Some $\Omega $-results related to the fourth power moment of the Riemann zeta-function and to the additive divisor problem

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

J

ERZY

K

ACZOROWSKI

B

OGDAN

S

ZYDŁO

Some Ω-results related to the fourth power moment of the

Riemann zeta-function and to the additive divisor problem

Journal de Théorie des Nombres de Bordeaux, tome

9, n

o

_{1 (1997),}

p. 41-50

<http://www.numdam.org/item?id=JTNB_1997__9_1_41_0>

© Université Bordeaux 1, 1997, tous droits réservés.

L’accès aux archives de la revue « Journal de Théorie des Nombres de Bordeaux » (http://jtnb.cedram.org/) implique l’accord avec les condi-tions générales d’utilisation (http://www.numdam.org/conditions). Toute uti-lisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit conte-nir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Some

03A9-results related to the fourth _{power moment} of the Riemann Zeta-function

and to the additive divisor

_problem

par JERZY KACZOROWSKI* ET BOGDAN SZYDLO

RÉSUMÉ. Nous donnons des estimations de

_{type 03A9}

relatives au

terme reste dans la formule

_asymptotique

_pour le moment d’ordre

quatre de la fonction zeta de Riemann et au terme reste dans le

problème

des diviseurs.

ABSTRACT. We

_improve

_{the "two-sided" omega results} in the

fourth _power mean

_problem

for the Riemann zeta-function and

in the additive divisor

_problem.

1. Introduction

Let

_E2(X)

denote the remainder term in the

_asymptotic

formula for the fourth _power mean of the Riemann zeta-function:

where P is a certain

_polynomial

of

_degree

four.

_Further,

let

E(x, k)

denote the remainder term in the additive divisor

_problem:

where k is a fixed

_positive

_integer,

d(n)

stands for the number of

_positive

divisors of _n, and

_Pk

is a certain

_quadratic

_polynomial.

In this note

_E(x)

will

_always

denote or

E(x,k).

Recent results on oscillations of these remainders can be written as fol-lows

They

are due to Y. Motohashi

_[10]

in the case of

E(x)

= and _to the

second author

_[11]

in the case of

_E(x)

=

E(x, k) .

* Supported by KBN Grant nr. 2 P03A 028 09.

The results stated in

₍₁₎

have in fact been obtained in an

analogous

_way. First of

_all,

the

_meromorphic

continuation of the Mellin transforms

is needed. In the context of the fourth power mean of the Riemann

zeta-function it was obtained

by

Y. Motohashi

[10]

and in the context of the additive divisor

_problem

for the first time

_by

L.A.

_Tahtadjan

and A.I.

Vino-gradov

[12] (cf.

also

_[5]

for some revision of

[12]).

The methods

adopted

to

achieve this

_goal

made essential use of the

_spectral

_theory

of the

_hyperbolic

Laplacian.

It turns out that the

_{only singularities}

of

_Z(s)

in the half

_plane

1/2

are

simple poles

at s =

1/2

₊

_ix,

where

1/4

₊

rc2

_are certain

eigenvalues

of the

_{hyperbolic Laplacian.}

It is well known that

_1/4+~2

&#x3E; 0. In

_particular,

_Z(s)

is

_regular

at s =

1/2.

The relevant

"nonvanishing"

of

the residues:

for

_infinitely

_many

_1/2

+

_irc,

was obtained

by

Y. Motohashi

[8],[9],

cf. also

[11].

This is crucial for a successful

_application

of a certain

general

result of E. Landau

_{[7] (cf.}

also

_[1]).

The issue is _{the two-sided omega result}

_(1).

In this note we indicate a

possibility

of further

_improvements

of

(1),

cf. Theorems 1 and 2 below.

Firstly,

we remark that

_quite

detailed information on the size of

_E(x)

is now

available,

see

[2], [3], [4]

and

[9].

Consequently,

the mentioned above theorem of E. Landau may

possibly

be

_{replaced by}

a more

_appropriate

means,

something

like a recent

_{generalization}

of it

_by

the first author

_[6],

Theorem 12. It states that if

_{f :}

_{[1, oo)}

-j R is a function such that

for

_{every A}

&#x3E;

_0,

and which Mellin transform

is

_regular

at s = 0 and is not

regular

in any

half-plane a

&#x3E; 8 - r~

with _q &#x3E;

_0,

then the

_Lebesgue

measure of both sets

is for every - &#x3E; 0 as T tends to

_infinity.

The first

_difficulty

we

are faced with when

_applying

this result to errors like

_E(x)

is that we are unable to

_verify

_(2),

_though

it

_{conjecturally}

holds in this case.

_Actually,

for _every

_positive

_ê, where c is an

_{effectively computable}

_positive

_constant,

coupled

with a

slight

modification of the

_just

mentioned Theorem 12 from

[6]

together

with

_analytic

_properties

of

_Z(s)

could

_produce

the

_following

estimates

in which _ii denotes the

_Lebesgue

measure on the real axis and e denotes an

arbitrary

positive

real number.

We prove more than this.

THEOREM 1. There exist constants c &#x3E; 0 and ro &#x3E; 0 such that

_for

_every 0 a ro we have as T -~ oo

THEOREM 2. _{For every}

_fixed

_{positive integer}

k there exists a constant

r(k)

&#x3E; 0 such that

_for

_{every 0} a

r(k)

and e &#x3E; 0 we have as T - oo

The main idea

_leading

to these results is that instead of

_using

individual estimates of

_type

₍₂₎

we rather resort to estimates _{in mean,} cf. Section

2,

formula

_(5).

This is crucial since A. Ivi6 and Y. Motohashi

_[4]

_proved

recently

the excellent

_inequalities

with an

_effectively

_computable

constant c &#x3E;

_0,

and

for _every

_positive

E. We _prove that

₍₃₎

and

₍₄₎

are

_sharp.

A. Ivi6 and Y. Motohashi

_conjectured

in

_[3]

that

as T -. _oo, and also that =

O(T3/2+£)

for

every

positive

E. The

above Theorem 3

_supports

to some extent this

_{conjecture. Finally,}

let us remark that Theorem 3 has the

_ordinary

Q-results

as corollaries. These were

proved

earlier than

(1),

cf.

[3], [9].

Acknowledgement.

We thank Professor Aleksandar Ivi6 for valuable re-marks on the first version of this work.

2. A

_Laudau-type

theorem for Mellin transforms

We formulate now a

_general

theorem which

_plays

the

_principal

role in this note. Let 0 be a fixed real number and

f (x)

be real for x &#x3E; 1.

_Suppose

that for U &#x3E; 1 we have

where _{q :}

_{(1, oo)}

certain

_positive

and

_{nondecreasing}

continuous function such that for _{every 6} &#x3E; 0

Indeed,

from

₍₅₎

and

₍₆₎

it follows for a &#x3E; 0 and 6 &#x3E; 0

if 6 &#x3E; 0 and ê &#x3E; 0 are chosen so that 6 + c

_{2(Q -}

_0).

Hence the Mellin transform

is

_holomorphic

in the

_half-plane

?5 = _{Q &#x3E;} 0.

_Moreover,

_suppose that for

real numbers r and _to

_~

0 we have

Under these

_assumptions

we have the

_following

result.

THEOREM 4. For _{every a} r

0)F(u)

and

for

_every b &#x3E; -r

+limsuPu-+9+(u -

0)F(u)

we have as T --~ oo

where denotes the

_Lebesgue

measure

of

a set A

_of

real numbers.

Let us observe that the above theorem is in fact a

_{tauberian-type}

result.

Indeed,

we do not need any information on F to the left of 0. In

_particular,

we do not assume that F has

_analytic

continuation to _any

_{region larger}

than the

_half-plane

_{of absolute convergence.}

_Applying

this theorem to

_E(x)

requires

less information on the

_analytic

_{character of}

_Z(s)

_{than is in fact} available. In

_particular,

we make no use of its

_meromorphic

continuation

on the

half-plane

u

1/2.

_However,

we

_require

that

Z(s)

behaves in a rather

_special,

_"singular"

_way when s

_approaches

to the vertical u =

1/2

from

_right.

_So,

it is sufficient and natural for us to

_employ

the fact that

Z(s)

has a

_{simple pole}

(

with a non-zero

residue)

on the line Q

= 1/2,

but no

singularity

at s =

1/2,

cf. section 1.

3. Proofs

Theorems 1 and 2 follow from Theorem 4

_by

_{taking f (x)}

=

E2(X)

or

=

E(x,

k).

Indeed,

the _{mean square} estimates

(3)

and

(4)

and the

analytic

properties

of Mellin transforms

_{Z(s) (cf.}

section

₁₎

show that all

_assumptions

of Theorem 4 are then

_satisfied;

we have B =

1/2,

_r =

SUPIER

I

&#x3E;

_0,

and

_0)Z(u)

= 0.

In order to deduce Theorem 3 from Theorem 4 let us

_put

and suppose the

contrary,

i.e. that

holds. Then is bounded as U -&#x3E; oo. From Theorem 4 we have for a certain sequence T - oo and c &#x3E; 0

with r &#x3E; 0 is as above. From this follows

which contradicts

_(10).

Hence it suffices to _prove Theorem 4.

Proof of Theorem 4. Observe that

₍₉₎

follows from

₍₈₎

_by

_replacing

f

by - f .

It suffices therefore to _prove

_(8).

Further,

let us denote

Then

Now, suppose that

(8)

is not true. With

it means that

For any 0 we have

Integrating by

parts

it follows that

But observe that for any 1 x, X2 we have

because the

_functions

and M are

_{nondecreasing.}

_Hence,

_by

well-known

properties

of the

_{Riemann-Stieltjes integral,}

it follows that

Next,

let us fix a

positive

5. From

(12)

it follows that there exists 1 such that

--From

₍₁₅₎

we have therefore

say.

Hence,

from

(13), (14)

and

(16)

we have for _any fixed 6 &#x3E; 0

Consequently,

(12)

implies

that

From

_{(7), (11), (17)}

and

₍₁₈₎

we deduce further that for a certain

se-quence of a

tending

to 0 from above we have

This contradicts our

assumption

that a r +

_B)F(Q).

Therefore

₍₈₎

follows.

REFERENCES

[1]

R.J.

_Anderson,

H.M.

_Stark,

Oscillation

_theorems,

_Analytic

Number

_Theory,

LNM _899,

_{Springer-Verlag,}

Berlin _1981, 79-106.

[2]

J.-M.

_{Deshouillers,}

H.

_Iwaniec,

Additive divisor

_problem,

J. London Math.

Soc.,

26(2),(1982),

1-14.

[3]

A. _Ivi0107, Y. _Motohashi, On the

_fourth

_{power moment}

_of

the Riemann

zeta-function,

J. Number

_Theory

_{51(1995), 16-45.}

[4]

A.

_Ivi0107,

Y.

_Motohashi,

The mean _square

_of

the error term

_for

the

_fourth

mo-ment

_of

the

_{zeta-function,}

Proc. London Math. Soc.

_(3)69(1994),

309-329.

[5]

M. Jutila, The additive divisor

_problem

and

_exponential

sums, Proc. Third

Conf. Canadian Number

_Theory

Association

_(Kingston

_1991),

Clarendon

Press,

Oxford _1993, 113-135.

[6]

J.

_Kaczorowski,

Results on the distribution

of

_primes,

J. reine angew.

_Math.,

446(1994), 89-113.

[7]

E.

_Landau,

Ueber einen Satz von

Tschebyschef,

Math. Ann.

61(1905),

527-550.

[8]

Y.

_Motohashi,

_Spectral

mean values

_of

Maass

_{waveform L-functions,}

J. Num-ber Th.

_42(1992),

258-284.

[9]

Y.

_Motohashi,

The

_binary

additive divisor

_problem,

Ann Sci. L’Ecole Norm.

Sup.,

27(4)(1994),

529-572.

[10]

Y.

_Motohashi,

A relation between the Riemann

_{zeta-function}

and the

hyper-bolic

_Laplacian,

Ann. Sc. Norm.

_Sup.

Pisa Cl.

_Sci.,

_22(4)(1995),

299-313.

[11]

B.

_Szydlo,

On oscillations in the additive divisor

problem, I,

Acta

_Arith.,

[12]

L.A.

_Tahtadjan,

A.I.

_Vinogradov,

The

_{zeta-function of}

the additive divisor

problem

and the

_{spectral decomposition}

_of

the

_automorphic

_Laplacian,

_Zap.

Nauch. Sem.

_LOMI,

ANSSSR

_134,(1984),

_84-116,

_(Russian).

Jerzy

KACZOROWSKI,

Bogdan

SZYDLO

Faculty

of Mathematics and

_Computer

Science

A. Mickiewicz

_University

ul.

_Matejki

_48/49

60-769

_Poznafi,

Poland