctb, where a|to and b \n, and (a, b) = 1. Hence

h(mn) 2 f(^ )simnld)d\mn

= H 2 f ( ab)g(mnlab)a\m b\n

= 2 ( i -22)a\m b\n

which is h(m)h{n) as required. Thus

d(m) = 2 1 . (1.23)d\n

the number of divisors of t o , and

(m) = 2,d, (1.24)d\m

the sum of the divisors of m, are multiplicative functions.

We can invert eqn (1.20) and return from h(m) to f(d ) by using

Mobius s multiplicative function fi(m), defined by

/x(l) = 1

n(p) = 1 for primesp }. (1.25)

p,(pa) = 0 for prime powersp awith a > ]


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1.1 A R I T H M E T I C A L F U N C T I O N S . 5

I f the positive integer m factorizes according to (1.1), then

2 p(d) = XI + ')}d\m i

= n a - i ) = o, (1.26)i

unless m = 1, when the product in eqns (1.1) and (1.26) is empty. We

have now proved the following lemma.

L e m m a . I f m is a positive integer, then

2 ^ ) = ( 1 % m = 1 > (1.27)cm ' |0 t f m > 1. K '

From the lemma we have the corollary:

C o r o l l a r y . I f h(m) and f(m) are related by eqn (1.20), then

f ( n) = 2 l*(m)h(nlm), (1.28)? n \ n

and if eqn (1.28) holds then so does eqn (1.20).

To prove the corollary we substitute as follows.

2 ji(m)h(njm) = 2 i (w ) I f (d)m\n ni\n d\{nfm )

= 2 m 2 K m) (i-29)d\n m\(nld)

when we interchange orders of summation. The inner sum is zero by

eqn (1.27), unless d = n,when only one term/(cZ) remains. The converse

is proved similarly.

We can also define an additive functionto be an arithmetical function

/(to ) with f(m n )= f(m )-\ -f(n ) when (in, n) 1. (1.30)

The simplest examples are log to and the number o f prime factors o f m.

There are useful arithmetical functions that are neither multiplicative

nor additive. W e shall make much use of A(m),given by

/[im\ i fmis a prime powerp a, a ^ 1, (131)|0 if mis not a prime power.

It satisfies the equation2 A(d) = log to. (1.32)

d\ m

We could have used eqn (1.32) to defineA(m)and recovered the defini

tion (1.31) by Mobiuss inversion formula (1.28).


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2

S O M E S U M F U N C T I O N S

T h e study o f the sum functions o f arithmetical functions is important

in analytic number theory. For instance, we shall treat many o f the

properties o f prime numbers by using the sum function

>ft(x) = 2 A(m). (2.1)

Our object is to express the sum function as a smooth main term (a power

o fxor o f logx,for example) plus an error term. In place o f the cumber -

SOme l/(OI = 0{cj(x)), (2.2)

we shall often write f (x ) g(x), (2.3)

and other asymptotic inequalities similarly. Some sum functions can be

estimated by writing the arithmetical function as a sum over divisors

and rearranging. In this chapter we shall give examples o f this method.From the theory of the logarithmic function we borrow the relation

where y is a constant lying between \ and 1. W e deduce the useful

formulaM ,

]T - = l o g ( M + l ) + y + 0 ( M ~ i ) . (2.5)111= 1

Our first example is an asymptotic formula for

() = 2 9 () (2.6)

as x tends to infinity. Since 9 (m) is the number of integers r with

1 ^ r ^ m and (r,m) = 1, eqn (1.27) gives

( )


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m

Hence 9 (to) = 2 2 (2.8)r=1 d|m

fZ|r

and 80 m = 2 2 2 mWm


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terra than in (2.12); in fact a little cunning enables us to improve

(2.13). W e let y be the positive integer for which y 2 ^ x ^ (? /+ l)2,

and write m = qr in eqn (2.12), so that

D(x) = 2 1QV X

= 2 2 i + 2 2 i - 2 2 iq^y r^xla r^yq^x/r q^y>'


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We have chosen to compare cl2(m) with cZ4(to) since, when m is a prime

powerp a, di (m) = J(a+ l) ( a+ 2)(a + 3), (2.19)

which is equal to d2(m)if a = 0 or 1. The next step is to find a function

b(m) for which d2(m) = ^ ch{u)b{v)' (2.20)

u v = m

Since \(a-\-l)(aJr 2)(aJr '&) l)a (a + l) = (a + 1 )2, (2.21)

the choice 6(1) = 1, b(p2) = 1, b(pa) = 0, for prime powers p a with

a not zero or two, satisfies eqn (2.20) when m is a prime power. I f we

complete the definition of b(m) by making it multiplicative, then

(2.20) holds for all m. The choice is thus

6(m) = H n) = (2.22)(0 if m is not a perfect square.

We now complete the proof. Equations (2.20) and (2.22) give

d2(m) __ ^ d{u)b{v)

1.2 SOME SUM FU NC TIO NS 9

m -4 uvuv^x

__. t2 Z-4 nuK;Xjt2

co

When we substitute (2.18) and the value 67r~2 o f 2 i11 0 this, wehave 70, , . , . 1

2 = ^ 2 + o (l))log% . (2.24)W \4t72 )

We require (2.24) and upper estimates for similar sums in the later work.

Any estimate for a sum involving divisor functions that we quote

will be a corollary o f (2.14) or o f (2.24), possibly using partial summation.

The method we employ in this chapter can be summarized as follows.

To work out a general sum functionF ( x ) = 2 / ( m), (2.25)

we try to write f(m ) = 2 a,(u)b(v), (2.26)u v = m

where we have an asymptotic formula for the sum function o f the a(m),

and b(m)is in some sense smaller than a(m). A necessary condition is that

b(p) = o{\a(p)|). (2.27)

Our determination of &(x) depended on the equation


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3

C H A R A C T E R S

T h e reduced residue classes mod # form a group under multiplication,

and the characters mod# correspond to the maps to->- x(m ) from this

group to the group of complex numbers of unit modulus under multi

plication. We define a group operation on the set of the cp(q) characters

mod#: the product X1X2 f two characters xi and Xzmod# is the map

m -* Xi(m)X2.{m)- The unit o f this group is the trivial character mod#.When the group of reduced residues and the group of characters are

each expressed as a direct sum of cyclic groups, we can see that they

are isomorphic and that the homomorphisms o f the group o f characters

to the complex numbers are given by %-> x(m)>where mruns through

the reduced residues mod#. Two finite Abelian groups related in this

way are said to be dual.

We have already seen that

2 ( ) = ( = (3.Dmmodg \0 11 Cl 0 (HIOQ^),

for the maps to h*eg(am) from the group o f residue classes to (mod #)

under addition to the complex numbers of unit modulus. W e have

similar results for the characters xim)mod#:

2 x M = l 9(q) ^ istrivial> (3.2)mmodg (0 if ^ is non-trivial.

Here x(m)is non-zero when the residue class to (mod#) is reduced, and

thus when mis a member o f the group o f reduced residue classes under

multiplication. Now eqn (3.1) has a dual interpretation, in terms of

residue classes a (mod#) and maps a -> ea(am). This corresponds to

T v im ) = (9{q) if m= 1 (m od2)> /o o\X'mod2 (o i f to ^ 1 (mod #).

The proof o f eqn (3.2) runs as follows. I f x is trivial, then x (to) = 1

when (to, #) = 1 and 0 otherwise, so that the sum on the left-hand sideo f (3.2) is the definition (1.10) of


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1.3 C H A R A C T E R S 11

a complete set o f residues mod#. Since (r,#) = 1, mralso runs through

a complete set o f residues mod#, and tlie sum is unchanged. This

contradicts the choice of rwithx{f ) ^ lifthe su m isno tze ro . Similarly,

in eqn (3.3) if mis not congruent to unity m od# there is a character xi

with Xi(m) 1- When we multiply the left-hand side of (3.3) bj^

2 x(m)xi(m) is a sum over all characters mod#, and again this summust be zero, for we have multiplied by a constant that is not xuiity

but have succeeded only in permuting the terms of the sum. Of course,

eqns (3.1), (3.2), and (3.3) are essentially special cases of a theorem on

dual finite Abelian groups.

An important notion is theproprietyof characters. Let #2be a multiple

of qv and xi a character m od#x. The group o f reduced residue classes

mod #2 maps homomorphically onto the corresponding group m od#x,and we define a character x%mod #2 by the equation

We note that xi and x% are different arithmetical functions. I f #x = 3,

#a = 6, and xi takes the values

since Xzis zero when mis a multiple o f 2 as well as when m= 0 (mod 3).

When Xi is constructed by eqn (3.4), we say that ^1 m od#1 induces

Xzmod #2, and, if #2 ^ #1( that y2 m od#2 is improper. A. proper character

mod # (also called a primitive character) is one that is not induced by

a character mod dfor any divisor d o i q other than # itself. The smallest

/ for which a character Xim od / induces xmod # is called the conductor

o f x The customary letter / is the initial of a German word for a tram

conductor.

We shall now discuss Gauss's sum r(x), defined by

In this curious, expression, the factors correspond to the multiplicative

group o f reduced residues mod # and the additive group of all residues

mod#. The absolute value of r(x) is found below. Gauss (preceding

Dirichlet) considered onty characters xfor which x(m)takes the valuesrb 1 and zero only. In this case r2is real, but it is still not easy to find

the argument of r(x).

(3.4)

1, - 1, 0, 1, - 1, 0

for m = 1, 2, 3, 4, 5, 6, then %2 takes the values

1, 0 , 0 , 0 , 1, 0,

(3.5)

(3.6)

t(x) = 2 x(m)eg(m).m mo cl g

(3.7)


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12 I N T R O D U C T O R Y R E S U L TS 1.3

W e shall use t (x ) to remove characters from a summation. With x ( m )

denoting the complex conjugate of x(m) (the inverse of xhi the group

of characters), eqn (3.7) gives

r(x)x(m) = 2 x(aK{am) (3.8)flmodg

whenever (to,#) = 1. I f r(x) is non-zero, we can use (3.8) to change a

summation over x(m)to a summation over the exponential mapsea(am),

which are easier to manipulate. A defect in eqn (3.8) is the condition

(to, q) = 1. If, however, xis proper mod qeqn (3.8) holds for all integers

m. W e must show that the sum on the right-hand side of (3.8) is zero

when m = tn, q = tr, t > 1. (3.9)

In this case, 2 xia)eq(am) = 2 xia)er{an)- (3.10)amoda amodg

Since % is not induced by any character modr, there is an integer b

with (b,q) = 1, b = 1 (modr), but x(b) ^ 1- Our standard proof now

applies. Multiplication by x(b)permutes the residue classes in the sum

on the left o f eqn (3.10), but multiplies the value o f the sum by a constant

that is not unity. The sums in (3.10) and (3.8) are thus zero ifx is proper

mod q and (to, q) > 1.

For eqn (3.8) to be o f use we must be sure that t (% ) is non-zero. When

Xis proper mod qthere is an elegant demonstration. For each mmod q,

|x(to)t(x)|2 = 2 2 x()x(6)ea(mM - (3.11)a m ods 6 mods

EE0IIC6

2 \x(m)?\T(x)? = 2 2 x(a)x(b) 2 ea(ambm). (3.12)w m odff am od g &mod 1. I f the lowest common multiple h = [/,g] o f fandgis not

# itself, x is not merely induced by some character Xa,mo(i but is

actually equal to X2>since any integer prime to h is already prime tofg q. Replacement o f m in the sum in eqn (3.7) by m-\-h permutes

the residue classes mod#, but multiplies r ( x ) b yeq(h),which is not unity.

In this case therefore r(x) is zero.


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1.3 C H A R A C T E R S 13

When (f,g) = 1 we invoke the Chinese remainder theorem. By

eqn (1.2) there are integers u, vwith

fu + g v = 1. (3.15)

Residue classes a (modfg) correspond to pairs of classes b (mod/),c (mod#) according to the relation

a = cfu-^bgv (modfg). (3.16)

In (3.16), a (modfg)is a reduced class if and only if both b (mod/) and

c (mod#) are reduced, and thus

H x ) = 2 * x ( a ) e a (a )amodtf

= 2* 2* X{cfu+bgv)ea(cu)ef{bv)bmodf cnioclf/

= 2 * x ( b ) e f ( b v ) 2 * e f, Mbmod/ cmodtf

= Xi(v)r(xiK(u)} (3.17)

where xi mod/ is the character inducing %mod #, and cy(u) is Rama

nujans sum, defined in eqn (1.11).

W e now proceed to compute Ramanujans sum. By eqns (1.11) and

^ '27 c(?0 = 2 eu(au) X ^ ( d)amod{/ d\a

mu

= 2P'W 2 %ld{bu), (3.18)d\g bmo&gld

where we have written a = bd. From eqn (3.1) the inner sum is zero

unless u is a multiple ofgjd. Writing h = g/d, we have

o(u) = 2 ( # ) (3' 19)Z

It is possible to continue and to express cu{u) in terms o f Eulers cp

function. In our application, eqn (3.15) ensures that (g,u) = 1 and

thus that ca(u) is /u.(#), which is itself zero if # has a repeated primefactor. Ramanujans sum with (#, %) = 1 can be regarded as Gausss

sum for the trivial character %0mod#, for which Xoim) 1 whenever

(g,m) = 1.

In this chapter, we have shown that r(x) is zero unless # = # // is

composed solely of those primes whose squares do not divide #, in

which case |r(X)|2 = / . (3.20)


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4

P O L Y A S T H E O R E M

F r o m eqn (3.2) we see that the sum function

x (v) = 2 x im) (4.1)

is bounded when x is a non-trivial character modg. Since the sum over

any qconsecutive integers is zero, the absolute value ofX (x) can be at

most \q. Polya (1918) proved the following sharper result.

Theorem. Let x be a non-trivial character m.odq with conductor f.

where the term o(1) is to be interpreted as f-> oo.

Polyas theorem was discovered independently by I. M. Vinogradov

(1955, chapter 3, example 12), with a different constant in the upper

bound. Later proofs have been given by Linnik and Renyi (1947) and

by Knapowski in an unpublished manuscript. We shall follow P olya s

argument, as it is the most precise and can easily be adapted to show

that the sum in (4.2) is frequently ! > / 3.

First we introduce some notation. I f a is a real number, we write [ct]for the largest integer not exceeding a, and ||a||for the distance from a

to the nearest integer, so that

where the maximum in (3.3) and the minimum in (3.4) are over all

integers m. We now state a lemma.

Lemma. The Fourier series

Then (4.2)

[a] = max m, (4.3)

(4.4)|a|| = min|ma|,

(4.5)

oom=0


24/138

converges to

1.4 P O L Y A S t h e o r e m

a__[a] _ i ifa no%an integer,

0 if a is an integer,

and the partial sums satisfy the relation

M e(mct) _ 1

-Mm7=0

2 t t w i

(4.6)

(4.7)

15

when a is not an integer.

Proof. We prove (4.7). SinceH(a)has period 1 andH( a) = H(a ),

we suppose that 0 < a < -J. Now

and thus

(2TTim)-1e(m


25/138

using the equation

0(a) = xjq-\-H(a.xjq)H(ot). (4.13)

We now have

X(l) + x(2) + ...+ l* (a0 = I x(m)0(mlq). (4.14)m 1

When we use eqn (4.7) to truncate the sums for H(axjq) and H(a) in

(4.14), the total error in modulus is at most

2J M [ , / ) - ! < 4 1 dh?m=l

< 277-~1i lf -1#logg. (4.15)

We now have finite sums to manipulate. Writing

CO

0(a) = 2 a(m)e(ma), (4.16) CO

we have to consider

q M

2 X(r) 2 es(TOr)- (4.17)J.=l M

Supposing first that x is proper mod#, we have from eqn (3.8)

m a M

2 a(m) 2 x0')e9(') = 2 a(m)x(m)r(x). (4.18)- M . > =1 M

Since xis non-trivial a(0)%(0) is zero, and by eqns (4.5) and (4.13), for

\a(m)\ < \TTin\-1. (4.19)

The modulus of the expression in (4.18) is now

/ M< ? i2 | (nm)-1

m =l

< 27 T ^^ ( logJf+ 0( l ) ). (4.20)

We choose M = qi+s, (4.21)

so that (4.20) is 77- 1gi lo g g (l -l- 0 (S)), (4.22)

and after (4.15) the tails of the series give

< ( n M ^ q l o g q = o(qHogq). (4.23)

Finally the omitted term \xix) is 0(1), and we have Polyas theorem

when x is proper mod q.



26/138

1.4 P 6 L Y A S t h e o r e m IT

I f x is induced by Xiproper m od/, where / < q, we could complete

the proof similarly, but it is easier to deduce this case.

2 x ( ) = 2 X i M X M d )m^cc d\q

d\m

= 2 ^(d)Xi(d) 2 Xi()d\q m ^x fd

< 2 / } l o g ' /{ ^ + o ( l ) } , (4.24)d\q

(d,f) = 1

since xi is proper m od /. The number of terms in the sum is at most

d(qjf), and we have completed the proo f o f Polya s theorem.

As a simple corollary we prove that for each primep > 2 there is an

integer mwith m < p i \ ogp (4.25)

for which the congruence

u2 = m (m od#) (4.26)

has no solution. Since 1, 4, 9,..., (# l )2 are distinct mod #, there are

\(p 1) reduced residue classes that are congruent to squares mod#,

and these form a subgroup of index 2. There is therefore a character

Xm od# with x(m) 1 when m is congruent to the square of a reduced

residue and 1 when (m,q) = 1 but m is not congruent to a square.We choose . i, ,.

x > p - \ o g p (4.27)

in (4.2). Not all terms in the sum (4.1) can be non-negative if p is

sufficiently large, and so there is an m < # - l o g # with x im) = !

The exponent \in (4.25) can be improved, but the conjecture that the

asymptotic inequality < # s (4.28)

for each 8 > 0 holds in place o f (4.25) has not yet been proved or

confounded.


27/138

5

D I R I C H L E T S E R I E S

Well, said Owl, the customary procedure in such cases

is as follows.

What does Crustimoney Proseedcake mean ? said Pooh.

For I am a Bear of Very Little Brain, and long words bother

me.

It means the Thing to Do.

I. 48

A Dirichlet series is an analytic function of the complex variable

s = c j+ itdefined by a series

/( ) = 1 a{m)m~s, (5.1)

or a generalization thereof. All the Dirichlet series that we need are

special cases of (5.1). I f eqn (5.1) converges at s0 = cr0-(-ii0, then

\a(m)m~8 o-0+ l . W e see that the region o f definition off(s) is a half-plane

bounded to the left by some vertical line; this line is called the abscissa

of convergence.

I f 4 ( ) = 2 a ( m ) (5.2)

is the sum function of the coefficients in (5.1), thenCO

f(s) = j s x - ^ A i x ) da;. (5.3)i

Formula (5.3) can be inverted: from /(s ) we can recover the sum function

A(x) o f the coefficients. Let a and ube positive real numbers. Then

ar-f- i oo (0 i f 0 1,

this is equal to the integral round the three remaining sides of the


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1.5 D IR IC H L E T S E R IE S 19

rectangle whose other corners are Rjlogu^iT^, R jloguiT^ The

modulus of the integral in eqn (5.4) is thus

R - ^ e - K i r o~R______ . ___________________________ (5 gx

2tt t 277^ log u 12irT2i\ogtia number which tends to zero as R, Tx, and T2 tend to -|-oo. When

u < 1, Rflogu is negative, and we must add the residue from the pole

of s-1 at s = 0; this gives unity. Finally when u = 1 we define the

value of the integral (5.4) to be the limit of the integral from ai Tto

a-\-iTwhen T-> oo. This reduces to an inverse tangent integral.

T

Of. Rjlogu

1' 1

F i g . 1

I ff(s) defined by eqn (5.1) converges uniformly in ton the line a a,

then for x > 0 term-by-term integration givesoi-J-ico

f x ^ f i s ) ds = 2 a(m)+a(x), (5.6)J m x) = 2m1

(5.8)

where x is a Dirichlets character to some modulus q, or aszeta functions

after Riemanns function

() = 2 m ~s>m1

(5.9)


29/138


which is the special case of Dirichlets definition when q = 1 and x is

trivial.

^-functions are defined by two properties. First, the coefficients a(m)

are multiplicative, so that Eulers product identity

a ( m ) _ -| r ( a ( p ) a ( p 2)

m=1

holds in a half-plane a ^ a in which one side of eqn (5.10) converges

absolutely. I f the product in (5.10) converges, f(s ) can be zero only

when one o f the factors on the right-hand side of (5.10) is zero. The

convergence of the left-hand side of (5.10) alone does not imply that of

the product; L(s, x)with xnon-trivial has a series (5.8) converging for

a > 0, but the function itself has zeros in a > preventing the product

from converging in 0 < a < J-.

The second defining property is thatf(s ) should have a functional

equation f{s)G (s) = f* ( r - s )G * (r - s ) , (5.11)

where r is a positive integer, G(s) is essentially a product of gamma

functions, and the operation * has (/ *) * = / and ((?*)* = G. As an

example, in the functional equation for L (s,x ) in Chapter 11, L*(s, x)

is L (s,x). An important conjecture about Z-functions is the Biemann

hypothesis that iff(s ) satisfies eqns (5.10) and (5.11) then all zeros of

f(s)G (s)have real part Jr. The truth or falsity o f this hypothesis is not

settled for any Z-function.

Two generalizations that are often called zeta functions are

CO

1 (m + 8)- s, (5.12)

m=l

where 8 is a fixed real number, and

002 r(m)m~s, (5.13)

m=1

where r(m)is the number o f representations o f to by a positive definite

quadratic form. Except in special cases these fail to have a product

formula o f the form (5.10), and not all of their zeros lie on the appropriate

line. Some authors even use zeta function as a synonym for Dirichlet

series.

In Chapter 11 we shall obtain analytic continuations o f (s) and other

.L-functions over the whole plane. Since the sum function X (x) formed


30/138

1.5 D I R I C H L E T S E R I E S 21

with a non-trivial character %is bounded, b y partial summation (5.8) con

verges for ct > 0 except when x is trivial. Similarly, the function

co

X ( l)m-1TO~~s = (1 21-S)(s) (5.14)m1

converges for a > 0 and provides an analytic continuation for (s).

When we make s 1 in (5.14), we see that (s) has a pole o f residue 1

at s = 1. When we p u t/(s ) = (s) in (5.6), the integrand has a simple

poleats = 1 with residue. The value of the right-hand side of eqn (5.6)

is betweenx 1andx. I f we deform the contour in (5.6) so that it passes

to the left o f the pole, the residue makes the main contribution, and the

contour integral left over is bounded. Let

(5.16)m < ( G

which are the coefficient sums of '(s)/(s) and o f l/(s) (we shall prove

this below). The function '(s)/(s) also has a pole o f residue 1 at

s= 1, but l/(s) does not. I f the corresponding contour integrals were

negligible, we should have

t (x ) = x + o ( x ) , (5.17)

M(x) = o(x). (5.18)

These are forms of the prime-number theorem, which we shall prove in

Part II.

Writing m= fg , we have

co , CO \ i c o N

2 m~s 2 ( /W W /) = ( 2 ( / ) / *) ( 2 H g ) r s)- (519)m = 1 f \m ' / = 1 ' ' ( 7= 1 '

I f b(g) = 1 and a(f) = /x(/) for each pair o f integers / , g,co CO

(5) 2 K m)m~a= 2 w s 2 M /) = 1 (5' 2)m = l m = l / |m

from eqn (1.27). Since eqns (5.3) and (5.6) imply that expansions in

Dirichlet series are unique, we have shown that the series on the left-hand

side o f (5.20) represents l/(s) wherever it converges. Similarly, using

eqns (5.19) and (1.32) we can check that '(s)/(s) has a Dirichlet series

with coefficients A(m).

For fifty years (1898-1948) the only proofs known o f eqns (5.17) and

(5.18) used contour integration and other complex-variable techniques.

In 1948, Selberg and Erdos gave a real proof of (5.18) (see Hardy and


31/138


Wright (1960), Chapter 22). The real-variable approach is not so well

understood, and the strongest forms of (5.17) and (5.18) (those in which

the error term is smallest) have been obtained by analytic methods.

The form (16.22) in which we shall prove (5.17) is a little stronger thanthe best so far obtained by Selbergs method.

Aj)art from the analytic arguments, studjr of log(A'r!) suggests the

form (5.17) as a conjecture. By eqn (1.32),

where weJiave written m= tiein the first sum. On the other hand, by

expression (2.5),

which agrees with the result of substituting (5.17) into (5.21). In this

way, Gauss was led to conjecture that

which is another form o f the prime-number theorem. By consideration

o f the binomial coefficient 2lvCjv> ^ can be shown that fj(N) lies

between boimded multiples ofN (Hardy and Wright, 1960); but there

are too many terms in the sum (5.21) to allow (5.17) to be deduced

from (5.22).

= 2 MN/e),e< iV

(5.21)

= N lo gN ~ N -}-0 ( logN ) (5.22)

2

(5.23)


32/138

6

S C H I N Z E L S H Y P O T H E S I S

And all the good things which an animal lilies

Have the wrong sort of swallow or too many spikes.

II. 30

M a n y problems in prime-number theory follow a similar pattern.

Various constraints are laid on a set o f integer unknowns, and we askwhether the integer unknowns can all be jsrime simultaneously, and

whether this happens infinitely often. Many of these problems are

subsumed under SchinzeVs hypothesis: i f1(x1,.. .,xn), .. ., fm(x1,.. .,xn) are

polynomials (with integer coefficients) irreducible over the integers, and

there is no primep for which J J / { = 0 (mod #) for all sets xn o f

residues mod#, then there are infinitely many sets xv ..., xnof integers

for which the absolute values o f / 1;...,f mare all prime.

There is a conjectiired asymptotic formula for the number of setsik-l,..., xn with 0 < xi ^ N for each i with each of f m prime, and

some bold authors have conjectured that the asymptotic formula can

be stated with an error term smaller than the main term by a factor

for each e greater than zero. Thus the simplest case is one poly

nomial,f (x ) = x, and the conjecture now states that tt(N),the number

of primes up to N, satisfies the relation

a very strong form o f the prime-number theorem (5.23). The accuracy

of (6.1) seems unattainable. W e shall prove later that the hypothesis

is true for one linear polynomial/(a;) = qx-\-a\this is the prime-number

theorem for arithmetical progressions, but the error term in the asym

ptotic formula will only be shown to be slightly smaller than the leading

term.

The next simplest case concerns two linear polynomials, f^ x) = x,

f 2(x) = x 2. Here the conjectured formula is

(

6.

1)

2

Ar

2 XJ {1 l ) " 2} I (lo g) 2 -f-error term, (6.2)

853518X


33/138

W e shall now describe how to write down the conjectured asymptotic

formulae. Let a/ \ v n. m o\S ( a ) = 2 , (6>3)

p^N

such an expression is called an exponential sum or a trigonometric sum.By the fundamental relation

/ / t (1 i fm 0,J e(m) * . = ( j j (6-4)

we see that the number o f primesp ^ Nfor whichp 2 is also prime is

l

J S(a)S(a)e( 2a) da. (6.5)o

We cannot, of course, work out this integral, but we can suggest a

plausible value for it. Writing

n ( N ; q , b ) = 2 1. (6.6)P


34/138

1.6 S C H I N Z E L S H Y P O T H E S I S 25

For small qthe argument above can be made rigorous; but then part

o f the range of integration in (6.5) does not support spikes. Aw ay from

a spike we cannot estimate S(a)except by replacing it by its absolute

lvalue; and the spikes with small q contribute very little to J |$(a) |2 da.

oIn the integral o f |$(a)|3 the spikes do dominate, and by this method

I. M. Vinogradov was able to prove that every large odd number is the

sum of three primes.

The approach to Schinzels hypothesis through exponential sums does

lead to an upper bound for the number o f sets x 1}..., xnof integers not

exceeding N for w h ich /lv ..,f m are all prime. To explain the method

we shall take n = 1, so we are considering integers * in the range1 ^ x Nfor w liich /1(a;),...,/m(a;) are all prime. W e now work modulo

a prime #. Apart from the finite number o fx for which one off t(x),...,

f m(x) isp, x must be such that none o ffx(x),...,fm(x) ~ 0 (modp). This

means that x must be confined to certain residue classes mo d# . We

therefore divide the residue classes mod# into a setH(#) of /(# ) forbidden

classes and a set K(p) o f g(p) = # /( # ) permitted classes; hmod# is

forbidden if and only if one of the po ly n om ia ls/^ ) is a multiple o f# .

I fxfalls into a forbidden class for any prime # smaller than each o f thefi(x), then one at least of th e /i (x) cannot be prime.

The values of x that makef x(x),..., f m(x) primes greater than some

bound Q form a sifted sequence, in the following sense. The increasing

sequence J fo f positive integers % , n2>... is siftedby the primes # < Q

if for each prime # < Q there is a set H(p) (possibly empty) o f /( # )

residue classes mod# into which no member ofJ rfalls. We shall show

in Chapter 8 that, ifJ/' satisfies the above condition, the number of

members ofJ fin any interval ofN consecutive integers is

Nv ... >,...wi, w ..+ error t e r m > (6 -12)I /**(?)/(?)/?(?)

a< Q

where f{q) = q H f ( p ) l p , (6-!3)

g(i) = ? IT { l - f i v ) h ? } - (6-14)v\a

We shall work out examples of this upper bound in Chapter 8; in each

case the leading term is a multiple o f the leading term in the conjectured

formula.

Upper bounds of the right order of magnitude were first found by

Viggo Brun using combinatorial arguments. Rosser used Bruns method


35/138

to obtain expression (6.12), which was found in a different way b y Selberg.

An outline of Selberg s method follows. It rests on the construction of

an exponential sum T(a)with the same spikes at rational points as

S( ) = 2 e K ) (6-15)!


36/138

1.6 S C H I N Z E L S H Y P O T H E S I S 27

Now = (6.23)

and so the coefficient of e(ma) in eqn (6,19) is

ii{d)d ^ ^{q)f{q) (6>24)fid) Z-, glq)

d=C_Q J ' g = o(mocld)meH(d) s


37/138

7

T H E L A R G E S I E V E

But whatever his weight in pounds, shillings and ounces,

He always seems bigger because of his bounces.

II.30

W e have seen that the behaviour o f a given sequence of integers con

sidered modg is reflected in the behaviour of the sums 8(a/q) where

JVS(ot) = (7.1)

1

in which amis 1 if m is in the given sequence, and 0 if m is not. An ujjper

bound for the sum ^ ^ |(g(a/?)|a (7 2)

q^Q amodg

(or for some related expression) is called a large sievefor residue classes.

Other large sieves will appear in Chapter 18. In this chapter we prove

what is probably the simplest of the upper bounds for the sum (7.2), and

in the next chapter we shall use it to prove an upper bound o f the form

(6.12) for the number of elements of a sifted sequence in a bounded

interval.

The p roo f does not require that am in eqn (7.1) takes only the values

0 or 1, and it treats (7.2) as a special case of the sum

f |S(*,)la (7.3)}.=i

where 0 ^ x x < x2< ... < xR ^ 1. Since ajqoccurs in the sum (7.2) only

if it is in its lowest terms, the points xr are distinct in our proposed

application. We write

S = min{*2a^Xgxz,. ..,xx+ l x ^ (7.4)

and suppose that S > 0. Before proving an upper bound for (7.3), we

consider what form it might possibly take. Certainly there will exist

sequences of coefficients am and points xrfor which

b 1

2 |S(,)|>8-i f |S()|d= ! J

= 8-1 K P - (7-5)1


38/138

1.7 T H E L A R G E S I E V E 29

On the other hand, any one term in the sum (7.3) may be as large as

(2 Kn!)2>ari(l ^ ail the are equal in modulus this is

t f f h J 8- (7.6)1

An optimistic conjecture is that the inequality

| \S(xr)\ * ^ (N + 8 ~ i) f\ a m\* (7.7)) =i i

always holds. Surprisingly, the right-hand side o f (7.7) has the correct

order of magnitude: we shall prove that

f \ S ( x , . ) \ ^ ( iV + p - W 3 + 0 ( l ) ) ir=l 1

this result is due essentially to Bombieri (1972).

To prove the relation (7.8) we use the language of ^-dimensional

vectors over the complex numbers. The inner product (g, h) o f two

vectors g = {g-s_,---,gN) and h (hv ...,hN) is given by

ml2; (7.8)

N

(g, h) = % g j i nsx

and the norm ||g|[ by ||g|| = ((g,g))*.

We can now state a fundamental lemma.

(7.9)

(7.10)

Lemma. Letu, f (1),..., f (fi) be N-dimensional vectors, and cv ..., cR be any

complex coefficients. Then

f c ,( u ,f W ) < ||u||(| | c,|f ( max f |(fM,fM)|)*. (7.11)r 1 ' 1 ' 1,...,R 8=1 '

Proof. The left-hand side of (7.11) is

(u, | c,fW)R

2 M W (7.12)

The square of the second factor on the right-hand side of (7.12) is

i f crcs(w,f)< 1 f | (Icri+Kia)i(fw,fw)ij = l s = l r = l s==l

= 2 K l2 2 l(f(,,)>f(8))lr=l s=1

(7.13)

from which the result follows.

In applying the lemma we choose c,, so that each summand on the left

o f (7.11) is real and positive. In Chapter 27 we shall apply the lemma


39/138

with cr of unit modulus to obtain Halaszs method for estimating the

number o f times 8(a) is large. To prove (7.8) we take crgiven by

cr= (fMf) (7.14)

and deduce the corollary.

C o r o l l a r y . We have

2 |(u,fW)|2 < jjujj3 max ^ |(f(,,),f (s))|. (7.15)r = 1 1


40/138

When we choose L to be an integer close to (SV3)-1, (7.24) becomes

< 2 iY + fS -1V3 + 0(1). (7.25)

We substitute (7.20), (7.21), and (7.25) into (7.15) to obtain (7.8).

Our inequality (7.8) represents an improvement of an inequality of

Roth (1965), which has led to much recent work. The best upper bounds

known for the sum (7.3) at the time of writing are

1.7 TH E LA RG E SIE VE 31

S-1(l + 270Ar3S3) f \ am\2, (7.27)i

and 2max(AT, S^1) 2 (7.28)i

Of these, (7.2 6) is the result o f this chapter, appropriate whenN> f 8 _1 V3,

and (7.27) and (7.28) are results of Bombieri and Davenport (1969,1968).

(7.27) is appropriate when S_1 > 3(10)i/V, and (7.28) for the inter

mediate range.

Note added in proof. H. Montgomery and R. C. Vaughan have nowproved the conjecture (7.7). This supersedes (7.26) and (7.28) but not(7.27).


41/138

8

T H E U P P E R - B O U N D S I E V E

1Its a comfortable sort of thing to have said Christopher

Robin, folding up the paper and putting it into his pocket.

II. 170

In this chapter we obtain the upper bound (6.12) as an application ofthe large sieve. The notation is that of Chapter 6. J r is a sequence

o f positive integers, and for each q ^ Qthere are setsH(q) and K(q) of

residue classes mod#. The f(q ) classes ofH(q) are precisely those that

are not congruent to any member ofJ r mod# for any prime# dividing q,

so that, if h is in a class ofH(q) and n e J f,

(nh,q) = l. (8.1)

Theg(q)classes ofK(q)are those that for each # dividing qare congruentmod# to some member ofJ r \their union contains all members of the

sequenceJ f.

We work with the exponential sum of eqn (6.15):

# ( ) = 2 ( ). (8 '2 )

where the sum is over members nv n2,...of the sequence J r . I f h is a

class ofH(q),

2 * S(alq)ea{ -a h ) = 2 c ^ h)a m o d g U;< N

= p{q)M, (8.3)

whereMis the number o f members o fJ r ,and we have used eqn (3.19) in

the sjiecial case (8.1). Hence

2* 2 8{alq)eq(-ah). (8.4)a m o d g heH(q)

Cauchys inequality now gives

,i*(q)f*(q)M* ^ ( 2 * | S ( / ? ) l a) ( 2 2 ^ ) 1 ' ( 8 -5 )' a m o d g ' ' a m o d g hef-T(n\ >heH( q)


42/138

1.8 T H E U P P E R - B O U N D S IE V E 33

The second sum over aon the right-hand side o f (8.5) can be rearranged

as follows.

2 * 2 ea(ah) 2 %iacJ) = 2 2 c fl(0 A)amodg heH(q) geH(q) geH(q) JisHiq)

= 2 2 2 M v ld)05H(q) heH(q)

d\(g-h)

= ' Z M ql d ) P (q ) l f ( d ) . (8.6)d\q

In eqn (8.6) we have a sum that is a Mobius inverse (in the sense of

(1.27)) of that in

' /*2(})fl,(r) _ m

rlm f (r) m y(8.7)

an equation that expresses the fact that for a prime modulus p, every

residue class is either in H(p) or in K{p). The terms involving d in

(8.6) therefore come to p?(q)g(q)lf(q),and we can put (8.5) into the form

j/J2 'V'*

qlq) " " Z -iJ w amodg

S a (8.8)

We apply the large sieve (7.8) with the rationals ajqwith q ^ Qand

(a,q) = 1 as the points xx,..., xR, so that

= (Q (Q - l ) ) -1 (8.9)

in eqn (7.4). The upper bound (7.8) now gives

2 2*|-S(/?)|2 < ( ^ + 0 ( e 2) ) ^ . (8-10)q


43/138

Two worked examples follow. First we consider the perfect squares

not exceedingN. These are a sifted sequence: the setH(p)contains the

residue classes mod# that do not contain squares, and thus

/ ( 2) = 0, g(2) = 2

f (p ) = U p - 1)> 9(P) = UP+ !) fo r# > 3

It is not difficult to sIioav that

'2,p2te)fte)lg(q) = ( < H - o ( i) ) Q , ( s . i 4 )

where c is a constant, as Q-> oo. Choosing Q N-, we have shown that

the number of perfect squares not exceeding N is 0(2V*). This is very

encouraging, since the sieve upper boimd is sharp, differing only by

a constant factor from the actual number o f squares. It is surprising

that we have not lost the correct order of magnitude in combining so

many inequalities.

Our second, less trivial example concerns the primes between Q and

N; these form a sifted sequence with

f (p ) = 1> v (p) = # - 1 = ( 8 -1 5 )

and thusg(q) Z-t 2 s > ' ^


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1.8 T H E U P P E R - B O U N D S I E V E 35

We derived the inequality (8.12) from Cauchys inequality; the

difference between the two sides of the inequality (8.12) is a measure

of how closely the values of S(a/q) are proportional to those of

O/ta))'1 2 e(cl&) (8'2)l x K ( q )

and this in its turn measures how evenlyJ f is distributed among the

g(q) residue classes mod# into which it is allowed to fall. W e could add

an explicit term on the right of (8.8) to measure the unevenness (what

statisticians might term a variance). The inequality (7.8) gives a strong

upper bound for this variance as well as for the main term. When we

use (7.8) to prove Bombieris theorem, it is the variance bound that is

important, not the bound for the main term.


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9

F R A N E L S T H E O R E M

Its just a thing you discover, said Christopher Robin

carelessly, not being quite sure himself.I. 109

T h e Farey sequence o f order Qconsists of the fractions ajqin their lowest

terms (i.e. (a, q) = 1), with q Q ancl 0 < a q. W e name them

f r = ar/qrin increasing order, so thatf x = 1IQ ,f2 = l/(Q l) ,. .. ,f F = 1.

For notational convenience we may refer tofF+r]this is to be interpreted

as 1 + / r Here Fis the number o f terms in the Farey sequence, so that

F = 2?(?)=377-*Q*+0(Qlog Q) (9.1)

from eqn (2.11). The properties of the Farey sequence are discussed by

Hardy and Wright (1960, Chapter 3).

W e shall sketch a proo f that

fr+l fr = ( M m ) - 1- (9 .2 )

Let us represent rational numbers ajq (not necessarily in their lowest

terms) by points (a, q)o f two-dimensional Euclidean space. Since/,, and

f r+1are consecutive, the only integer points in the closed triangle with

vertices 0 (0, 0),Pr (ar, qr),andPr+1 {ar+1, qr+1)are its three vertices. By

symmetry, the only integer points in the parallelogram OPr TPr+1 are

its vertices, where Tis (ar-\-ar+1, qr-\-qr+1). W e can now cover the planewith the translations of this parallelogram in such a way that integer

points occur only at the vertices o f parallelograms. It follows that

OPr TPr+1has unit area, which is the assertion (9.2).

Before stating FranePs theorem we introduce some notation. For

0 < a < 1 we write

E{a) = 2 1- a F , (9.3)

so that E(a)is the excess number of Farey fractions in (0, a] beyon d theexpected number aF. Franel considered the sum

S Imfr)I2 (9.4)r = l


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1.9

>

F R A N E L S T H E O R E M 37

and showed that it is o (Ql)i f and only i f eqn (5.18) holds; and an upper

bound for (9.4) with 3 < A < 4 is valid if and only if

\M{x)\ (9.5)

We can also connect M(x) with

l

J |2?(o:)|2da. (9-6)o

In fact, the ratio of (9.4) and (9.6) lies between bounded multiples o fF;

but this fact requires proof, as the Riemann sum corresponding to the

integral (9.6) and the points/i,...,/F is

f (fr+l-fr) i m W , (9.7))=1

and from eqn (9.2) we see that the differencef r+1f r varies from Q-1

almost down to Q~2. Franel (1924) produced a curious identity for the

sum (9.4). W e shall show in (9.20) that (9.4) is less than a bounded

multiple o f Franels expression involvingM(x),and deduce the only if

clause o f Franel s result by a method of Landau (1927, Vol. II, pp. 169-

77).

We use the function H(a) of eqns (4.5) and (4.6):

t t, \ M \ i f a is not an integer,# ( ) = n -f + 9-8(0 it a is an integer.

We have

_ (F(a.) if a is in the Farey sequence, . .

\ J ?()- i i f not, (J 'J)

for 0 < a -


47/138

By (9.2), the consecutive Farey fractions are at least Q ~2apart, and so

the sum of the error terms in (9.11) is

< Q ~z X Q 2l t < log -P < log Q, (9.12)t = i

where we have used expressions (2.5) and (9.1). We can now replace

the term t= r (since H (0) = 0) and rearrange the first term on the

right of (9.11) as

F Q2 Q2

X X {27nm)-1e{mfrmfi) = 2 X X* e (m / ^ )e ( a m )t~ 1 Q1 Q2 3


48/138

1.9 F R A N E L S T H E O R E M 39

Squaring (9.11) and substituting (9.12) and (9.19), we have

2 W ) I 2 < Q2 2 (*)( 2 c i - m ( Q i d ) Y + F i o g * Q r = 1 t^Q ' d


49/138

P A R T I I

T h e P r im e -N u m b e r T h e o r e m

10

A M O D U L A R R E L A T I O N

Winnie-the-Pooh read the two notices very oarefully, first

from left to right, and afterwards, in case he had missed

some of it, from right to left.I. 46

Inthis second part we study (s) and L ( s , x ) as functions of the complex

variable s, and work towards the prime-number theorem. Our investigations are based on the functional equations for 'C(s)and for L ( s , x ) . The

first step is therefore to prove these.

We need a lemma from the theory of Fourier series.

Lemma. ( P o i s so n s sum ma t i o n f o r mu l a . ) L et Tc, I be i n t eg er s. L e t f ( x )

be a d i f f er en t i a b l e f t m c t i o n o f a r ea l v a r i a b l e t v i t h

\ m \ < A

o n [ k , I ]. T h e n

> k i lX f m e ( r n x ) d x = i m + f ( } c + l ) + . . . + f ( k + l ~ l ) ^ f ( l c + l )

m = - c o

(10.1)a n d m o r e over t h e -p ar t i a l s um s sa t i sf y t h e i n eq u a l i t y

M l,e + 1

f ( x ) e ( m x ) d x - y ( k ) - . . . - y ( k + l ) ^ l A M - H o g M .

M k ( 10 .2)

P r o o f . Equation (10.1) is additive on intervals: its truth for [/, t ] implies its truth for [ k , t ]; so we may suppose 1 = 1. By


50/138

a change of variable we may suppose also Jc = 0. We use the function

H ( x ) of eqn (4.4):

H ( x ) = y M = H if o < * < i,Z -t 27rim 10 if x ~ 0 or I ,

m co vm^O

and in particular if 0 < x < 1 we have (4.6):

2.10 A MO DU LAR RE LA TI ON 41

^ > + 2 5. 27rim M7)1#0

Hence

/ M M V2H ( x ) Jr 2 (277im)_1e(m3;) da; 2 m-1 da;+ da;

o 0 1 ' 1 I'M

M -1 log M . (10.5)We now have

jif, }2 I f ( x ) e ( m x ) da;-m Jm#0 0

= [/(%) 2 (277-iTO)_1e(?wa;)j J jV'O) 2 (27rim)_1e(OTa;) d*. (10.6)

m 0 0

The first term on the right of eqn (10.6) is zero, since the terms for m

and m cancel, and by (10.5) the second is

1

J f ( x ) H ( x ) d x - { -0 ( A M ^ 1 logM ) . (10.7)0

The integral in (10,7) is

1

j / '( * ) ( * - * ) da = i / ( 0 ) + i / ( l ) ~ J /(* ) d. (10.8)o o

Combining (10.6), (10.7), and (10.8), we have

M }

2 f f { x ) e( m x ) d x = | /(0 )+ i/( l) + 0 (^ ilf-1log Jf), (10.9)

and letting M tend to infinity we have the case k= 0, 1 = 1 of (10.1).

A general method of proving functional equations is to write the

required function as an infinite series, apply Poissozis summation

formula to a partial sum, and then let the length of the sum tend to

infinity. This entails a change in the order of summation in a double

infinite series. We could prove the functional equations for ( ( s ) and


51/138

L ( s , x ) directly by this method, but it is more troublesome to justify

the interchange of summations and more difficult to identify the

functions that arise. We shall therefore prove the identity

co oo

2 exp{(to-|-8)27t2:-1} = x i 2 exj)(i7i2Trx)e(m8), (10.10)m = co m= co

where x is real and 0 s ' S


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2.10 A MODULAR R ELA T IO N 43

If we combine (10.15) and (10.12) we have proved that

CO CO

2 exp{ ('-)- 8)27rai-1} = j C7r~%*exp(7rm2x) e(m S). (10.17)? = CO 7)1 co

Putting S = 0, x = 1 verifies that

c = ttI , (10.18)and we have proved (10.10).

Equation (10.17) should not be let pass without some comment. Letus put m

(w) = 2 exp(7rmaai) (10.19)m = oo

for values of to for which the series in (10.19) converges, that is, for

complex towith positive real part. Then we have

6 ( t o + 2) = 8 (w), (10.20)

and (10.10) with 8 = 0 gives us

02( - l / ) = to82{to), (10.21)

provided w is pure imaginary. However, since eqn (10.21) holds along

the imaginary axis, the two sides of (10.21) have the same derivatives

at points iywith y> 0, and, since power-series expansions of regular

functions are unique, (10.21) must hold whenever 8(to)and 6( l / to)are

b o t h defined, which is whenever the imaginary part of oj is positive.

From eqns (10.20) and (10.21) we see that

04H dw (10.22)

is invariant under the group o f transformations of the upper half of the

complex plane generated by co-> w + 2 and w-> 1/to .

We are now in the realm of the elliptic modular functions. A mo d u l a r

f u n c t i o n is one that is invariant under the group of transformationsgenerated by a>->oj-|-1 and oj -> 1/co or under a subgroup of finite

index in this group. The derivatives of a modular function are not

invariant under these transformations, since dw itself is not invariant;

functions that satisfy the transformation law for a power of a derivative

of a modular function are called mod u l a r f o r m s . The name elliptic

modular functions arises as follows. The periods of an elliptic function

form a free Abelian group on two generators t ox and to2. A modular

form corresponds to a function of two complex variables and w 2

which is homogeneous and whose value does not change when we replace

o j land t o2by another pair of generators of the same free Abelian group

(or, more generally, which takes a finite set of different values when toJ


53/138

and a )2 are generators of the same Abelian group). Here co

Thus we may consider

f { a >i ,co2) = i o ^ e ^ i o j c o x ) ,

w i t h / ( o j 2, OJjJ = U)2~ 1 02 ( W j / o j g ) = 162(co2/co1)

= / K . w 8)

and /(oij , a)2+2co1) cox102(co2/a)1-)-2) = /(aij, co2).

44 THE PRIM E-NUM BER THEORE M 2.10

= w2/wi-

(10.23)

(10.24)

(10.25)


54/138

T H E F U N C T I O N A L E Q U A T I O N S

When he awoke in the morning, the first thing he saw was

Tigger, sitting in front of the glass and looking at himself.

Hallo! said Pooh.

Hallo! said Tigger. I ve found somebody just like me.

I thought I was the only one of them. II . 21

W e return to (s) and L ( s , x ) The definition

CO

r ( ) = / e - v y ^ d y , (11.1)0

valid for cr > 0 (we recall s cr+itf, cr and treal), becomes

CO

A is)= J e~~m7rxx is~1da; (H -2)o

when we write y = -nmPx. Summation over m gives

co >7t - } s r ( \ s ) i ( s ) = 2 q - 2itxx Is - 1 da;. (11 .3)

m = 10

Since the sum and integral in eqn (11.3) each converge absolutely, we

can rearrange the right-hand side of (11.3) as

CO co

J 2 d x = J 4(0 (i )- l) **-1 da;, (11.4)

o 1 o

where 6(a>) is the function of eqn (10.19). Next we write

1 >| i( 0 (i )-l) * -1 d a != J (0(i/f) l)$-ls-1 d t (11.5)

0 1

and use eqn (10.10) to put (11.5) into the form

CO co

f ^ (^ (it J -lJ H 8- 1 d t = f I0(it ) t ~ ^ i d t -K 2 S -1)

i iCO

= J 4 ( 0 ( i i ) - l ) H - * d i - s - 1- ( l - a ) - 1. (11.6)

11


55/138

The left-hand side of (11.3) has now been expressed as

CO

J (0(ia;) l ) ( J - i + a r + ' -* ) d { ( i a) } - i ( 11 .7 )

i

The expression (11.7) was obtained under the assumption a > 1, but,

Since 0 ( i ) - l < e - (11 .8 )

for x > 1, the integral in (11.7) converges for all complex s. Since

r(|s) is a known function, and ( m s ) ) - 1 is integral (single-valued and

regular over the whole s-plane), we can take (11.3) with (11.7) as theCO

definition of (s), knowing that X m ~s agrees with our new definition

iwhen the series converges. We have now continued l ( s ) over the whole

plane. Further, (11.7) is unchanged when we replace sby 1s,so that

^ r ( i s ) U s ) = t^ -^ T (! -1 S) (1 -S), (11.9)

the promised functional equation. Since

' M = 21 7r^T(s)cosis7r, (11.10)1 I 2 2)

an alternative form of (11.9) is

(1s) = 21- s7T~sr ( s) c o s %st tt,(s). (11-11)

We now list some properties of F ( s ) (see for example Jeffreys and

Jeffreys 1962, Chapter 15). The product

1

46 THE PRI ME -NU MB ER THE ORE M 2.11

r ( s + i )

where yis tbe constant of (2.5), converges for all s, and defines F ( s ) as

a function tliat is never zero and has simple poles at 0, 1, 2,....Using this information in (11.7) we see that the pole of (11.7) at 1 comes

from (s), the pole at 0 from -T(^s), and that (s) must have zeros at

s = 2, 4,..., to cancel the other poles of F ( ^ s ) . From eqn (11.12),

T(s+1) = s F ( s ) , (11.13)

and Jrr( 1 -)s)_Z ( 1s) = i t scosec 77s, (11.14)

where we have used the product formula for sin7rs. We can verify

eqn (11.10) by showing that the ratio of the two sides is a constant.Equation (11.1) is obtained by evaluation of the limit of

N

C*10+1(1t /N ) N d t (11.15)


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in two ways as Ntends to infinity. We can also obtain from (11.12) the

asymptotic formulae

logics) = (s-P og s-s + llog2 7 r+ 0 (l /|s | ), (11.16)

and ! / / = lo gs+0(l/|s|) (11.17)

which hold as |s| co uniformly in any angle 7i+S < args < t tS

for any 8 > 0.

Next we consider an L-function L ( s , x ) with x a proper character

mod#. There are two cases. I f x(~~l) is 1> we argue as above up to

2.11 THE FU NC TIO NAL EQ UA TION S 47

f 00Tr-*sqissr(^s)L(s, x) = a;is-1 2 x(m)e~m*n!>:lQ

o m=100

= j x i ^ ' ? ( x , x ) d x , (11.18)

0co

where ( p ( x , x ) = 2 x ( m )e ~m*nx a' (11.19) CO

We approach cp(.T,x ) through (10.10):

CO CO

2 e - ( n + 8 ) M * = x h2 e ~ m S 7 r -r e ( m S ) . ( 1 1 . 2 0 )co co

We put S = cijqand use eqn (3.8):

so that

x ( m ) T ( x ) = 2 * x ( a )ea (a m )> (H.21)amodg

r(xMx>x)=* 2* r n 2amodg m = co

co

= 2* x{a)(


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The analogue of (11.7) is now seen to be

co co

TT~lsqisr(^s)L{s, x ) = i J x) ~ x~^-^(x, x)da.

1 1 (11.25)

As before, the right-hand side of (11.25) converges for all s, so that

L ( s , x ) has an analytic continuation over the whole plane, with no

singularities. Moreover, L ( s , x ) must have zeros at 0, 2, 4,... to

cancel the poles of F ( \ s ) . We proceed to deduce the functional equation.

We have

t(x) = 2 X (m )eq(m ) = 2 x ( - ) e g{ m) mmodg mmodg

= r ( x ) , (11.26)

since it was assumed that x( 1) = 1. By eqn (3,14), since xis proper

m 0 d q g*/r(x) = r ( x ) l r . (11.27)

We now see that the right-hand side of (11.25) is r(x)g,_i times the

corresponding expression with s replaced by 1-s and x by x> which

gives the functional equation

i r - b q i * r ( $ s ) L ( 8 , x ) = T (x )q ~l TT-l +*sq ^ sr ( ~ s ) L ( l - s , j f ) .(11.28)

We now consider characters xim)proper modg with %( 1) = 1.Since we want to consider a sum from co to co, we use m x ( m ) in place

of x(m). Writing s+ 1 for s in (11.2), we have

OD7T-Ks+I) i(s+I)p (i(s-|-I))i(s, x ) = 2 me-'! % !s- da:

o m=1co

= iJp (a?,X) **-*da!, (11.29)0

co

where p ( x , x ) 2 (11.30) co

We find a functional equation for p ( x , x )by differentiating (10.10) with

respect to 8. We get

t(x)p(x > X) = i ^ x - i p ( l f x , x ) . (11.31)

Arguing as before, we find

w-*-tgi+*r(i(*+l))i(s,x)CO CO

= i| />(*, x ) ^ d x + ^ ) f P(X> x ) * - * * d. (11.32)

48 THE PRIM E-N UM BER THE ORE M 2.11


58/138

2.II THE FUNCTIONAL EQUATIONS 49

Again, the integrals on the right o f eqn (11.32) converge for all s,so that

L ( s , x ) has an analytic continuation; it must have zeros at 1, 3,

5,... to cancel the poles of r(| (s+ l)) , and satisfies the functional

equation

There is also an analytic continuation of L ( s , y ) when %mod qis not

proper. If Xiproper m od/ induces %modg, then for cr > 1

when we write m = d r . The sum over rin (11.35) is L ( s ,^i), which has

an analytic continuation since Xim od/ is proper, and the sum over d

is defined for all / . The corresponding functional equation for L ( s , x )

contains the sum over d explicitly. We shall not need this case again.

A number of proofs of the functional equation can be found in

Chapter 2 of Titchmarsh (1951).

i s lq i i s r ( l %8)L(ls,x)

= ^ -s+1 (s+1)r(| (S+ l) )L (S, x). (11.33)

To check this, we note that when %( 1) = 1

t(x) = (x)- (11.34)

(11.35)


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12

H A D A M A R D S P R O D U C T F O R M U L A

Suddenly Christopher Robin began to tell Pooh about some

of the things: People called Kings and Queens and something

called Factors.II. 174

In proving the prime-number theorem, Hadamard studied i n t e g r a l f u n c t i o n s o f f i n i t e or d er , that is, functions f ( s ) regular over the whole

plane, with log|/(s)| < || (12.1)

for some constant A , as |s|-s*oo. The order of/(s) is the lower bound

of those A for which an inequality of the form (12.1) holds. Hadamard

showed that an integral function of finite order can be written as an

infinite product containing a factor sp corresponding to each zero p

of the function. This generalizes the theorem that a polynomial can be

written as a product of linear factors. Weierstrasss definition (11.12) of

the gamma function is an example. The product is especially simple

when/(s) has order at most unity. The order of l / r ( s-\ - l ) is unity,

from (11.16). We shall obtain the product formulae for (s) an(l ( s>x)

g i v e n b y m = 8 ( l - 8 ) n - * r ( i 8 ) Z ( a ) (12.2)

and (,*) = ( q M ^ r { U s + a ) ) L ( s , x ) , (12.3)

where % is proper modg and a= 0 or 1 accoiding to the relation

%( 1) = ( 1). Note that (11.9) is just the assertion that (1s) is

equal to ( s) ,and (11.28) or (11.33) implies that

l f( l - . X )l = \(S,X)\- (12.4)

First we show that |(s, x ) has order one. By eqn (5.3), if a > 0,

L ( s , x ) = f 2 x ( m) d x . (12.5)j

By Polyas theorem (4.2), the sum over mis bounded, and thus00

!-(*> X)I


60/138

and for cr we have

i o g l ^ x ) !


61/138

52 THE P RIM E-N UM B ER TH EOREM 2.12

Now (0) = 1, so we can apply (12.13) to ( s)at once. We shall prove

later that (0, x) is non-zero, so that (12.13) can be applied to ( s, x)/(0, x).

To avoid a circular argument, we choose a 8 for which (S, x) is non-zero

and apply Jensens formula to g(s,x)/(&> x)- (12.7) or (12,12) wehave B

J r ~ xN ( r ) dr < ^ . B l o g B , (12.15)o

and as T oo N ( T ) T l o g T . (12.16)

Here, N ( T ) is the number of zeros of ( s) or of ( s, x) with |s| ^ T .

When we examine the formulae (12.2) and (12.3) for ( s)and |(s, x),

we see that any zero of (s, x) must be a zero of L ( s , x), and similarly

for ( s) . The converse is not true, because L ( s , x) has extra zeros atnegative integer values to cancel the poles of the gamma function in

(12.3). If s = o-\ -r l with a > I, Eulers product formula

^ . X ) = = n { l - X ( ^ - S} -1 (12.17)p

converges absolutely and so is non-zero. By the functional equation,

(s, x) is therefore non-zero for cr < 0, since g(s,x) is non-zero for a > 1.

Thus all zeros p = /3+iy of ( s, x)have 0 ^ ^ 1, and the same is true

for ( s)by a similar argument. Riemanns hypothesis is that /3 is always -J.Riemann stated the hypothesis for ( s) ,but it is difficult to conceive a

proof of the hypothesis for (s) that would not generalize to (s, x).

We shall prove later that 0 < j8 < 1: this statement is equivalent to the

prime-number theorem in the form (5.17) in the sense that each can be

derived from the other.

For later use we now prove a result more precise than (12.16).

Lemma. T h e n um ber o f zer o s p = /3-f-iy o f(s, x) i n t h e r ec t a ngl e B ,

0 < j 8 < l , T ^ y s ^ T + 1 , (12.18)

i s a t most ^log (g(| y|-fe)), (12.19)

a n d o f i ( s) i n B i s at m ost

< lo g (m + e). (12.20)

P r o o f . Let s0 be the point 2 + i(T + | ). Then

|(*o>x)l > i - i - i - A - - > i (12.21)

We apply Jensens formula (12.13) with B = 3 and/( ) = ( - * , x)/(*0.X). (12.22)

By (12.7) and Stirlings formula (11.16) the left-hand side of eqn (12.13) is

< l o g g+log(|T|+e). (12.23)


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2.13 H A D A M A R D S PRODUCT FORMULA 53

The circle radius 3 and centre s0clears the box Bby a distance at least \,

so that, if N is the number of zeros required, the right-hand side of

eqn (12.13) is > t fl o g 6 /5 . (12.24)

This proves (12,19), and (12.20) follows similarly.

We shall not need the following more accurate formula. I f T ^ e,

the number of zeros of (s, x )with 0 - (12.26)P# o

where if 0 is a zero we add a factor sat the beginning. P ( s ) is a regular

function with the same zeros as f ( s ) . We should like f ( s ) j P ( s ) to be a

constant or some other simple function. Certainly

g(s)= log { f ( s ) l P ( s ) ) (12.27)

can be defined to be single-valued and regular over the whole s-plane.

We shall prove that g{g) = A + B s _ (12 28)

By (12.19) or (12.20), there is a sequence of Btending to infinity with

i?-|p| > (lo g i? )- i (12.29)

for each zero p . We want a lower bound for log P ( s ) on the circle |s] = B .

N o w

- 2 log|(l//>)exp(/p)| < B 2 l/>l_1 < ^ lo g 2E,0< \ p \ < i B 0


63/138

54 T H E P R IM E -N U M B E R T H E O R E M 2.12

the power-series expansion of g(s) makes g ( R c i s 9)a Fourier series in 6:

g (R c i s 6) = ( a (n ) Jr i b ( n ) ) R n c i sn d > (12.34)

so that B , e g ( R c m d ) = R n ( a ( n ) c osn 6b( n ) si n n d ) (12.35)

and

2tt

r a ( n ) R n = f cos(?i0)Reg { Rcis 6) |dd . (12.36)

Hence*7/

\ a (n ) \ Bn J |Ree g ( Rcis 6), 0} ddo

< 1 -f R l o g Z R . (12.37)

Since (12.37) holds for an infinite sequence of R , a ( 2) , tt(3),... must be

zero, and similarly so must 6(2), 6(3),..., and we have proved eqn (12.28).

We have therefore

(s) = eBsJJ (1s//>)exp(s/p), (12.38)

pi ( s , x ) = C ( x ) eB{x)s I I ( ! s l p ) e x p ( s l p ) , (12.39)

p

with the modification mentioned above i f p= 0 occurs as a zero. These

products converge for all s. By taking logs and differentiating, we get

^ 1 = B -----j + i l o g2n I F ' j f S + 1) + y ( ~ h-V (12.40)(s) s1 2 r ( 2s + l ) ^ \ S~ P pj

and

(S>X) _ R/,,\ 110"^ l-^'(2(S + ffl)) | X ' / 1 I fl^ 41)

where pruns over all zeros of ( ( s ) or L ( s , x ) that do not coincide with

the gamma-function poles. We can substitute the equation

- l m = i r + (12'42)

which follows from the corresponding product formula (11.13) for F ( s ) ,

and obtain a sum over all zeros (except p 0) of (s) or L ( s, x ) . It can

be shown that = I0g 2 + il o g w - l- iy , (12.43)

but no simple expression for G(x ) and B ( x ) is known in general.


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13

Z E R O S O F f ( s )

You remember how he discovered the North Pole ; well, he

was so proud of this that he asked Christopher Robin

if there were any other Poles such as a Bear of Little Brain

might discover.

I. 131

W e saw in the last chapter that i ( s ) and g(s, x)have 110zeros p /3+iy

with /3 > 1. The prime-number theorem corresponds to the fact that

no zeros of (5) (these include the zeros of ( s) ) have j8 = 1. All the

direct proofs that j8 < 1 at a zero are based on the following argument.

The function '(s)/(s) has poles of residue 1 at the poles of (s) and

poles o f residue 1 at the zeros. Now

where A ( m ) is the function o f eqn (1.31). At s = 1, the series diverges

to + 00, corresponding to the pole of residue -(-1. Now, if 1+iy were

a zero of (s), we should expect the series in (13.1) to diverge to co,

and the partial sums to be as large as those for the case s= 1, butnegative. To achieve these, the numbers to-1? must be predominantly

near 1. The values of to2i>/ are therefore predominantly near + 1 , and

there is a pole at l + 2iy with residue -(-1) aild so a simple pole of (s)

at s = l + 2iy, which we know does not occur.

To make this argument rigorous, we use (13.1) with s= a + ii where

a> 1, so that the series converges. For all real 9,

p

2 log p i l p - * ) - 1

(13.1)

3 + 4cos0 + cos29= 2(l + cos0)2 0. (13.2)

(13.3)

00

Since Re('(s)/()) = 2 A ( m ) m ~ a c o& ( i t \ ogm ),

we have (13.4)

853518 X E


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We now make cx+if tend to a zero j8+iy. Since (s) has a pole at 1, there

is a circle centre 1 and some radius r , within which (s) is non-zero.

(Calculation shows that r= 3 has this property.) I f we suppose that

j3 > 1 - f r , (13.5)

then |y| ^ r , and so is bounded away from zero. In eqn (12.40),

t'W P, 1 W l ) V / 1 ,1)m = ~ B + i " ll082 - i T ( i i + i ) - 2 , 1 ^ + W (13'6)

we shall assume 1 < a < 2, \ t\ ^ fr > 0. Here the sum is over zeros p

of ( s) , not over all zeros of (s), and, since sp and phave positive

real part, we have / -, ^Re _ i_ + > 0 (13.7)\ S~~-P PI

whenever a > 1 and p= /3+iy has 0 ^ / 3 ^ 1. By (11.17) the term in

^ (i + l) is


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for some constant c2 (Titchmarsh 1951, theorem 5.17). The latest

result is

2.13 ZERO S 03? ( s ) 57

ft < 1,nTT TIVfTe (13'14)C3(e)

log(|y|+e)^

where the constant c3depends on e. This was proved in 1958 by Korobovand by I. M. Vinogradov independently. Intermediate improvements

on (13.13) used the intricate methods of I. M. Vinogradov (1954).


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14

ZEROS OF |(s , x )

What do you think youll answer ?

I shall have to wait till I catch up with it, said Winnie-

the-Pooh.I. 34

In this chapter and the next we prove results like (13.12) for the zerosp= /3+iy of L ( s, x ) ,with uniform constants in the upper bounds. We

actually work with (s, %),where %is proper mod#; the zeros of (s, x )

are those zeros of L ( s , x ) that are not at negative integers and so are

not cancelled by gamma-function poles. We use the product formula

in the differentiated form (12.41),

L ( s , x ) TT 2 i ( ! ( + ) ) Z - 4\ sp pj

where the sum is over zeros pof (s, x)> the term 1/pbeing omitted ifp = 0. We shall assume throughout the chapter that s = a -\ -i t with

1 < (7 < f.

The first complication is the elimination of B ( x ) from eqn (14.1). We

subtract from (14.1) its value at s 2, noting that

CO

\ L ' ( 2 , x)\ I \ L(2>x) \ < 2 (14.2)m=1

which is bounded independently o f x Estimating the gamma-function

term from (11.17), we have

- K e tM 5 5 ~ 2 R6( ^ , - ^ ) +0(log(|,1+e)) ( 1 4 '3 )where the term yo = 0, if it occurs, is now included in the sum. We now

note that ^ ^ - f ) - 1 = 2 ( 2 ^ ) \ 2 ~ p \ - 2 < l o g ?, (14.4)p p

by (12.19). Writing l (t ) = l o g{ q ( \ t \ +e) } , (14.5)

we have Be ^ ^ < "V Re - \ -0 ( l ( t ) ) (14.6)JjyS} ^ j S p

for 1 < a^ |, the implied constant being absolute.


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2.14 ZEROS OF { { s , x ) 59

When we apply (13.2) we obtain the relation

_ 3 o) - 4Re 7; ' ( X2) and _Z7 (s, X i ) ! L { s , X -d

differ only by terms involving powers of those primes that divide qbut

not/. For a > 1, these terms give at most

m=2?p\qp\Xf _

^ 2 lospKp 1 ) < log ?- ( u -8)p\q

The inequality (14.8) applies also fo r / = 1, X2 = Xo- conclude that

(14.6) is valid for any non-trivial xmodg, possibly with a different

O-constant, and that

_ R e ^ ^ Z jL _ R e V _ i |- 0 ( l ( t ) ) (14.9)L ( s , X o ) Is 1|

for the trivial character Xo mod q.

If xais non-trivial, substitution of (14.6) and (14.9) into (14.7) gives

4(uj8)_1 < 3(ct 1 )-!+ O(Z(0), (14-10)

implying that / ? < 1 - c J H y ) (14.11)

for some absolute constant cx when we choose aappropriately. If x2is

trivial, then4(ctjS) -1 < 3 ( a - l ) -1+ ( a - l ) / { ( a - l )2+ 4 y2}+0{Z(y)}, (14.12)

which is consistent with j8= 1 when a- 1. However, if

\ y \ > c j l ( y ) (14.13)

for some positive c2, then by choice of a in (14.12) we can show that

0 < 1 - c 8/Z(y), (14.14)

with a smaller absolute constant c3. We have now shown that either

(14.14) is true or |y| < 8/log?, (14.15)

where is an absolute constant. The absolute constant c3 in (14.14)

depends on the choice of 8 in (14.15). When (14.15) is satisfied with


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60 THE PRIME-NUMBER THEOREM 2.14

y ^ 0 we can still deduce an

m.n. huxley-the distribution of prime numbers_ large sieves and zero-density theorems (oxford...

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