Another application of Linnik dispersion method Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 19. Выпуск 3.

УДК 512.54 DOI 10.22405/2226-8383-2018-19-3-148-163

Новое применение дисперсионного метода Линника

Etienne Fouvry — Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclav, 91405 Orsay, France. e-mail: Etienne.Fouvry@u-psud.fr

Maksym RadziwiH^ Department of Mathematics, McGill University, Burnside Hall, Room 1005, 805 Sherbrooke Street West, Montreal, Quebec, Canada, НЗА 0B9 e-mail: maksym.radziwill®gmail.com

Аннотация

Пусть am и — две последовательности вещественных чисел с носителями на отрезках [M, 2М] и [N, 2N], где M = X1/2-á и N = X 1/2+й. Мы доказываем существование такой постоянной 5o, что мультипликативная свертка ат и имеет уровень распределения 1/2 + 5 — е (в слабом смысле), если только 0 ^ 5 < So, последовательность является последовательностью Зигеля-Вальфиша, и обе последовательности ат и ограничены сверху функцией делителей. Наш результат, таким образом, представляет собой общую дисперсионную оценку для "коротких" сумм II типа. Доказательство существенно использует дисперсионный метод Линника и недавние оценки трилинейных сумм с дробями Клоостермана, принадлежащие Беттин и Чанди. Также мы остановимся на применении полученного результата к проблеме делителей Титчмарша.

Ключевые слова: равнораспределение в арифметических прогрессиях, метод дисперсии.

Библиография: 10 названий.

Для цитирования:

Etienne Fouvrv, Maksvm Radziwill, Another application of Linnik dispersion method // Чебы-шевский сборник, 2018, т. 19, вып. 3, с. 148-163.

CHEBYSHEVSKII SBORNIK Vol. 19. No. 3.

UDC 512.54 DOI 10.22405/2226-8383-2018-19-3-148-163

Another application of Linnik dispersion method

Etienne Fouvry — Laboratoire de Mathématiques d'Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclav, 91405 Orsay, France. e-mail: Etienne.Fouvry@u-psud.fr

Maksym Radziwill^ Department of Mathematics, McGill University, Burnside Hall, Room 1005, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 0B9 e-mail: maksym.radziwill@gmail.com

Abstract

Let am and be two sequences of real numbers supported on [M, 2M^d [N, 2N] with M = X N = X1/2+5. We show that there exists a S0 > 0 such that the multiplicative

convolution of am and jSn has exponent of distribution ^ + S — e (in a weak sense) as long as 0 < S < So, the sequenee is Siegel-Walfisz aid both sequences a^d are bounded above by divisor functions. Our result is thus a general dispersion estimate for "narrow" type-II sums. The proof relies crucially on Linnik's dispersion method and recent bounds for trilinear forms in Kloosterman fractions due to Bettin-Chandee. We highlight an application related to the Titchmarsh divisor problem.

Keywords: equidistribution in arithmetic progressions, dispersion method.

Bibliography: 10 titles.

For citation:

Etienne Fouvrv, Maksym Radziwill, 2018, "Another application of Linnik dispersion method" ,

Chebyshevskii sbornik, vol. 19, no. 3, pp. 148-163.

In memoriam Professor Yu. V. Linnik (1915-1972)

1. Introduction

An important theme in analytic number theory is the study of the distribution of sequences in arithmetic progressions. A representative result in this field is the Bombieri-Vinogradov theorem [2], according to which for any A > 0,

£ (mi*-, £ 1 — ¿y £1 «4 ^ w

p<x p<x

p=a (mod q)

provided that Q < y^logx)-B to some constant B = B(A) depending on A > 0.

Nothing of the strength of (1) is known in the range Q > x1/2+e for any fixed e > 0 and already establishing for any fixed integer a = 0 and for all A > 0 the weaker estimate,

£| £ i — ¿y £ i *0og (2)

q<Q p<x p<x

p=a (mod q)

with Q = x1/2+s and some 5 > 0 is a major open problem. If we could show (2) then we would say that the primes have exponent of distribution 2 + 5 in a weak sense. However we note that there are results of this type if one allows to restrict the sum over q < Q in (2) to integers that are xe smooth, for a sufficiently small e > 0 (see [16, 5]).

Any known approach to (2) goes through combinatorial formulas which decompose the sequence of prime numbers as a linear combination of multiplicative convolutions of other sequences (see for example [13, Chapter 13]). If one attempts to establish (2) by using such a combinatorial formula then one is led to the problem of showing that for any A > 0,

^ ampn — ^ ampn|< X(logX)-4 ,X := MN (3)

q<Q M<m<2M M<m<2M

N<n<2N N<n<2N

mn=a (mod q) (mn,q) = 1

with Q > X 1/2+e for some e > 0. In [14] Linnik developed his "dispersion method" to tackle such expressions. The method relies crucially on the bilinearitv of the problem, followed by the use of various estimates for Kloosterman sums of analytic or algebraic origins. For a bound such as (3) to hold one needs to impose a "Siegel-Walfisz condition" on at least one of the sequences am or

Definition 1. We say that a sequence @ = (@n) satisfies a Siegel-Walfisz condition (alternatively we also say that /3 is Siegel-Walfisz), if there exists an integer k > 0 such that for any fixed A > 0, uniformly in x > 2, q > |a| > 1,r > 1 and (a, q) = 1, we have,

1

E

Pn Y] Pn = 0A(rk(r) ■ x(log x)-A).

<£>(q) '

x<n<2x x<n<2x

n=a (mod q) (n,qr) = l

(n,r)=1

where tj.(n) := n =n 1 is the kth divisor function.

It is widely expected (see e.g [3, Conjecture 1]) that (3) should hold as soon as min(M, N) > X£ provided that at least one of the sequences an, ¡3m is Siegel-Walfisz, and that there exists an integer k > 0 such th at |aTO| < Tk (m) and < Tk (n) for all integers m, n > 1. We are however very far from proving a result of this type.

WThen Q > X1/2+e for some £ > 0, there are only a few results establishing (3) unconditionally in specific ranges of M and N (precisely [9, Théorème 1], [3, Theorem 3], [11, Corollaire 1], [12, Corollary 1.1 (i)]). All the results that establish (3) unconditionally place a restriction on one of the variable N or M being much smaller than the other. We call such cases "unbalanced convolutions" and this forms the topic of our previous paper [12].

In applications a recurring range is one where M and N are roughly of the same size. This often corresponds to the case of "type II sums" in which one is permitted to exploit bilinearitv but not much else. This is the range to which we contribute in this paper.

Theorem 1. Let k > 1 be an integer and M,N > 1 be given. Set X = MN. Let am and be two sequences of real numbers supported respectively on [M, 2M] and [W, 2N]. Suppose that fl = (@n) is Siegel-Walfisz and suppose that |aTO| < Tk(m) and |^ra| < Tk(n) for all integers m,n > 1. Then, for every e > 0 and every A > 0;

E | E am@n - E am@nX(logX)-A W

Q<Q<2Q mn=a (mod q) (mn,q) = l

(q,a)=l

uniformly in N56/23X-i7/23+e < q < NXand 1 < < X.

Setting N = X1/2+5 and M = x1/2 5 in Theorem 1 it follows from Theorem 1 and the Bombieri-Vinogradov theorem that (4) holds for all Q < NX-e with 0 < ô < So := jj2. Previously the existence of such a ¿0 > 0 was established conditionally on Hoolev's R* conjecture on cancellations in short incomplete Kloosterman sums in [8, Théorème 1] and in that case one can take ¿0 = Similarly to our previous paper, we use the work of Bettin-Chandee [1] and Duke-Friedlander-Iwaniec [7] as an unconditional substitute for Hoolev's R* conjecture. In fact the proof of Theorem 1 follows closely the proof of the conditional result in [8, Théorème 1] up to the point where Hoolev's R* conjecture is applied. Incidentally we notice that the largest Q that Theorem 1 allows to take is Q = X17/33-5e provided that one chooses N = X17/33-4e.

Unfortunately the tvpe-II sums that our Theorem 1 allows to estimate are too narrow to make Theorem 1 widely applicable in many problems (however see [15] for an interesting connection with cancellations in character sums). We record nonetheless below one corollary, which is related to Titchmarsh's divisor problem concerning the estimation of J2P<X T2(P — 1) (f°r the best results on this problem see [10, Corollaire 2], [3, Corollary 1] and [6]). The proof of the Corollary below will be given in §5.

Corollary 1. Let k > 1 and let a and fl be two sequences of real numbers as in Theorem 1. Let ô be a constant satisfying

0 <^< ÏI2 •

and let

X > 2, M = X1/2—, and N = X1/2+&. Then for every A > 0 we have the equality

E E ampnT2(mn - 1) = 2 ^ -1- ^ ^ ampn + 0{X(logX)-A).

(mn,q) = 1

2. Conventions and lemmas 2.1. Conventions

For M and N > 1, we put X = MN and C = log2X. Whenever it appears in the subscript of a sum the notation n ~ N will means N < n < 2N. Given an integer a = 0 and two sequences « = (&m) m<m<2m and P = (Pn) n<n<2n supported respectively on [M, 2M^d [N, 2N] we define the discrepancy

E(a, /3, M, N, q, a) := EE am@n - -E E am@n,

mn = a mod q (mn,q) = 1

and we also define the mean-discrepancy,

A(a, p, M, N, q,a) := E lE(a, P,M,N,q,a) |. (5)

q-Q

(q ,a) = 1

Throughout rj will denote any positive number the value of which may change at each occurence. The dependency on ?y will not be recalled in the O or ^-symbols. Typical examples are Tk(n) = 0(nv) or (logx)w = 0(xv), uniformly for x > 1.

If f is a smooth real function, its Fourier transform is defined by

/<x

f (t)e(-£t)dt,

-<x>

where e(-) = exp(2^f-).

2.2. Lemmas

Our first lemma is a classical finite version of the Poisson summation formula in arithmetic progressions, with a good error term.

Lemma 1. There exists a smooth function ip : R —> R+ with compact support equal to [1/2, 5/2]; larger than the characteristic function of the interval [1, 2]; equal to 1 on this interval such that, uniformly for integers a and q > 1, for M > 1 and H > (q/M) log4 2M one has the equality

£ O=+ 7 E «(f)O+<6>

m=a mod q ^ ^ 0<lhl<H '

Furthermore, uniformly for q > 1 and M > 1 one has the equality

E K S) = ^(0)M + 0(T2(q)log4 2M). (7)

(m, q) = 1

Доказательство. See Lemma 2.1 of [12], inspired by [4, Lemma 7]. □ We now recall a classical lemma on the average behavior of the r^-function in arithmetic progressions (see [14, Lemma 1.1.5], for instance).

Lemma 2. For every к > I, for every e > 0, there exists С (к, e) such that, for every x > 2, for every x£ < у < x, for every 1 < q < yx-£, for every integer a coprime with q, one has the inequality

E ^(n) (k>e)Jr-)(i°g2x)

\k-l

x—y<nKx n = a mod q

<p(q)

The following lemma is one of the various forms of the so-called Barban-Davenport-Halberstam Theorem (for a proof see for instance [3, Theorem 0 (a)].

Lemma 3. Let k > 0 be an integer. Let @ = (/3n) be a Siegel-Walfisz sequence such that Ifinl < Tk(n) for all integer n > I. Then for every A > 0 there exists B = B(A) such that, uniformly for N > 1 one has the equality

E

£ £ a

1

g<N(log 2N)-B a, (a,q)=l

n = a mod q

v(Q)

n~N (n,q) = l

ßn =Oa{N(\og2N)-A).

We now recall an easy consequence of Weil's bound for Kloosterman sums.

Lemma 4. Let a and b two integers > 1. Le11 an interval included in [1,a]. Then for every integer I for every e > 0 we have the inequality

n ~

nei

(n,ab)=l

Доказательство. We begin we the case b = 1. We write the factor ^y as

n

ip{n)

Z = E

K,(V) \n

where k(v) is the largest squarefree integer dividing v (sometimes k(v) is called the kernel of v) This gives the equality

£

nei

(n,a) = l

v> l

nei k(V)\n

(n,a) = l

v >l

(v,a) = l

('П)1^ ££ *a)l= S £ )

mel/^(v) (m,a) = l

In the summation we can restrict to the v such that k(v) < a. Applying the classical bound for short Kloosterman sums, we deduce that

nei

(n,a)=l

E ^f Ю l<<£ (£,a) ^a2+£ П (1 - 1)-l V>a) 2a2

+2e

p<a

This proves Lemma 4 in the case where b = 1. When b = 1, we use the Möbius inversion formula to detect the condition (n, b) = 1. □

Our central tool is a bound for trilinear forms for Kloosterman fractions, due to Bettin and Chandee [1, Theorem 1]. The result of Bettin-Chandee builds on work of Duke-Friedlander-Iwaniec [7, Theorem 2] who considered the case of bilinear forms. These two papers show cancellations in exponential sums involving Kloosterman fractions e(am/n) with m x n. We state below the main theorem of Bettin-Chandee.

Lemma 5. For every e > 0 there exists C(e) such that for every non zero integer j, for every sequences let a, P and v be of complex numbers, for every A, M and N > 1, one has the inequality

E E E ct(m)P(n)v(a) e )|<C (^H^m \\Ph,N \\u ^

nJ - —lal

a—A m—Mn—N

x (1 + ) 1 ((AMN) 20 +£ (M + N)1 + (AMN) 8+(AM + AN) 8 3. Proof of Theorem 1

All along the proof we will suppose that the inequality 1 < | a\ < X holds and that we also have

3 1

X8 < M <X 2 < N and Q < N. (8)

3.1. Beginning of the dispersion

Without loss of generality we can suppose that the sequence P satisfies the following property

n | a ^ ¡3n = 0. (9)

Such an assumption is justified because the contribution to A(a, P,M,N,Q, a) of the (q,m,n) such that n | a is

< QX^ +X^ EE r2(lmn - al) + MX£ < (M + Q)XV.

n\a m-M

mn = a

By (5), we have the inequality

A(a, P,M,N,Q,a) < E E ^ E $n - E

q~Q m-M ' n-N V(0)

(m, q) = 1 n = am mod q (n,q) = 1

Let ^ be the smooth function constructed in Lemma 1. By the Cauchy-Schwarz inequality, the inequality |am| < Tk ( m) and by Lemma 2 we deduce

A2(a, p, M, N, Q, a) < MQ£k2-1{W(Q) - 2V(Q) + U(Q) j, (10)

u(Q)= E (Eft.)2 EHM)• <">

with

(q,a) = 1 r n-N (m,q)=1

( n, q)=1

v W) = E MQ (E,ni ) E

(q,a) = 1 ni~N n2~N m=an1 mod q

(ni,q)=1 ("2,<?)=1

w(q) = E WQ)( E ^)( E ^) E Kjj). (12)

(q,a) = 1 n1~N T12 m=an1 mod q

("1, i)=1 ("2, i)=1 m=an2 mod q

3.2. Study of U(Q)

A direct application of (7) of Lemma 1 in the definition (11) gives the equality

^(q/Q) ' —

q<p(q)

= Umt(Q)+0(N2Q-1X1I) , (13)

U(Q)=mM E ( E $n)2 + 0(N2Q-1Xv)

( o,a)=1 q^(q) n-N

( q,a)=1 1

( n, q)=1

by definition.

3.3. Study of V(Q)

H = M-1QX£. (14)

This leads to the equality

V (Q) = V MT(Q) + V Errl(Q) + V Err2(Q), (15)

where each of the three terms corresponds to the contribution of the three terms on the right hand-side of (6). We directly have the equality

V Err2(Q) = 0(M-lN2X71). (16)

For the main term we get

VMT(Q) = mM E ^ (i>)2. (it)

(n,q) = l

Bv the definition of VErrl (Q) we have the equality

^(q/Q)

(g,a)=

V™(Q) = M E f/f ( E A*)

(q,a)=l "2-N

(n2,q)=1

( E «i E *(q/MM^))

"l-N 0<lhl<H

(ni,q) = 1

from which we deduce the inequality

\VErrl(Q)\<MQ-2 £ If3ni | £ l/3n21 £ \V(ni,n2,h)\ (18)

ni^N n,2~N 0<lhl<H

with

V(n.i,n.2,h}= E „(,/Q)JQLi(JmM,af )•

(q,anin2) = l

( , nl ) = 1

ahnl . q ah .

-= -ah--\--mod 1.

nl nl

1 < | a| < X H

is ^ X£t-1 when i x Q. This allows to make a partial summation over the variable q with the loss of a factor X£. After all these considerations, we see that there exists a subinterval J C [Q/2, 5Q/2] such that we have the inequality

\V (m,n2,h)\^X£ V -l-e(ah^-) 1 1 ^ ^(q) \ in)

(?>"1"2)=1

Lemma 4 leads to the bound

ii 1 1 | V (n1 ,n2, h) | « Xs (ah, n1) 2 (n1 n2 )v nf .

Inserting this into (18), we obtain

VErr1(Q) «MN2q-2X£+ E IPm | E (h,Ul)1,

n1~N 0<lhl<H

which finally gives

VErr1(Q) « N5Q-1X2£+ (19)

using the inequality |^n| < Tk(n) and the definition of H. Combining (15), (16), (17) and (19) we obtain the equality

V (Q) = V MT(Q)+0£((M-1N2 + N 2Q-1)X 2e+ ). (20)

where VMT(Q) is defined in (17) and where the constant implicit in the Oe-symbol is uniform for a satisfving 1 < |a| <X.

4. Study of W(Q)

4.1. The preparation of the variables

The conditions of the last summation in (12) imply the congruence restriction

n1 = n2 mod q and (n1n2, q) = 1. (21)

In order to control the mutual multiplicative properties of n1 and n2 we decompose these variables as

(ri1,n2) = d,

n1 = dv1, n2 = dv2, (v1,u2) = 1, (22)

v1 = d1 v[ with d1 | and (,d) = 1.

Thanks to |^n| < rk(n) and to (9) the contribution of the pairs (n1, n2) with d > X£ to the right-hand side of (12) is negligible since it is

«X £ £ £ £ £ 1

X£ <d<2Nm~M vi-N/d q-Q v2-N/d

dmi^i —a=0 q\dvim-a i2=vi mod q

«XV E E E T2(ldu1m -a|)(N + 1)

X £ <d<2Nm~M vi-N/d

d m i 1— a=0

T2^i—1vn — £

« MN2Q-1Xn-£ + X1+n. (23)

Now consider the contribution of the pairs (n1, n2) with d < X£ and d1 > X£ to the right-hand side of (12). It is

«x-'E E E E E E 1

d<Xf xf<d1<2N m^M v^^N/(dd1) q^Q V2^N/d

dl |d~ dd1mv'1-a = 0 qlddlV'm-a modq

«X E E E E T2(lddiv[rn -a|)(N + 1)

d<Xf X^<d1<2N rn^M v'1^N/(dd1) Q

di |d^ , , / 11 dd^mV' -a = 0

« XVMN2Q-1 £ 1 E 7" +X"MN Y, d E ±

d<X- d d1>xc dl d<Xf d d1>xc dl

d1|dTC d1|dTC

« MN2Q-lXf + Xf. (24)

Consider the conditions

d<X£ and di <X£, (25)

and the subsum W(Q) of W(Q) where the variables nl and n2 satisfy the condition (25). By (23) and (24) we have the equality

W (Q) = W (Q) + 0(MN2Q-lX^-f +X l+r>). (26)

4.2. Expansion in Fourier SGFIGS

We apply Lemma 1 to the last sum over m in (12) with H defined in (14). This decomposes W( Q)

W (Q) = W MT(Q) + W Errl(Q) + W Err2(Q), (27)

where each of the three terms corresponds to the contribution of each term on the right-hand side of (6). _

The easiest term is WErr2(Q) since, by |fn| < (n) and (8), it satisfies the inequality

WErr2(Q) «M-l £ £ £rk(ni)rk(n2)

Q/2<q<5Q/2 n1-N

n1=n2 mod q

« M-lN2Xv (28)

According to the restriction (21), we see that the main term is

WMT(Q) = mM £ ^IQ E ( E E ^), (29)

(q,a) = 1 (¿,ç)=1

ni=n2 =S mod q

where the variables n1 and n2 satisfy the conditions (25). By a similar computation leading to (23) and (24) we can drop these conditions at the cost of the same error term. In other words the equality (29) can be written as

W MT(Q) = W MT(Q) + 0(MN2Q-1X^-2 + X 1+1i ), (30)

where WMT(Q) is the new main term, which is defined by

WMT(Q)=mM £ ^^ £ ( £ pny. (31)

(q,a) = 1 q (S,q) = 1 _n~N

n = o mod q

4.3. Dealing with the main terms

We now gather the main terms appearing in (13), (17), (26), (27), (30), and in (31). The main term of W(Q) - 2V(Q) + U(Q) is

WMT(Q) - 2VMT(Q) + UMT(Q)

=mM E mo. E ( E - -1) E ,„)2.

(Q,a)=1 q (S, g)=1 *(q)

n = o mod q (n,q) = 1

Appealing to Lemma 3 we deduce that, for any A, we have the equality

WMT(Q) - 2VMT(Q) + UMT(Q) = o(M ■Q-1 ■ N2(log2X)-A) (32)

provided that

Q <N(log2N)-B, (33)

for some B = B(A).

4.4. Preparation of the exponential sums

By the definition (27), we have the equality

W-HQ) = ME E E$nift,2 E KmH^r)

n " m . nn~N 17 "

ni, n2-N 0<\h\<H

ni =n2 mod q

where the variables (n^^) are such the associated ^d d1 satisfy (25).

This implies that any pair (n1, n2) satisfies n1 - n2 = 0 and since we have n1 = n2 mod q (see

(21)) these integers cannot be near to each other, indeed they satisfy the inequality

N1 -W2| > Q/2.

Since we have (n1n2, q) = 1, we can equivalently write the congruence n1 - n2 = 0 mod q as

V1 - V2 = d1u'1 -U2 = qr, (34)

and instead of summing over q, we will sum over r. Note that 1 < |r| < R/d, where

R = 2NQ-1. (35)

In the summations, the pair of variables (n1, n2) is replaced by the quadruple (d,d1,u'1, v2) (see

(22)). The variables d and d1 are small, so we expect no substantial cancellations when summing over them. Hence for some

d, d1 < X£, d1 |

we have the inequality

WErr1(Q) «X2£MQ-1iwi, (36)

where W = W(d, d1) is the quadrilinear form in the four variables r, u2 and h defined by

W V" V" V" R R ^((d1U1 - V2)/(rQ))

w = L L L ^[^2 (d ' )/{rQ)

1<M<R/d ddiV[ ,dV2~N K 1 1

mod r

J^(M - )>( )' <37)

where e(-) is the oscillating factor

/ ahdd\v' \ &(') = )'

and where the variables satisfy the following divisibility conditions:

(d\, u2) = 1, (,d) = 1 and (ddir, d\— u2) = r. Using Bezout's reciprocity formula we transform the factor e(-) as follows:

ahddiu' , (d\u' — u2)/r ahr , _ 1 = — ah1-^—-j-^--+ ——i—,—;-r mod 1.

(d\— u2)/r dd\dd\(d\— u2)

Since ( ddi, ) = (r, ) = 1 we can apply Bezout formula again, giving the equalities

ah(dlV'\— ?)/r = ahU' ((hV\; V2)/r + ah^^imod 1 dd\u[ ddi

= ahU'(dlV]; V2)/f — ah1-^ mod 1 dd\

The first term on the right-hand side of the above equality depends only on the congruences classes of a, h, r, and u2 modulo ddi. As a consequence of the above discussion, we see that there exists a coefficient £ = ((a, h, r, , u2) of modulus 1, depending only on the congruence classes of a, h r, and u2 modulo ddi such that we have the equality

ah ah d 2

e(') = ^ ' 6 I ddi (dl — U2)J ' 6l ).

Returning to (37), and fixing the congruences classes modulo ddi of the variables h, r, and u2, we see that there exists

0 < ai,a2,a3,a4 < ddi

such that W satisfies the inequality, |WI < X6£

E I E E E ^^ h ,U2)e (^W^)!, (38)

1<IA<R/d dd1u'1 ,dv2~N 1<W<H 1

r=ai mod ddi diV'=i>2 mod r h=a^ mod ddi

v-^ =a'2 mod ddi

V2 =a3 mod ddi

where is the diflerentiable function

y (h , ) = H(diui — u2)/(rQ)) j/_h_> ( ahr \

r(h U2) (div[ —V2)/(rQ) (diu[ — V2)/(rM)je\ddlv[ (div[ — U2)J '

In order to perform the Abel summation over the variables u2 and h (see for instance [9, Lemme 5]) we must have information on the partial derivatives of the ^-function. Indeed for 0 < e0, e1, e2 < 1, we have the inequality

) £0+ei+ £2

dh«dv Fto? ^(h,V'i, U2) «50£ |h|-M-1 U2-62 (N/(T Q))ei+2, (39)

as a consequence of the inequality - v2\ > rQ/2 (see(34)), of the definition of H (see (14)) and of the inequality 1 < |a| < X.

Since (d1u'1 v2, r) = 1 we detect the congruence d1u'1 = u2 mod r bv the <p(r) Dirichlet characters % modulo r. By (39) we eliminate the function in the inequality (38) which becomes

|Wi <x™-n2Q-2 £ ' E

1<IA<R/d X mod r

r=a1 mod dd1

E E x(^1)%(E (40)

dd1v1eM1 dv2eM2 hen 1

v1=a2 mod dd1 h=a4 mod dd1

V2=a3 mod dd1

• where N1 and N2 are two intervals included in [ N, 2N],

• and where % is the union of two intervals in eluded in [-H, -1^d [1, H ] respectively.

Denote by W1(r, %) the inner sum over u'1, u2 and h in (40). Remark that the trivial bound for W1( r, x) is 0(XvHN2/(d2d1)). We now can apply Lemma 5 to the sum W1(r,%), with the choice of parameters

j ^ ar, A ^ H,M ^ N and N ^ N.

We obtain the bound

W1(r, x) « Hl1N2N1 + ^P) ((HN2)20 +£N4 + (HN2)8+(HN)8) . By the definition (35), (14) and the inequality 1 < |а| < X we deduce the inequality

A - i „ / 17 39 15 ,

W1(r, x) «X1o N 39 +HN15),

and using (14) we finally deduce

/ 10 39 10 -i 15 „ \

W1(r,x) «X5£+v(M-noN29Q20 +M-1N15Q).

x

wErr1(Q) «X67e+^(M20NooQ-20 +NHQ-2). (41)

4.5. Conclusion

We have now all the elements to bound A(a, ¡3, M, N, Q, a). By (10), (13), (20), (26), (27), (28), (30) and (41) we have the inequality

A2 « MQ£k2-1{(wMT(Q) - 2VMT(Q) + UMT(Q)) + N2Q-1Xn

+ (M-1N2 + N 5 Q-1)X 2£+n + (MN2Q-1Xn-2 +X1+n )+X 67£+n (M N 5OQ" i + N ^Q-2)} ,

which is shortened in (recall (8))

A2 « MQ£k2-1{MN2Q-1(\og2N)-A+

+ MN 2Q-1Xn-2 +X 67£+n (M N 20Q-§ + N ^Q-2)) ,

bv (32) and (33) if one assumes

Q < NX—. (42)

To finish the proof of Theorem 1, it remains to find sufficient conditions over M, ^^d Q to ensure the bound A2 ^ M2N2C-A. Choosing rj = e/5, we have to study the following three inequalities hold

' MQ ■MN2Q-1X-4 < M2N2X-4,

„ 3 79 „ 43 co 0 <-, £ „ ,„.

MQ ■M20N7oQ 20X68s < M2N2X-4, (43)

MQ ■N31Q-2X68 < M2N2X-4.

The first inequality is trivially satisfied. The second inequality of (43) is satisfied as soon as

56 17 ,

Q > N53 X-17 +65e. (44)

17

This inequality combined with (42) implies that N < X33. The last condition of (43) is satisfied as soon as

Q > N fx-1+69£.

17

We can drop this condition since it is a consequence of (44) and of the inequality N < X 33. The proof of Theorem 1 is now complete.

5. Proof of Corollary 1

Let S(M, N) be the sum we are studying in this corollary. We use Dirichlet's hyperbola argument to write

mn — 1 = q r, (45)

and by symmetry we can impose the condition q < r. This symmetry creates a factor 2 unless mn — 1 is a perfect square. The contribution to S(M, N) of the (m, n) such that mn — 1 is a square is bounded by 0(X 12+v) with rj > 0 arbitrary. This is a consequence of l(5nl < (n).

The decomposition (45), the constraint q < r and the inequalities X — 1 < mn — 1 < 4X imply that q < 2X2. In counterpart, if q < X2 we are sure that q < r. Thus we have the equality

S(M, N) = 2 £ £ £ amPn + 2 E E E E + 0(X2+v)

q<X 1/2 n^N •mn-1 = qr,q<r

- mn = 1modq m~M, n-N,X1/2 <q<2X1/2

= 2 Sq(M,N) + 2S\(M, N)+0(X1+), (46)

by definition. A direct application of Theorem 1 with Q = X2 gives the equality

SQ(M, N) = £ -j- ^^amf3n + 0(XC-C), (47)

q —X1/2 - m — M n-N (mn, q) = 1

for any C.

For the second term S\(M,N), we must get rid of the constraint q < r. A technique among

others is to precisely control the size of the variables m, n and q. If it is so, then r = (mn — 1)/q is

>

1

A = 2 ],

where B = B(A) is a parameter to be fixed later, and where [y] is the largest integer < y. If we denote by Lo = [CB] we see that

ALo

= 2 and that A = 1 + 0(C B). We denote by Mo, No and

Q0 any numbers in the sets

Mo No Qo

= {M, AM, A2M, A3M, ■■■ , ALo-1M } = {N, A N, A2N, A3N, ■■■ , ALo-1N } = {X i, AX i, A2X 2, A3X 2, ■■■ , ALo-1X 2 },

respectively. We split S1(M,N) into

S1 (M, N ) = E £ £ S1(Mo,No,Qo), (48)

moemo nogao qo£qo

where S1(Mo, No,Qo) is defined by

S1(Mo,No,Qo) = E £ £ am&

q~Qo n~N0

mn = 1 mod q

• where the notation y ~ Y0 means that the integer y satisfies the inequalities Y0 <y < AY0,

• where the variables m, ^^d q satisfy the extra condition

mn - 1 > q2. (49)

Note that the decomposition (48) contains

0(C3B), (50)

terms.

Since mn-1 > M0N0-1 and q2 < Q^A2 in each sum S1(M0, N0, Q0), we can drop the condition (49) in the definition of this sum as soon as we have

MoNo - 1 > Q0A2. (51)

WThen (51) is satisfied, the variables m, n and q are independent and a direct application of Theorem 1 gives for each sum S1(M0, N0, Q0), the equality

S1(Mo,No,Qo)= ^ E £ + Oc(XC-C), (52)

Qo ^(C') ™~Mo, n~No (mn, q) = 1

where C is arbitrary.

It remains to consider the case where (51) is not satisfied, which means that ( M0, N0, Q0) £ £0 where

So := {(Mo, No, Qo); MoNo - 1 < Q^A2}. (53)

We now show that the variable «considered in such a S1(Mo, No,Qo) varies in a rather short interval. More precisely, since MoA > m, NoA > n and Qo < q we deduce from the definition (53) that q2 > mnA-4 - A-2 which implies the inequality q > (mn) 2 A-2 - 1. Combining with (49), we get the inequality

(mn) 1 A-2 - 1 < q < (mn)2

which implies

(q2/m) <n< ((q + 1)2/m)A4.

Using the inequality

X1/2 <q< 2X1/2 < (Q2/M )(A4 - 1)X- 2, and I(5n\ < Tk (n) we apply Lemma 2 to see that

£ £ ^Si(Mo,No,Qa)

(Mo,No,Qo)e£o

< E Tk(m) £ £ Tk(n)

m~M q-xV2 (q2/m)<n<((q+1)2/m)A4

( q,m) = 1

« (A4 - 1)£k-1 ^ rfc(m) £

1 q2

<p(q) m

« C2k-2-BX.

Actually, by introducing a main term back, which is less than the error term, we can also write this bound as an equality

£ £ Y,Si(MO,NO,QO)

(Mo,No,Qo)e£o

= EEEE ¿y EE '^n+o(c2k-'2-B), («J

(Mo,No,Qo)e£o q^Qo m~Mn, n~Nn

(mn, q) = 1

( m, n, )

Gathering (46), (47), (48), (50), (52), (54) we obtain S(M,N)=2 £ -L £ £ amPn

a<X 1/2 m~M n-N

— (mn,q) = 1

+ 2 E E EE -L E E^

MoeMo NoeMo QoCQo q~Qo y ' m-M0, ™-No

(mn, q) = 1

+ 0(CiB-CX) + 0(C2k-2-BX) + 0(X1+),

( m, n, )

together, we complete the proof of Corollary 1 by choosing B and C in order to satisfy the equalities -A = 3B - C = 2k-2-B.

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Получено 22.06.2018 Принято к печати 10.10.2018