The Atkinson type formula for the periodic zeta-function Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 14 Выпуск 2 (2013)

УДК 519.14

THE ATKINSON TYPE FORMULA FOR THE PERIODIC ZETA-FUNCTION

In the paper an explicit formula for the error term in the average mean square formula for the periodic zeta-function with rational parameter in the critical strip is obtained.

Keywords: Atkinson formula, generalized divisor function, periodic zeta-function.

ФОРМУЛА ТИПА АТКИНСОНА ДЛЯ ПЕРИОДИЧЕСКОЙ ДЗЕТА-ФУНКЦИИ

В статье получена явная формула для остаточного члена в формуле для усредненного второго момента периодической дзета-функции с рациональным параметром в критической полосе.

Ключевые слова: периодическая дзета-функция, обобщенная функция делителей, формула Аткинсона.

1. Introduction

Denote, as usual, by Z(s), s = a + it, the Riemann zeta-function. In the theory of the function Z(s), the moment problem occupies an important place. It consists of finding the asymptotic behavior for

S. Cernigova (Vilnius), A. LaurinCikas (Vilnius)

Abstract

С. Чернигова ( Вильнюс), А. Лауринчикас ( Вильнюс)

Аннотация

as T ^ то. Many attention is devoted to the mean square

Ja(T) = f \((a + it)\2dt,

J 0

of Z(s) for 2 < a < 1. The asymptotics of Ja (T) as T ^ to is well-known. Let y0 denote the Euler constant, and

E(T) = J. (T) - T logT - (270 - 1)T.

In [1], F. V. Atkinson obtained an interesting explicit formula for the error term E(T) in the formula for Ji (T). Let 0 < c1 < c2 be two fixed constants such that c1T < N < c2T, and

Ni = N (T ) = T + N-\l[ t +

\( N 2 NT )

VU + ^I

Moreover, as usual, denote by d(m), m E N, the divisor function, and define

arsinh(x) = log(x + Vl + x2)

and

f (T,m) = 2Tarsinh ^ + V2nmT + n2m2 — 4.

Then Atkinson proved [1] that

e>t>=v t '-^vmp (^f)f (iTm+.................

— 2 E^m (*jim )-i “ (T —T+?) + "■ 111

m<N1 v \ / \ /

The proof of the Atkinson formula is also given in [4]. The papers [5], [6], [14], [15], are devoted to modified versions of formula (1).

K. Matsumoto [11] and jointly with T. Meurman [12] obtained the analogue of the Atkinson formula in the critical strip. The second author [9], [10] gave a version of the Atkinson formula near the critical line.

The Atkinson formula is very useful in the theory of Z(s). This formula allows to obtain various estimates for the error term Ea (T) in the formula for Ja (T), to

study the mean square of Ea (T) and to continue other investigations of Ja (T). In

[3], the Atkinson formula has been applied to obtain an estimate for the twelfth power moment of ((s).

Analogues of the Atkinson formula are also known for other zeta-function, for example, for Dirichlet L-function [13], and for the periodic zeta-function (x(s) [7], [8]. The function (x(s), A E R, is defined, for a > 1, by the series

m=1

and by analytic continuation elsewhere. For A E Z, the function (x(s) reduces to the Riemann zeta-function. In view of the periodicity of the coefficients e2mXm, we may suppose that 0 < A < 1. In the above mentioned papers [7] and [8], the Atkinson type formula has been studied for the error term of

|(A(a + it)\2dt,

where A = a with integers a and q, 1 < a < q. In [8], the case a = \ has been investigated, while the paper [7] deals with the case 2 < a < 1. Let, for 2 < a < 1,

E*(q,T) = \Ca(a + ^dt - q((2a)T

a=1

(qT)

C, (2a - 1)r(2a - 1) sin(^a^^2_2CT

1-a

Then, in [7], an explicit formula for Ea(q,T) with a certain error term has been obtained. However, the error term of that formula with respect to q is not right. Therefore, the present paper is devoted to a more precise Atkinson type formula for Ea(q,T), and removes some inaccuracies of [7]. We limit ourselves to the case 2 < a < 4. We note that the method of investigation is analogical to that used for the Riemann zeta-function, however, some new problems arise from the involving of the parameter q.

Let c1T < N < c2T with some positive constants c1 < c2. Define

N1 = Ni(q, N,T ) = q| T + qN -I (qN)2 +

((?)'+'‘N)!)'

ms

denote by aa(m), a E C, m E N, the generalized divisor function, i.e.,

aa(m) = ^2 d° d\m

and let

5>T)=2-q- (ly- E m_»(m) (arsing /f))

v 7 m<N v v v 77

x (2nqm +1) cos (2Tarsin^^/IP) + ^2nqmT+T2q2m2 —1)

and

_1

X

£>T>- -v-(i)’ L (Tk*(£) -T+j)

-1

x cos ( Tl og ( ^— ) — T + 4 Theorem 1. Suppose that 2 < a < |. Then, for q < T,

E-(q,T) = £i(q,T) + £2(q,T) + R(q,T),

7

where R(q,T) = O(q4_- logT), with the O - constant depending only on a.

If q = 1, then we have the Atkinson formula for the Riemann zeta-function obtained in [11].

2. Lemmas

Lemma 1. Let a =1, 5, y and T E R+, k E R, \k\ > 1, 0 < a < 2, a < and b > T. Then, for every £ > 0,

rb exp{iT log + 2nkiy}dy

Ja ya(1 + y)P (log ^)Y =

= 8(k)(2kVn)-1T2V_YU_2 (u — 0 (u + 2j

x exp jiTV + 2nikU — nik + + O(a1-aT-1) + O(bY_a_?\k\-1) + R(T, k)

uniformly for \a — 1\ > £, where

R(T, k)

and

U={2k+4)

V = 2arsinh \/—-

{ lnk\

[m)

{

t —2^_4\k\_1--1 _I if \k\ < t, t_2_a\k\a_1 if \k\ > t,

2

6(k) = j 0 if k > 0'

w [0 if k < 0.

The lemma is Lemma 2 of [1], see also Lemma 15.1 of [4]. In the above form, the lemma is stated in [11].

For a, b, a E R+, and m, q E N, define

I ^a, b;

r x_ (i,rsinh) (2X+1 y((2k+4)2+() x

exp | i ^±4nxy/— — 2Tarsinh ^x^2Tp ^J — (2nx2T + n2xA^ | dx.

Lemma 2. Let c^^/qT < a < c2^/qT with fixed 0 < c1 < c2. Then

a-1 3 ^

_1 / rn \ 2 _a

I (a'b±ma .4,,T-1( my („(Tm))- (Z — m

x ex^ i|T — T*|£) — ^ + I

i

-O T( T. — Tt _')

1

(

+ O T 2 min 1

a-(az-— | ±2Jm

+ol4-“( n)1+O (T V)

1

+ O e_CT _C

with a large constant C > 0; where

if

0 otherwise.

— {in — mq^ — mb2 and the double sign takes +,

m — Tjnq, ma2

The lemma is a slight modification of Lemma 3 from [1], see also Lemma 15.2 of

[4]. The statement of the lemma follows that of Lemma 4 of [11].

The next lemmas are related to the function a1_2a (m). Let

Da (x) = ^ a1_2a (m),

m<x

where the sign "'"means that the last term in the sum is to be halved if x E N. Define A1_2a (x) by

/ ^ v Z(2 — 2a)x2_2a Z(2a — 1)

A1_2a (x) = Da (x) — Z (2a)x---------- ---- --------+

2- 2a

2

Lemma 3. For every £ > 0;

A1_2a (x) = O(x ^+£).

The lemma is Lemma 2 from [11].

Lemma 4. We have

A1_2a (x)

3- a x 4 a

X /

a1_2a (m) x

x (cos ^4n\/mx

mx — 4 I —

j — (32nVmx) 1 (16(1 — a)2 — 1) sin ^ +O(x_1 _a),

n

mx —-4

the series being boundedly convergent in any fixed finite interval of x. The lemma is Lemma 1 of [11], and is a result of [16] and [2].

+

q

1

8

3. A formula for Ea(q,T)

Let u and v be complex variables, Reu > 1 and Rev > 1. Then we have

q X 2niam X e_2ni-n

q e

EZ--<v> = EEn

a=1 a=1 m=1 n=1

‘z (u+v) + ]>]]C]C

1=1 n= m=n

munv

a=1 m=1 n=1

Since

^ e2n-(m_n) = j q if m = n (mod q),

a=1 v

0 if m = n (mod q),

we have from (2) that

J2Z - (u)Z_ - (v) = q(Z (u + v) + fq(u,v) + fq (V'U))'

a=1

where

oc oc

mU(m1 + qm2)v

mi = 1 m2 = 1 1

Using the Poisson summation formula and properties of the gamma-function r(s), we find that, for Re(u + v) > 2 and Reu < 0,

Z (u + v — 1)r(u + v — 1)r(1 — u)

fqv) = ----------------quн-lГ(v)-~ + gq(u v),

where

1

X cos(2nmqy)dy

Vu(1 + y)v '

We need the analytics continuation for gq(u,v) to a certain region lying in 0 < Reu < 1, 0 < Rev < 1. Suppose that we have such an analytic continuation. Then, in view of (3) and (4), we find that

gq(u,v)

qu+v_

X „

I ^ ^ a1_u_v(m) I m=1 J0

2

(A, 4 Z (u + v — l)r(u + v — 1)Г(1 — u)

Y,Za(u)Z-a(v) = H z(u + v) + -------,,+,-ir(v) -------

a=1 ' ' '

Z (u + v — 1)Г(и + v — 1)Г(1 — v)

+ g (gq(u,v) + gq(v,u)).

qu+v-1r(u)

In the latter equality, we take u = a + it and v = 2a — u = a — it. Then, using the estimate [12]

rT ( r(1 — a — it) + r(1 — a + it)) dt = sint^a) T2_ъ, + _2a

Jo \ r(a — it) r(a + it) ) 1 — a

we obtain that

± fT к.(a + utfdt = q<(2a)T + Z(2a — ^ ~ (qT)—

a=1 a

p a+iT

— iq gq(u, 2a — u)du + O(qT-2a). (5)

J a-iT

Now we consider the function gq(u, 2a — u). Define

f~ cos(2nxy)dy h(u,x) = 2 — ------^-----.

Jo Уи(1+ У)2а-и

Then, by the definition of gq(u,v),

1 Ж

gq(u, 2a — u) = -2a-1 ai-2a(m)h(u, mq). (6)

q m=1

Suppose that N G N, and let X = N + 2. Then, by the definition of D1-2a(x) and A1-2a (x), we have that

^2 a1-2a(m)h(u,mq)= h(u,qx)dD1-2a(x)

m>N JX

= (Z (2a) + Z (2 — 2a)x a )h(u,qx)dx

J X

r<x>

+ h(u,qx)d^1-2a (x)

JX

= —A(X)h(u.qX) — [ А-a(x)dhu^

+ I (Z (2a) + Z (2 — 2a)x1-2a )h(u,qx)dx.

X

This and (6) show that

gq(u, 2a — u) = Y] a1_2a(m)h(u,mq)----20~[A1_2a(X)h(u,qX)

q m<N q

1 fX A ( \ дh(U' qx) j

- Jx A1_2*(x)—g^dx

1 fX

+ _2^T / (Z (2a) + Z (2 — 2a)xl_2a )h(u,qx)dx q Jx

= gq,1(u) — gq,2(u) — gq,3(u) + gq,4(u). (7)

By the definition, the function h(u,x) is analytic in the Reu < 1. Therefore, the functions gq>1(u) and gq,2(u) also are analytic in the latter region.

Using Lemma 3 and estimate [1]

dh(u, x) dx

we obtain that

O(xReu_2)

0^ [ A1_2a(x) qx) dx < qReu_2a [ xReu+ 4k1 _2+£dx,

q a JX dx

IX JX

and the integral is convergent for Reu < 1 — 4a+1. Since 1 — > a for a < |, we

have that the function gq,3(u) is analytic in the region including the line Reu = a. It is easily seen that

1 fX

gq,4(u) = (Z (2a) + Z (2 — 2a)x1 _ 2a)

q Jx

( rix e2niqxydy f_ix e_2niqxydy ) ,

x ( I —/------------+ I —/------------ I dx.

\Jo yu(1 + y)2a _u Jo yu(1 + y)2a _uJ

Suppose that Reu < 0. Then

1 CX / CiX e2niqxy dq, \

(Z(2a) + Z(2 — 2a)x1 _ 2a) —-— ) dx

q2a_1 Jx V Jo yu(1 + y)2a_uJ

1 fiX e2niqxy dy,

:(Z (2a) + Z (2 — 2a)x1 _ 2a) 1 e dy

2niq2a J0 yu+1(1+ y)2a _ u

X

DC

/ rX e2niqxy dn. \

((Z(2 — 2a)(1 — 2a)x J0 yu+l(1+ y)2a_uj dx

2nq2aJx\ ’Jo yu+1(1+ y)2a~

1 fiX e2niqXy

1 (Z (2a) + Z (2 — 2a)X 1_2a ) ' e dy

2niq2a v J J0 yu+1 (1 + y)2a"

Z(2 — 2a)(1 — 2a) /*X, rx e2niqydy

dx

2niq2a JX J0 yu+1(x + y)2a~

1 fX e2niqXy dy

(Z(2a) + Z(2 - 2a)X1_2a' ' V

2niq2a v J J0 yu+1 (1 + y)2a"

Z(2 — 2a)(1 — 2a)X 1_2a fX e2niqXydy

2niq2a (2a — u — 1) J0 yu+1 (1 + y)2a_u_1'

Similarly, we find that

1 O ^-2<n fX e_2niqXydy

— (Z (2a) + Z (2 — 2a)X 1_2a)

+

2niq2a_2 J0 yu+1(1 + y)2a~

Z(2 — 2a)(1 — 2a)X 1_2a fX e_2niqXydy

2niq2a (2a — u — 1) J0 yu+1(1 + y)2a_u_1 ‘

The later two qualities yield

1 , M* n ^ v1 — 2a\ fX sin(2nqXy)dy

gq,4(u) =----------2a (Z (2a) + Z (2 — 2a)X a )

nq2a ' ' J0 yu+1 (1 + y)2a_u

Z (2 — 2a)(1 — 2a)X 1_2a rX sin(2nqXy)dy (8)

nq2a_2(2a — u — 1) J0 yu+1 (1 + y)2a_u_1 ‘

The above integrals are convergent absolutely for Reu < 1. Thus, we have analytic continuation for gq,4(u) to the suitable region. Consequently, (5) is true for 1 < a <

3

4.

From (5) we find that, for 2 < a < 4

na+iT

Ea(q,T) = —iq gq(u, 2a — u) + O(qT a)

a iT

Therefore, in view of (7),

CO

1

Ea (q,T) = —iq2 2a (Gq,1 — Gq,2 — Gq,3 + Gq,4) + O(qT 2<T)' (9)

where

«--* e -.mf n

cos(2nqmy) sin(T log )dy

rr-

4i a--2a (m) /

m<N J0

ya(1+ y)a log ^

fx cos(2nqXy) sin(Tlog — )dy

Gq,2 = 4iA i-2a (X ) / --------TZ----T— ----Tyy-----

Jo ya(1+ y)a log l-+-

d f fx cos(2nqxy) sin(T log -+y )dy

„ cc>s(2nqxy) sin(Tlog )dy\

Gq,3 — 4i A - -2a (x)— / --------- ------—--------T------- dx

' dx\ Jo ya(1 + y)a log^ )

x a dx I Jo ya(1 + y)a log ^

r~ d ( fx cos(2nqy) sin(T log )dy\

i Jx 1 -2a X dxy Jo ya (x + y)ax1 -2a log J x

. . cos(2nqy) ((2a — 1)x2a 2 sin(Tlog ^)

4W A - -2a (x) ' '

f ~ A ( ) f ~ cos(2nqy) ^

Jx A --2a (x)L —aa—{

ya \ (x + y)a log ^

x2a—1T cos(T log2^) ax2a—1 sin(T log ^) x2a- 1 sin(T log ^) \

I____________s w y '______________________s w y '_________________s w y ' dxdy

(x + y)a+1 (x + y)a+1 log (x + y)a+1 log2 ^ J

4r™ A—Ax)lr ms(2xqxy) / coJ T logi±y'

Jx x \Jo ya(1 + y)a+1 log l-+ V V y ,

+ sin ^T log ~+~y^ ^(2a —1)(1 + y) — a — log1- j dyj dx,

2i /"^ sin(2nqXy) sin(T log 111)

Gq4 - — — (C(2a) + C(2 — 2a)Xi—2a) ---- ----^

’ nq Jo ya+1 (1 + y)a log ^

(1 — 2a)C(2 — 2a)Xi—2a [~ / sin(2nqXy) /*a+iT (^)udu \ d

~~ Jo yy(1 + y)2a— - Ja—it u — 2a + ^ y

nq Jo \y(1+ y)2a 1 Ja—it u — 2a + 1

4. Proof of Theorem 1

By (9), it suffices to evaluate Gq,i — Gq,4. For evaluation of Gq,i, we apply Lemma 1 with a — 3 — a, 7 —1, k — qm and k — —qm. Then taking T ^ n ^ T gives

Gq- — 2a—lqa—l(n)a—2 ix

^ ai—2a(m)ma -V -U 2 sin(TV + 2nqmU — nqm +4) +

m<N

17 1

• 1 —ar,— 4 +a ^— 2 ^ —

+O(max(T 4—a q— 4 +a ,T— 2)) —

— 2a—^1 (T)a— 5 i £ (— 1)qma- —2„(m)ma— 1 (arsinh f 7^)) x

m<N ' ' * n '

x (— + 1V 4 cos f 2Tarsinh + 2nqm (-^- + 1) 5 — n)

\2nmq 4J y \\j 2n J \2nqm 4J 4j

+O (max (T4—aq—4+a,T — .

For Gq,2, it is sufficient to obtain an estimate. Lemma 1 implies that

:10)

Gq,2 — O(A-—2a(X)^T~X^ ^arsinh ^

,

x( 2X+i)"4+O(A -^(X )T—2 qa—1 >•

Therefore, in view of Lemma 3,

Gq,2 — O (T+£qa—1 (log q)—1) + O (T— 2+£qa—-j — O (t21-+) +£qa—1) .

(11)

Now we will deal with Gq,4. First we observe that, in virtue of the residue theorem, for 0 <y < 1,

u

r-a+iT ( —) du

I V—Z —T — 2niReSu=2a— i (•••) —

a-iT u — 2a + 1 ( )

du ( 1 + y ) 2a-

a

ur+0 m‘u—t+1 -(+OT).

y u — 2a + 1 y

Moreover, for y > 1,

r+iT 11+y\u du — / r+iT du \

la—iT\ y J u — 2a + 1 V Ja—iT u — 2a + 1 )

O(log T).

Thus,

r*CO

(

sin(2nqXy)

ai iT

>o \y(1+ y)2a—iJa—iT "1 sin(2nqXy)

(^ur—£m) iv — (I+D

2ni

2a

dy + O T

+

(o y ( sin(2nqXy) fa+iT

- \y(l + H)2’-- Ja-i.T

-

u — 2a + 1

| sin(2nqXy)|dy

' o

y

/1 + y\ “ du \ V y ) u — 2a + 1/

ai-

dy•

We have that

(...)dy

2nU sin^Xy)dy — 2ni p sinC^-rgXy)(% + o(t -iq-i)

2a 2a

/o y Jo y

2ni(2nqX)

2a—-

sin ydy

(2n)2a (qX )2a—-i

oy

n

2a

2r(2a) sin(na)

+ O(T— lq—l) — + O(T—lq-l)

-

T

-

sin(2nqXy)dy

ai-

Jo y

°^qX fr’S + O (t"'Jxy^

dy

)

and, in view of the estimate

O(qa T a—i)

du

r+iT( 1 + y\u___________

la—iT\ y ) u — 2a + 1

O(log T),

I"(

ai iT

/1 + y\ “ du \ V y ) u — 2a + \)

y

ai iT

dy

sin(2nqXy)

y(1 + y)2a—iJa—iT

cos(2nqXy)

27rqXy(1 + y)2a—iJa—iT\ y ) u — 2a + 1

3 ( cos(2nqXy) fa+iT (1 + y)u du

I- \2nqXy2(1 + y)2a—iJa-iT\ y ) u — 2a + 1

/1 + y\u du \

V y ) u- 2a + 1/

(1 + y )u du \ V y ) u — 2a +1 /

dy

, n 2 \ ( cos(2nqXy) fa+iT (1+ y

+ (1 -2a\h U,Xy(1 + y)^"

r“ ( cos(2nqXy) fa+iT

Jl y2nqXy (1 + y)2a—iJa—iT

O(q—lT—l logT).

a—iT ) y 1 + yN ' 1 y

du

u — 2a + 1 du

y2(u — 2a + 1)

dy

dy

u

All these estimates show that the second term in the formula for Gq,4 is

in(2n)2a-l(1 — 2a)q2a-2 * ^ + O(qa-lT—). (12)

I(2a) sin(na)

For the evaluation of the first term of Gq,4, we apply the second mean value theorem and Lemma 1. We write the integral as

r-<x / /• (2qX) 1 r-<x \

/ (•••)dy — / + / (•••)dy.

>o \Jo J(2qX)-1 J

Then

[(2qX) ^ fV sin(Tlog^)yl a(1 + y)l a,

lo {-)dy < 2nqX I -------y(1 + y)log ^--------dy

— 2nqX3l—a(1+ 3)l—a sin(Tlog )dy

log -+£ (a y(1+ y)

_2nqX3l—a(1+ 3)l—a r_- ( 1+ y

l+V

V

>og- (T—l cos (rlogi+y))

V

O(qa T a—l)

where 0 < a < 3 < (2qX) l. Moreover, an application of Lemma 1 gives the estimate

/ (•••)dy — O(qa Ta—l).

J(2qX )-1

From these estimates and (12), we obtain that

GqA — in(2n)2a-l(1 — 2a)q2a-2 * ^ + O(qa-lTa-1 )• (13)

I(2a) sin(na)

The most complicated is the integral Gq,3. We apply Lemma 1 again and find that, for x ^ T,

r” coS(2,Tqxy) '+ (Tcos (T1+y)) + sm (Tbgi+1)

Jo ya(1+ y)a+l log V y )) V y )

a

CO

x ^(2a — 1) (1 + y) — a — ^log j dy — i^~^a~2qa~lxa~lT3

(arsinh([T)) (2nqx+4) ‘ ((^+4)2+2)

x cos ^arsinh ^+ 2nqx ^^nqx + ^ — nqx + 4^ + O(qa lT1 axa l)

Hence,

-i

r<x>

3

Gq3 — ir~n~2q°~vT2-" I Alx-—r((x) (^arsinh ^ J~qx

lX x2—a\ W 2T

i / i \ —-/ T 1\— 4 / / T 1\ 2 1\

^2nqx + ij y \2nqx + V +2y

x co^2T arsin^ + 2n?x( 2nqx + 4) — nqx+1)dx

+ O (qa—lT5 —a J A-—x(x) dx) . (14)

In remains to evaluate and estimate the latter integrals.

Using Lemma 3 and the restriction 2 < a < 4, we obtain that

qa—lT5 —a f Al—2a(x^ — O(qa—lT21+&) +£). (15)

Jx x2 a

For the evaluation of the first integral in (14), we apply Lemma 4 and the argument proposed in [11] to avoid the problem arising from the bounded convergence of the series in Lemma 4. Thus, by (14) and (15),

n -a— 3 ( T\ 2 % a-—2a (m) f — 3 f f fm n\

Gq 3 — iq 4 7:— lim-----5---- x 2 cos 4nxW--------------------

q’ ^ \2n) ^ m5—a JvqX V V V q V

-i

m ^ lim ai-2;(m)

\2n J b^<x m 4— a

— ^32nxy/—^ x (16 (1 — a)2 — 1) sin ^4nx —

v V V2T// \2nx2 V \\2nx2 V 2

x cos ^2Tarsinh ^x^+ (2nx2T + n2x4)2 — nx2 + 4^ dx

+ O (qa—lT+£)

In the notation of Lemma 2, this can be rewritten in the form

a-—2a (m)

3 / T \ 2—a “

Gq,3 — iqa— 4 — ) ,lim

\2n/ m 4—a

N 7 m=l

x ^Re/ ^ VqX, b; —, m—, 0 + /m/ ^VX, b; +, —, 0 + x (16(1 — a)2 — 1) (/m (VX b; +, —, 5) + Re/ (rfX, b;+,—, 5)))

+ O (qa—lTW40).

Define

1 — 4a

a—ln1271+40) +M (16)

H( X’+X)'

Then an application of Lemma 2 with a — | and a — 2, and a — VqX for (16) yields

3

n ■ a— 3 ( T\ 2 a 1- I /1 " 5T"' a' —2a(m)

Gq 3 — iq 4 — lim 4nq 4 T > ----------

q ’3 q \2n b^\ q ^ ml—a

v 7 \ m<Z

x {*(£—)"') ™( rlt^(i=)—T+j)

+O (•- >T-1E ’p—? (- ( i—)f( T - — )-1)

+o (q-1 t - 3 e (|——(-)

+o (-2 E ’p"? ((m )■+o ( t ))

+ O (e"CT E a'-2, (—) e^^l ( ^ —4-a /

<«X >5—(-x+f)2+Vm—

+ O (qa—lT27r+4a7. (17)

I It1 — 3 ^ a'—2a(m) . | -■

+O I t 4 ^min | 1

m=l

1 — 4a

Since 2 < a < 4, we have that

as b —— to.

From the definition of Z, it follows that Z ^ T. Thus, ^ — Z ^ Tq. Therefore,

T

m

£

mZ

a-—2a (m)

1a

WSO)" (T—?)"'

a-—2a (m)

mZ

m

1a

19)

in view of the estimate

a-—2a (x) C x, x > 0. (20)

m<x

Similary, we find that

T-2q-4 E a-a—llT — —Y2 « r°-2q-4. (21;

ml—a \2n q )

m1—a

mZ

Since q T,

a-—2-—] e— O(e"C1 T) (22)

m=l

— —a

m4 a

with some c' > 0. We have that

(2/

1 12X + 2Tq

2 Vq2X + T

)!

q2X

+ Tq

+ 2n — q

/

q2 T2 + qXT

4

2n

Z.

2

Thus,

m=l

5-a

m4 a

q

2T

<q,X >2 -I -x+n) +V q

-

<X , s

^ q5T— 4 ^ a-—2a m min(1, Ijm — \[ZI"l)

m=l

+

E

+

+

^m< 77 77 <m<Z—\fZ Z—^JZ<m<Z+^fZ Z+^fZ<m<2Z m>2Z j

X

al—2a(m) . | j— ^I — l'*

----5------------min(1, lyjm — vZ| ).

m 4

(23)

Clearly, in view of (20) and Z ^ T,

2

q2t-3 E (■■■) « q2T"5 E (jZ —V—)~l

— —a

7 r- 7 r- m 4 a

Z <m<Z—\/Z Z <m<Z—y/Z

\-l

< q2Ta 2 ^2 a-—2a(m)(Z — m)

Z <m<Z—^Z

q5Ta— 3 ^2 al—2a (Z — m)m—l ^ q5Ta—2 log T, (25)

VZ <m< Z

,3 ^ a-—2a (m)

, (...) ^ qA i 4

<Z+VZ

by using Lemma 3,

q2T— 4 V (...) < q2T— 4 V zr< q2Ta—2 (26)

' J J m 4 a

Z—yfZ<m<Z+yfZ Z—\fZ<rm<Z+\fZ

and

q 2 T"

E (-)«q2 T°

Z+VZ<m<2Z

log T,

(27)

m>2Z

m>2Z

a l—2a

m 4 a

3

2

Finally, combining (17) - (19) and (21) - (28), we obtain that

G- I)' 'Е^НЙ))

v 7 m<Z v v 77

x cos(Tlog (2m) “ T + j) + °('r 1 logT)■

Thus, from this, (9) - (11) and (13), Theorem 1 follows because Z can be replaced by N' with a negligible error.

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Vilnius University Поступило 3.05.2013