a sieve result for farey fractions

$: A sieve result for Farey fractions$
Lithuanian Mathematical Journal, Vol. 39, No. 1, 1999

A SIEVE RESULT FOR FAREY FRACTIONS

Villus Staktnas

Abstract. This paper is concerned with the sieve problem for Farey fractions (i.e., rational numbers with denominators less than x) lying in an interval (LI, k2). An asymptotic formula for the sifting function is derived under the assumption that (~.2 - Ll)X ~ oo as x -+ co. Two applications of this result are made. In the first one, the value distribution of the vector o(m/n) = (~(m), ~(n)) is considered; here, for k = PIP2"'" Ps, Pl >1 P2 >1 . . . . ~(k) is defined by r = (log pt / log k, log P2/log k . . . . . log ps/log k, 0 . . . . ); all pi are prime numbers. It is shown that the limit distribution is 7r x zr, where zr is the Poisson-Dirichlet distribution. The asymptotical behavior of finite-dimensional distributions of ~(k) for natural numbers was studied by Billingsley, Knuth, Trabb Pardo, Vershik, and others; the result of weak convergence to the Poisson-Dirichlet distribution appears in Donnelly and Grimmett. The second application is concerned with the density of sets {re~n: f (m/n) = a}, where f is a function with the almost squareful kernel.

Key words: Farey fractions, Poisson-Dirichlet distribution.

A SIEVE ESTIMATE FOR RATIONAL NUMBERS

We introduce the system of intervals of rational numbers (Farey fractions)

{m } Ux = : (m, n) = 1, n ~< x N (~q, ~'2), (1)

where 0 ~< ),1 < X2, and ~1, )"2 may depend on x. Let Q0, Ql, and Q2 be some natural numbers depending, possibly, on x and having no common prime factors. In this section, our main purpose is to prove an asymptotic estimate for the sifting function

S(~x, Qo, QI, Q 2 ) = #{ m E ~x: (m, n ) = (m, QoQt) = (n, QoQ2) = 1 }.

THEOREM 1. Uniformly in Qo, QI, Q2, and 0 ~ ~.1 < ~'2, we have

S(.TCx, Qo, QI, Q 2 ) = ~--~(~-2- ~.t) x2 I'-I (1 2 p + l )

plQo

• p+1) {I + 8"R(x' Q)}' (2)

R(x, Q)= 2(2+E)'fQ)( l~ 1 ) + \ x x0.z -- ~-l)

Supported by the Lithuanian State Science and Studies Foundation.

Vilnius University, Naugarduko 24, 2600, Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 108-127, January-March, 1999. Original article submitted September 14, 1998.

0363-1672/99/3901-0086522.00 �9 1999 Kluwer Academic/Plenum Publishers

86

A sieve result for Farey fractions 87

where Q = QoQl Q2, oJ(Q) denotes the number of distinct prime factors of Q, e > 0 is an arbitrary number, arkt the quantity B~ is a bounded function with the bound depending only on ~.

It is easily seen that, with w(Q) bounded, (2) is not trivial for intervals with the condition x()~z - )~l) ---> as x --+ oo. This yields an improvement of the sieve result ([7], Lemma 1), which gives a non-trivial asymptotic for S(.~'x, Qo, QI, Q2) with the assumption x Q . 2 - ~q) / logx --+ oo as x -* oo.

The proof of this theorem is based on the following simple lemma.

LEMMA. Let Jcx be the set of Farey fractions defined by (1), and let f (m/n) be a function taking a complex value for any m/n such that (m, n) = 1. Then, for the sum

E { (nm~) m } S(f , -,~x) = f : - - E a~'x n

the following equality holds:

where

d<~x

M(u) = Z Iz(m)' m~u

T ( n ) = E { f ( n ) : X l n < m < X 2 n } .

(3)

(4)

Note that, in the definition of T(n), the coprimality of m and n is not required;/z(m) in M(u) is the Mrbius function.

Proof of the lemma. It is quite straightforward. We have

dl(m,n)

=EE/z(d) E f : Xin/d<m <X2n/d} n<~x dln

n<~x din d<~x

and the lemma is proved.

Proof of Theorem 1. We apply the lemma for f (m/n ) defined by

f ( m ) _-- {1,0, otherwise.if (m, QoQl)= (n, Q o Q 2 ) - - 1 ,

In this definition, we suppose that (m, n) = 1. Let qo(k), ql (k), and q2(k) be multiplicative functions defined on powers of primes by

pro, p l Q i , qi(pm) = 1, otherwise.

Now we write the sum (4) as

T(n) = #{m: m = qo(n)q2(n)dm', `kin < m < `k2n, d t ql(n), (m', QoQl) =- 1}

Z { )'In ~.2n (m, QoQI)= I }. = # 1: qo(n)q2(n)d < m < qo(n)qz(n)d'

d[ql(n)

88 V. Stak~nas

For the summands, we use the standard asymptotic equality

#{m: u < m < o , ( m , P ) 1 } ( v - u , H ( 1 - 1 ) = = + B2 ~(P),

pIP

(5)

which holds uniformly for all 0 ~< u < v and P ~< 1. Here and in the sequel, B is some absolutely bounded function, usually not the same at each occurrence.

We then have

T(n) = (~'2 -- ~'1) H plQoQl

qo(n)q2(n) E "d + Br(qt(n))2~~176 dlqt (n)

where r(n) means the divisor function. Replacing the sums T(n) in (3) by these expressions, we obtain

S(.~x, Qo, Q~, Q2)= (~.2- ~-1) H ( l - 1)Sl(x ) + B2'~176 plQoQi

(6)

where (x) .

s,(x) = S ,M qo(.Tq;(,.) S , ~, n<~x dlql (n)

S2(x)=EIM(x) r(ql(n,). n<~x

Let us first estimate the sum S2(x). We bound it as follows:

d,ql (d)=d n <~x/d

(7)

Using the prime number theorem in the form

M(u) << u exp{ - r ~o,/i~}, u ~ 2,

we establish the bound

lm( )l <<o+ <<o+o lexp{-cl /l~ n<~ v l ~n<. v/2 l ~n<. o/2

[log(v/2)/log 2] 1

-exp" { - r m=l o/2=+1 <n<~ v/2 m

oo

<< v Z e-C2"/-~ << v; m=l

here and in what follows, we denote by r some positive constants. With this estimate at our disposal, we deduce from (7) that

S2(x) << x

d,ql (d)=d

= x H 1+ + + . . . <<xl-I ~+ d F 7 ~

plQz plQI

(8)

A sieve result for Fareyfractions 89

To deal with the sum Sl (x) we invert the order of summation and rewrite it as

SI(X) = E / z ( d ) qo(n~q2(n) E = E l z ( d ) V x d<~x n..zx/d 8lql (n) d<~x d '

where

( " ,'-) v( . )=~ q0(.~2(.)2 �9 n<~u ($[ql (n)

We introduce the set of positive integers

~D = {dodld2: qi(di) -" di, i = O, 1, 2}

and take Q = Qo Q IQ2. We then have

V(u) = E E mdl ~: m <<. dodld2' (m, Q) = 1 aoa ~ d2 ~ v ,{ u }

= Z d l Z s E m: m<~dodlld2'(m'Q)=l �9 dodld2~'D ($[dl

We replace the inner sum by

1 ( u '~2 ( 1 _ 1 B2o~(Q ) u ,~0~,~, H ~)+ ~o~,~

plQ

(9)

which can be easily obtained from (5) by integrating by parts. This leads to the following expression for V(u):

pIQ

1 1

dodld2E~ d~dld~ a[dl

1 1

Z~ ao< ~ a dodld2$D

The quantity Vl is the product of three sums that can be easily written as follows:

1 (1 - plOo

1 1 Z z ~ x = E dl pIQl

= I - I pIQI

Z ~ = 1-I - d2 pl Q2

By some standard calculations we obtain the following upper bound for V2:

n plQoQ2 p]Ql

90 V. Stal~nas

In view of these results, we have

v(u,=�89 FI ( ,-~)FI(t-~) -'+B2"(~'" 1-I plQoQ2 plQ plQoQ2 plQt

Applying this expression in (9), we derive

pIQoQ2 pIQt a<~x d

Using

we finally obtain

6 ( p~) /~,(d) B ~-~=1-I , - = ~ 7 + - , p d~x

+B2~'(Q)xl~ H ( ' - I - 1 ) H ( 2 + l ) �9 (10) ~IQoQ2 plQI

Now let us replace &(x) and S2(x) in (6) by their expressions (8) and (10). Omitting the absolutely bounded products in the remainder term, we deduce

S(.Tx, Qo, Q,,Q2)=~-~(xz-~.,)x2 E (1 - --~)p+ QYIQ (1 p +i-)l pIQo p[ 2

plQt

Using some routine technique for products over primes, we reduce the last equality to

S(~x, Qo, QI, Q2)= ~2 (~.2- ~.l)X2 1-" I (1 2 plQo p + l ) I"[ plQI Q2

(1- 1---"~) { 1 + B2~ P +

where ( : ) ( , ) x L = l~ ~ 1+ E 2 + + ~ I - I 1+

x 7 7 z2 ~, 7 pIQoQ2 plQl pIQ

Using the obvious inequalities log2(1 + x) ~< (ln2)-lx and log2(2 + x) <~ 1 + x, x ~< O, we have

l~ E 1 + P ~< ~ pin----2 + ~ ln'---2 ~< cl(e) + eog(QoQ2), P[QoQ2 2/(p In 2)~e 2/(pln2)<~ P

PIQoQ2

(11)

( ' ) ( ' ) l og2H 2 + p ~<~-~ l + p <~c2(e)+(l+e)co(Q,). plQI plQI

Hence the product in the first summand of (11) can be replaced by BE2 (t+~)~ The same arguments hold for the rest. We reduced the remainder term to the form required. The proof of Theorem 1 is complete.


T H E P R I M E F A C T O R S AND T H E P O I S S O N - D I R I C H L E T D I S T R I B U T I O N

For a natural number m, let Pt ~ P2 ~< "-" ~< Ps, s = f2 (m) , be its prime divisors in decreasing order; then m = p i p 2 " . . . " Ps. We take Pi = 1 for i > s and define

= [ l o g p t logp2 "e ( m ) ~ logm ' 1o--"~ ' " " ")"

The function ~(m) takes its values in the set

A = {(am,X2, ...): X l ~X 2 ~ -.., X I --[-X2 ~t -.-- -" I},

which we shall consider as a subset of the product space Ioo = [0, I] x [0, I] x --- endowed with the Borel o-algebra B(A). It was shown in [2] that the probability measures

B e ~< x"

converge weakly to the Poisson-Dirichlet distribution 7r as x --> oo. The Poisson-Dirichlet distribution can be characterized (see Donnelly and Grimmett [2]) as the measure

= ) . f - l ,

r(A) = ) ` ( f - l (a ) ) , A e B(A),

where )`: 13(Ioo) --> [0, 1] is the Lebesgue measure defined on the Borel it-algebra o f / co , and f = h o g is the composition o f two functions continuous on their sets of definition. The function g is defined on Ioo by

g((xl , x2 . . . . )) = (Yl,)'2 . . . . ), Yl = Xl, Yn = (1 - Xl)(1 - x2) �9 �9 �9 (1 - - Xn-l)Xn, I"1 <~ 2,

and takes its values ).-almost surely in

T = { ( X l , X 2 . . . . ) ~ Io0: Xl + X 2 + ' ' " = 1},

and h: T ---> A rearranges the components of (xl, x 2 , . . . ) ~ T in decreasing order. For a rational number, we define

We adopt the approach of Donnelly and Grimmett [2] to the case of intervals of Farey fractions and, using Theorem 1, prove the weak convergence of the probability measures

= - - ~.Tx: 17 ~ B , B E B ( A ) x B ( A ) . n

THEOREM 2. Let 0 <~ )`l < )`2 and le t . for each y > O,

(1 + ~.l)t-~(~2 - ~q)x v ~ oo as x --> oo.

Then the probabi l i ty measures vx converge weakly to the product o f Po i s son-Dir i ch le t distributions Jr x rr as x--+oO.

P r o o f o f Theorem 2. We first give an outline of the proof. We have

r r x n ' = ( ~ x ~ . ) F -1, F: / o o x / o o - - + A x A ,

92 V. StakAnas

where

F ( ( x , y ) ) = ( u , v ) , u = h o g ( x ) , v = h o g ( y ) ,

with the functions g and h defined above. We express the measure Vx in the form

Vx = IzxF -l, /Zx: B(A) • B(A) --, [0, I],

and show that/x~ converge weakly to ~ x ~. as x --+ cx~. This yields the weak convergence of Vx to Jr x a" as

We start now with the construction of/zx. Let Q+ be the set of all positive rational numbers re~n , (m, n) = 1.

For any m / n E Q+, m = P I P 2 �9 . . . " Ps, n = qlq2 " . . . �9 qt , where Pi, q~ are all primes, we define

{ ( m . ) o. } ~ m / n - " , t r , t r , : = ( P f i , Pi2 . . . . . P i , ) , t r , -~ ( q h " qJ2 . . . . . q J , ) ;

here tr*, tr. run over all rearrangements of prime factors of m and n, respectively. Let now f2 be the union of I m tT* all ~2m/,. We define the function ~: f2 --+ I ~ x I~ taking, for o) = ,~-, , tr.),

=

where ~ is defined as follows on tr* = (Pil . . . . , Pi , ) , tT, - - (qh . . . . . qh):

._. [ l o g P i t logpi 2 Iogpi.__..._L ) ml

~(o'*) \ l o g m l ' logm2 . . . . ' l ogms ' 0 . . . . . m l - - m , ml+l = Pi--~l '

) " ' " - / l ~ l~ l ~ , h i = n , n t + l = - - . ~(0".) \ l o g n l ' logn2 ' " ' " logn/ qh

Now we introduce the probability measure Px on f~. For o~ = (~, o'*, or,), we define

l [ ~ m , Px(w) = . . - ~ l l - - , U x l r ( ~ r )r(tr.) ,

~j-x X n / ( 1 2 )

I m where (~-, ~'x) is the indicator function of the interval of Farey fractions 5r~, which is assumed to be nonempty, and r(tr*), r(tr.) are defined below.

If p is a prime and n a natural number, then we denote by t~(p, n) the greatest positive integer k such that pkln. Now we set

where

r(cr*) = P(Pq)P(Pi21Pi t ) " . . . " P(Pis lPi l , Pi2 . . . . . P i , - i ) ,

log Pil p (pq ) = ot(pq , ml) lo--O~l,

�9 log Pit P (P i t [ P i l , Pi2 . . . . . P i t - l ) - " c t ( p i t , m l ) ~ o g - ~ l l ,

and r(tr,) is defined quite similarly. We now take/zx = px~- l , i.e.,

m I ~--- m ,

m m l

P i l Pi2 " " " P i t - l

m ( 8 ) = z 8) , B Z(A) x

By straightforward calculations we check that

= - - E .F,,: ~ ~ B = l z x F - l ( B ) , n


and the relation vx = IzxF - I is established. Now we are going to prove that/Zx converge weakly to the Lebesgue measure on loo x Ioo, which we denote

by ~. x ~. using the notation ~. for the Lebesgue measure on Ioo. To prove the weak convergence of some probability measures ~.x on a countable product of unit intervals

I-I{Ij: j E T} to the corresponding Lebesgue measure ~., it suffices to establish the weak convergence of finite dimensional distributions

LxZr~ -1 ~ )~zr~ -1, x -~ oo, (13)

where rrr((Xl,X2, . . . ) ) = (xtt,xt2 . . . . . xtm}, for r = (tl, t2 . . . . . tin), ty E T. As noted by Donnelly and Grimmett [2], to achieve this, it suffices to prove that

m

lim infLxrrr t (E) ~< ~.rr~'l(E) = H ( b t k -- ark), k = l

(14)

for each set E = (aq,bt l ) x . . . x (at,.,bt,,) C Iq x . . . x It,., 0 < art < btl < 1, l = 1 , . . . , m . For a detailed proof of this useful fact, see Donnelly and Grimmett [2]; we show here only that (14) implies

(13) in the special case r = (t). Indeed, if

is satisfied for all 0 condition

liminf~.xZr~I ((a, b)) ~< b - a

< a < b < 1, then (13) holds, unless there is some (c ,d) , 0 < c < d < I, with the

l iminf ~xZrrt ((c, d)) ~< d - c + 6e,

where e > 0, and it can be chosen to satisfy e < min(c, 1 - d). Then, for all x sufficiently large, we should have

1 = X:r~- ' ( l t ) <~ XxTrU'((s, c)) + Xx~r~-l((c, d)) + X:r~- l ( (d , 1 -- ~)),

1 < ~ ( c - 2 ~ ) + ( d - c + 5 e ) + ( 1 - d - 2 e ) = l + e ,

a contradiction. We now have to prove that, for every pair of sets E = (al, bl) x . . . x (as, bs), E ' = (cl, dl) x . . . x (ct, dr), 0 <

ai < b i < 1, 0 < c ) < d j < 1,

$ t

liminfP(x, E, E') .< X(E)-X(E'), X(E) = 1-I(bi - a D , L(E') = H(dj -cj) , i= t j = l

where we denoted

P(x , E, E') = Px ( (m/n , r or.), ~(cr*) ~ E x /co , ~(tr.) ~ E' x Ioo).

Using the definition of Px, we rewrite P(x , E, E') as

P(x , E, E') =

where �9 (pi, qj) = 1, k <~ s, l <~ t satisfying

05)

�9 f / 1 S ' * 1 7 o t ( p i . m i ) l o g p i ct(qj, nj) logqj

#~-x z_., x t log mi log nj i=1 j = l

indicates that the sum is taken over the collections of prime numbers (Px, - - - , Pk), (q l , - - - , ql),

m - - ~ U x tl

w i t h m = p l - . . . . p k , n = q l . . . . . q t ;

pi E (lFlai,Fl2~i), qj E (Fl?,rl dj) f o r l ~ < i ~ < s , l <~ j <~ t;

94 V. Stak~nas

here ml = m, nl = n, mi ..~ m / ( p l . . " P i - l ) , nj = n / ( q l . . "q.i- l) , i, j <~ 2. ' We now look for a good lower bound for P ( x , E, E') . Let us first replace all quantities a (p i , mi) , a(q.i, n j )

by 1. We then omit the terms corresponding to ~ 6 f x with n <~ ex , where e > 0 is an arbitrarily chosen number.

Hence, we shall consider only those summands that correspond to m / n satisfying

~ln < m < ~2 n , 8x < n ~ x .

-q It is easy to see that iL with s = x / ( q t . . , q j - l ) , the inequality xj < qy < (es holds, then the inequality

n~ i < qj < n ; j with nj = n / ( q l . . "q j - l ) is satisfied as well. Similarly, i f xi = n / ( p l . . " P i - l ) and (~.2xi) al <

Pi < (~.lgXi) bl, then m7 i < Pi < mbi j. Using these facts and the obvious inequalities nj <~ s mi <~ ~.2xi, we bound P ( x , E , E') from below as follows:

1 s log Pi ( ' I log qj P (x , E, E') ~ ~ x Z * H log(&2xi) l ogs

i=1 j = l

(16)

where by * we now indicate that the requirements

m

n m = pl . . . . . pk, n = ql . . . . . qt <~ ex ,

- q ()~2Xi) al < Pi < (eLlxi) hi, xj < qj < (es di, 1 <<. i <<. s, 1 <x j <<. t,

must hold for all the summands. We denote the sum on the right-hand side of (16) by S (x , E , E') and inverting the order of summation rewrite

i t as $

S(x , E , E') = ~_.* I - I logpi i=1 1og(~.2xi)

(17) x ti_i logqj # i r a ~ U x : n <~ ex, p , . . . ps lm, q , ' " q t l n }

j=l logs t n

where the sign �9 now means that the sum is taken over the collections of primes (Pt . . . . . p,) and (ql, . . . , qt), (Pi, qj) = 1, satisfying the requirements

_cj (~'2Xi) ai < P i < (8~ ' lX i ) bi , Xj < qj < (e.~j) ai i = 1 . . . . . S, j = 1 . . . . . t.

For fixed Pl, . . . , Ps, ql . . . . . qt, (Pi, q)) = 1, denote u = Pl �9 . . . �9 Ps, v = q, �9 . . . �9 qt. Then we have

{ m } { k k v x ( k , l ) = ( k , v ) ( l , u ) 1] ~x: ulm, vln = : -[ E-Qq,u ~ .2) , l ~ < - , v = = "

We are now ready to apply Theorem 1. This yields

-; ulm, = 1-I ( I - pluv

1) ( x) �9 - + ( 1 8 ) p + l + B (~.2 ~ q ) x l o g x

u v '

where p in the product runs over distinct prime factors only. The condition n <~ ex in (17) introduces an error term not exceeding e L ( x , u, v), where by L we denote the

expression on the right-hand side of (18). We then have

S(x , E, E') <~ (1 - e){S1 + BSz + BS3}, (19)


~2 s log Pi ( SI = (~-2 - ~.1) x2 E * I-I Pi log(~.Xxi) 1

i=1

1 ) j i l l logqj (1 1 ) p~ + 1 qj log ~j qj + 1 '

fl log Pi f l log qj $2 = Q.2 - )~ l)x log x E * Pi log Q.2xi) log ~j '

i= l j = l

log Pi t log qj $3 = x E * l"I logQ.2xi) H log.~j ' qj

i=1 j l '=

where the sign �9 has the same meaning as above. We now show that though the quantities ~.2Xi, .~j depend on Pi, qj, they grow unboundedly, uniformly in i, j

and all possible collections of Pi, qj. Since

~.2Xi = Pi <~ (~.2xi) bi,

~-2Xi+ t

we have ~.2Xi+I ~< (~2Xi) l-hi , and, by induction,

~,2Xi+I ~ (~.2X) (I-bl)'''(l-bs) ~ (~,2X) a ,

with tr = (1 - b l ) ' - - (1 -- bs) for all i = 0, 1 , . . . , s - 1. A similar uniform bound holds for :~j. We use the fact just established calculating the sums on the right-hand side of the inequality

log Pl S1 ~<)-~(1 - 8)(X2 - kl)X 2 ~--'~* Pl log(k2Xl)

Pt

x ~ - ' * logql ...~)--~. logqt , - , qSgi ,' ql qt

log Ps

"'" ~ * Ps Iog(X2x~) Ps

which holds for x ~< x0 with 8 = 8(xo) > 0, 8(x0) ~ 0, as x ~ oo. Here the sign * in the sum over Pi and qj, respectively, indicates the requirements

_cj (~,2Xi) al < Pi < (~,l~Xi) bi , Xj < q] < (E~j)dj .

In view of the unbounded growth of the ranges, we have

log Ps 1 * Ps log(~.EXs) = log(3.2xs) ( log(~qexs)b~ -- 1og(~'2xs)as) + O(1),

Ps

X ----> OO.

We further have

log(~.leXs) bs - log(3.2xs) as (bs as) log(~.2Xs) - bs log ~.2xs = - + o ( 1 ) ~,lEXs ~2 1

= (bs -- as) log(X2Xs) - bs log ~ l - bs log -e + O(1).

We now suppose that ~'l ~ e()~2 -- ~'1) holds for x <~ xo(e). Then

log L2 ),2 - X1 1 Ll ~-i e

96 V. Stak~nas

and we obtain log ps

E * - (b, - as) + o(1), log(L2xs) Ps P,

Applying similar arguments to all remaining sums in St, we have

x --+ oo. (20)

St ~ ~2(I - 8)(~.2 - )~I)X2{L(E)3.(E ') + o(1)}, x ~ oo.

Estimating $2, we use (20) for the sums corresponding to primes Pi, while for those corresponding to qj, some less precise treatment is sufficient. We then obtain

t

$2 << (~2 -~.,)~.(E)xlogx E * H logq/ ql ..... qt j = l Iogxj ~ ( ~ - 2 - ~ ' i ) ~ ' ( E ) x l o g x q l E q t *1,

(21)

where the sign * stands for the requirements on qj. Using the obvious bound

EX

qt

we derive , 1 , 1

E * l ~ < ( ' ~ x ) a ' E ~ , ' " E q~_ " ql,...,qt ql. "11 qt-I

We further use the following simple bound:

(22)

E p~ 1 l - - a < U , p~<u 1 -

which is valid for 0 ~< a < 1. Applying it to the last sum in (22), we obtain the estimate

1 1 1

ql,...,qt ql q l qt-2 qt--2

By induction, we derive , 1 1

E "1 ~< c(u)(ex) ~(') E ,~a(,,) " " E ' q : ( , , ) ' q I ..... qt ql "11 qu

where ~(u), I ~< u ~< t - 2, are defined recursively by

(23)

a ( t - 1 ) = d t , u ( u - 1 ) = ~ ( u ) + ( 1 - ~ ( u ) ) d u , u<<. 1;

all constants c(u) are positive. Then (21) and (23) yield

$2 << (>,2 - ~q))~(E)x logx(ex) ~', 0 < ~ < 1, t~ = ct(O).

We apply the same procedure for estimating the sums in $3. This leads to the bound

$3 <<x(sLlx) 'L(E' ) , 0 < fl < 1.

A sieve result for Farey fractions 97

Now the bounds for Sl, $2, and $3 used in (19) imply

S(x, E, E') <~ 3 ( 1 - e)(l - 8)0.2 - ~q)x2{)ffE))`(E ') + o(1)

+ B { : - ' logx + (~)`1)~x~-~()`2 - ) ` 0 - 1 } } .

Since (1 +)Ò~'-t()`2 --)`t)x ~' ---> oo for any }, > O, we have (e)`l)#X#-l(X2 --)`t) -1 --> 0 and

S(x, E, E') <~ 3 ( 1 - e ) ( 1 - 8)()`2 -)Ì)x2{)`(E))`(E ') +0(1)} , x --+ 00.

Then, together with (16), this gives

l iminf P(x, E, E') <~ ( 1 - e)(1 - 8))`(E))`(E'),

where we used #: 'x "" 3 ~()`2 - )Òx 2, which easily follows from Theorem 1. Since the numbers e and 8 can be chosen arbitrarily small, we have proved that

l iminf P(x, E, E') <~ )`(E))`(E'),

under the assumption that, for each e > O,

)`1 ~ S()`2 -- )` l) as x <<. Xo(e). (24)

Now we remove this constraint. Let (24) fail for all e > 0, for some unbounded sequence of values of x depending on e. Let us fix e > 0 and for x not satisfying (24), define ~.] = s(X2 - ~.t). For this sequence of x, we now have two intervals ()`'1,),2) C (M,)`2) and two sets of Farey fractions

= n.< n () '̀1, )`2), G = , : n - < x n(xl,)`2), ~ c G .

We denote by Pff, Px the probability measures defined in (12) for ~ and ~'x, respectively. It is almost obvious that, for any available set B,

, # ~ ~ : # ( G \ ~ ) , , Px(B) = P~(B)~-~x +u~, ~ ). (25)

Since )`2 - )̀ ] = ()`2 - )Ì) + )̀ I - e()`2 - )`t) and )̀ I < e()`2 - M), we obtain

)`2 - - )` t l = ( ) ` 2 - - )`1)(1 -- y) , 0 ~< y < e.

Then, using Theorem 1, we see that

#~ = 1 - y + o(1), #G #( .G\~)

#G < < y , X---~ OO.

Let us denote by P*(x, E, E') the function defined in (15) with P*, instead of Px- Then (25) yields

P(x, E, E') = P*(x, E, E')(1 - y) + By, x --+ oo.

Since, obviously, )`'t ~< e()`2 - )`'t), by the result established above we have

l iminf P*(x, E, E') ~< (1 - e ) ) ` ( E ) L( E ' ) .

Then it follows from (26) and the inequality 0 ~< y < e that

l iminf P(x, E, E') ~< (1 - e)(1 - e))`(E))`(E') + Be.

(26)

98 V. Stak2nas

Since we are free in the choice of e, we get

l iminf P(x , E, E') <<. X(E)X(E'),

and the proof of Theorem 2 is complete.

FUNCTIONS W I T H ALMOST SQUAREFUL KERNELS

Theorem I gives a non-trivial asymptotics for the sifting function with the assumption x(~2 - ~t) ~ oo as x --+ (x), that is, the intervals (A.l, ~-2) are not allowed to vanish too fast. However, we have proved Theorem 2 under a more restrictive condition for intervals of Farey fractions. In this section, we prove a result on the value distribution preserving, for the intervals, the same condition as in Theorem 1.

We consider a function defined for the positive rational numbers and taking values in some (abstract) set A. For prime p, we denote by u t, the maximal integer u such that, for each v, [vl < u, the inequality

holds for all rational numbers m / n provided that (p, ran) = 1. If (27) holds for all integers v, we take up = +(x~. Then u t, ~< 1 for all primes. We say that a rational number m / n belongs to the kernel of f (denoted by K ( f ) )

~t a2 ~" the inequality oti ~< ut, ~ holds for all primes Pi. if, in the canonical representation mn = Pl P2 "'" Ps ,

Definition. We say that a function f ( m / n ) has an almost squareful kernel if

E : - - ~ K ( f ) < oo. (28) n

It is obvious that condition (28) is equivalent to

1 E pU; < (x).

P

If f2 (n) (a~(n)) denotes the number of all prime (all distinct prime) factors of a natural number, then f ( m / n ) = ~2(mn) - co(ran) is a function with an almost squareful kernel. It was studied over the set of positive integers by many authors (see an account in Wu [4]). Any additive function satisfying f ( p ) = 0 for all (or almost all primes) has an almost squareful kernel. This class of functions is considered in the monograph of Kubilius [6]. We denote, as before, by Y'x the set of rational numbers lying in the interval (~.t, ~.2) with denominators less than x.

THEOREM 3. Let 0 ~< ~.1 < ~-2, ~-1 ( ( x(~.2 -- ~.1) ~ ~:~ as x --+ o0. Then, for every function f with an almost squareful kernel and its value a, there exists a number v(a, f ) such that

#Yx # n e Y:x: f -- a --+ v(a, f ) , x --+ cx~.

Moreover,

I): a A}=l.

a n d / f f - l ( a ) is nonempty, then v(a, f ) > O.

From this theorem it follows immediately that if f is a function with an almost squareful kernel, then for any value a and a function p(x) --+ (x~ (x ~ ~ ) , there are infinitely many rationals satisfying

O~ - - ~ ' , ~ a , n

for every positive number or. A slight modification of the proof shows that this statement is true with p(x) replaced by some constant C( f , a) > O.


Proof of Theorem 3. Let y <~ 2. For a positive integer m, we set

= [-[ y}.

For an arbitrary function g(m/n), we define

=

Then gy is always a function with an almost squareful kernel, the set of its values a is finite, and, moreover, any nonempty set g-~l(a) is a union of disjoint sets

{ Qtrn. (ran, QtQ2' = (re, n, = 1 } , A Q I ' Q 2 = Q2n"

where [QiQ2]y = QIQ2. It follows from Theorem 1 that

{ m m } 1 Ql~IQ ( 2 ) Vx n: -- ~ .Tx tq AQbQ, > 1 , x --+ oo; , , O.i02 2 P + l

here and further, for a subset of rational numbers A, we denote by vx{A} the density #{re~n: m/n ~ .TxNA}/#.Tx. Hence, Theorem 3 holds for each function gy.

Now let f be a function with an almost squareful kernel. We first sketch the proof. We are going to prove that

[m f ( m ) f y ( m ) } vx --" r << E(y) (29) n "

and e(y) --+ 0 as y ~ oo. This gives, in particular,

m m m Vx{n: f ( m ) = a ] - - V x { n : f y ( n ) : a l <<e(y), (30,

o r m

VX{n: f ( m ) = a ] - - v ( a , fy)<<e(Y)+8(x,Y),

with 8 (x, y) ~ 0 as x ~ oo. We then establish the bound

(31)

v(a, fy) -- v(a, fz) << e(y, Z), y ~< z, (32)

oo. As a consequence, we obtain that v(a, fy) tends to some limit v(a, f ) as where 6(y, z) --+ 0 as y --~ y -+ oo. Then (31) yields :(m)

iim vx : =a = u(a, f) . x-..-~ oo

Let A(y) = {a: f i l ( a ) ~ 0}. If (29) is established, then we have

m m S (Vx{ m: f ( m ) = a ] - v x l n : f Y ( n ) = a / ) < < e ( y ) .

aEA(y )

Passing to the limit in this relation x --+ cx), we obtain

{v(a, f ) : a E A(y)} - I << e(y).

I00 V. Stakenas

Hence,

E v(a, f ) = I a

holds. Thus, it remains to to prove (29) and (32). We use the obvious relation

m m

l~ ~ ~-: : (~ ) ~ :,(~)1 ~ U { m ~ . : p"'lmn}. n

p,pup >Y up<O0

(33)

Let us consider the set

in j 9rx: p"lmn = Dl(p",x) UD2(p",x),

{m } {m : } D l ( p " , x ) = n ~ Y ' x : p"lm , 7 9 2 ( p " , x ) = n ~ Y ' x : I n .

We obviously have

J : - - ~ (k t ,~ .2 ) ,n~<x , (m, n) = (n, p) = l , n

and, by Theorem 1, we obtain

: ( , ){ .,o~x " i #Dt(p", x) = (~.2 - Xl)~'-~ 1 P + 1 1 + ~ x + x(~'~--- ~-1) "

Since pU << XEX we have #Dl(p u, x) << (X2 - - X I ) x 2 p -u , provided that ~.2/(~.2 - - X I ) <(<~ 1, or , equivalently, Xl/(~2 - Xl) << 1. Let us now consider the case where the last condition is not satisfied.

Let el(X2 - ~.t) < ~.1 < r - ~-1) X with some positive constants. We then have

#DI(pU'x) <<" E E {I: "'~2 <~n <~ mpu mpU]x, , m<~.2x/pU

;~ ~ + 1} ~ -;~ p ~-71 + p, <~ l . , , . , , .d t

m<~.2x/p u

�9 . x 2 k2 k2x x 2

here we have used the inequalities ~.2/LI = l+(~.2--~.l)/Xl < l + l / C l , and X2x = >,lx+(k2--Xl)X << (L2--XOx 2. Hence,

x 2 #79~(p", x) << (;~2 - ~.~)-- pU

holds provided that kl << (~-2 - - ~-1) X. For 792(p u, x), we have

/mm ~u J D2(p ~,x)= -- : - - E p " ( L 1 , L 2 ) , n ~< , ( m , p ) = 1 , n n

and Theorem 1 now yields

#792(pU,x)=-~(~.2--~.l)X~--~(1

pU log x /p u X { I + B x

1 p + l )

1 } x 2

"q- X(~-2 ~-1) <~( (~'2 '~-1) pU


Using the bounds for #DI ( p ' , x), #D2(p", x), we have

# --n E.T'x: f =/:fY n << (3.2 -- ~.tlx 2 ~ pup" pup > y

Hence, (30) holds with

1 t ( y ) - + 0, y --~ c~ . e(y) = pUp'

pup > y

Using the same arguments with fz instead of f , we have

and

with

m I

y<pUp <Z

vx --" fz = a --vx -- : fv = a <<e(y,z) II"

1 e(y, z) --~ 0, y - ~ ~x~. e(y, z) = ~ pUe , y<pUP <z

Taking x --+ oo in (34), we establish (32) and complete the proof of Theorem 3.

(34)

R E F E R E N C E S

1. E Bilhngsley, On the distribution of large prime factors, Period. Math. Hungar., 2(1--4), 283-289 (1972). 2. E Donnelly and G. Gdmmett, On the asymptotic distribution of large prime factors, J. London Math. Soc., 47(2),

395-404 (1993). 3. D.E. Knuth and Lo Trabb Pardo, Analysis of a simple factodzation algorithm, Z Theoret. Comput. Sci., 3, 321-348

(1976). 4. J. Wu, Sur un probl~me de R6nyi, Mlt Math., 117, 303-322 (1994). 5. A.M. Verslfik, The asymptotic distribution of factodzations of natural numbers, Dokl. Akad~ Nauk SSSR, 289(2),

269-272 (1986). 6. J. Kubilius, Probabilistic Methods in the Theory of Numbers, Providence, R.I. (1964). 7. V. Stak~nas, Additive functions and rational approximations, Lit& Math. J., 28(3), 260-272 (1988).

(Translated by V. Stalcdnas)

a sieve result for farey fractions

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