Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

NTANAS

L

AURIN ˇ

CIKAS

Limit theorem in the space of continuous functions for the

Dirichlet polynomial related with the Riemann zeta-funtion

Journal de Théorie des Nombres de Bordeaux, tome

8, n

o

_{2 (1996),}

p. 315-329

<http://www.numdam.org/item?id=JTNB_1996__8_2_315_0>

© Université Bordeaux 1, 1996, tous droits réservés.

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

Limit Theorem in the

_space

of continuous

functions for the Dirichlet

_polynomial

related with the Riemann zeta-funtion

par ANTANAS

LAURIN010CIKAS

RÉSUMÉ. Dans cet article on _prouve un théorème limite dans l’espace des

fonctions continues _pour le _polynôme de Dirichlet

$$

où

_{d03BAT (m)}

sont les coefficients du _{développement} en série de Dirichlet de

la fonction

_{03B603BAT (s)}

dans le demi-plan 03C3 &#x3E; _1, 03BAT =

(2-1

log

lT) ½,

_03C3T =

½ +

log2 lT/lT,

lT &#x3E; 0, lT ~ log T et _lT ~ ~ _lorsque T ~ ~.

ABSTRACT. A limit theorem in the space of continuous functions for the Dirichlet _polynomial

$$

where

_{d03BAT (m)}

denote the coefficients of the Dirichlet series _expansion of the function 03B603BAT

(s)

in the _{half-plane 03C3} &#x3E; _1,03BAT=

_{(2-1 lnlT)-½,}

03C3T =

½ + ln2/

lT

and lT &#x3E; 0, lT~ ln T and lT ~ oo as T ~ _~, is proved.

Let s be a

complex

variable and

(8),

as

_usual,

denote the Riemann

zeta-function. To

_study

the distribution of values of the Riemann zeta-function

the

_{probabilistic}

methods can be

used,

and the obtained results

usually

are

presented

as the limit theorems of

probability

theory.

The first theorems

of this

_type

were obtained in

_~1~,~2~,

and

they

were

_proved

in

_[3]-[5]

_using

other methods. In modern

_terminology

we can formulate it as follows. Let

Manuscrit _regu le 20 octobre 1993

*This _paper has been written _during the author’s stay at the _University Bordeaux I in the framework of the _{program "S~T Cooperation} with Central and Eastern _European Countries" .

C be the

_complex

space and let

B(S)

denote the class of Borel sets of the space S. Let

meas{A}

be the

Lebesgue

measure of the set A and

...

where in

_place

of dots we write the conditions which are satisfied

by

t. We define the

_probability

measure

THEOREM A. For u

_{&#x3E; 2}

there exists a

probability

measure P on

(C,

13(C))

such that

PT

converges

weakly

to P as T ~ oo.

More

_general

results were obtained in

[6~.

Let M denote the _space of

functions

_meromorphic

in the

_half-plane

Q

&#x3E; 2 ,

_equipped

with the

_topology

of uniform convergence on

_compacta.

Define the

probability

measure

THEOREM B. There exists a

probability

measure

_Q

on

_(M,

_B(M))

such that

_QT

_converges

_weakly

to

_Q

as T -~ oo.

Note that the

explicit

form of the measure

Q

can be

indicated, and,

obviously,

Theorem A is a

corollary

of Theorem B.

The situation is more

_complicated

when a

_depends

on T and tends

to 2

as T - _oo, or cr

= 2.

It turns out that in this case some _power

_norming

is _necessary.

_{Let lT}

&#x3E; 0 and let

lT

tend to

_infinity

as T 2013~ _oo,

or lT

= _oo.

We take

The case

iT

= _oo

corresponds

to

8T

_{= 2 ~}

The function

is called the characteristic transform of the

_probability

measure P on the

space

(C,13(C))

_[7].

The

_{lognormal probability}

measure on

(C,B(C))

is defined

_by

the characteristic transform

THEOREM C . The

_probability

measure

converges

weakly

to the

_{lognormal probability}

measure as T 2013~ oo.

Here if

_{~(s) ~}

_{0, a}

E

_R,

then

_(a(s)

is understood as

_{exp{alog~(s)}}

where

_{log ~(s)}

is defined

_by

continuous

_displacement

from the

_{point s}

= 2

along

the

_{path joining}

the

_{points 2,}

2 _{+ it} and (1 + it.

ivhen %T

_{= 1}

Theorem C was

proved by

A.Selberg

(unpublished),

see

also

_(8],

and for different form of

_lT,

it was obtained in

_{[8]-(10], [5].}

Now it arises the

_problem

to obtain some results of the kind of Theorem

C in the _space of continuous functions.

Let

_Coo

= C _U

{oo}

be the Riemann

sphere

and let

d(si, S2)

be _{a metric}

on

Coo

_given

by

the formulae

Here s, sl, s2 E

C. This metric is

_compatible

with the

_topology

of

Coo.

Let

C(R)

=

C(R, Coo)

denote the

space of continuous functions

f :

R - Coo

equipped

with the

_topology

_{of uniform convergence} on

_compacta.

In this

topology,

sequence

C(1f8)}

converges to the function

f

E

_C(R)

if

as n -~ oo

_uniformly

in t on

_compact

subsets of R.

The functional

_analogue

of the

_probability

measure in Theorem C is the

measure

Does this measure _converge

_weakly

as T ~ oo to some

_probability

measure

on

(C(R),

,ri

(C(II8)))?

At this moment this

_question

is open and it seems to be _very difficult.

In the

_proof

of Theorem C an

_inportant

role is

played by

the Dirichlet

where

_{d,~ (m)}

denote the coefficients of the Dirichlet series

_expansion

of the

function in the

_half-plane

u &#x3E; 1

_(see

_{[11], [12] ) .}

Therefore the aim of this _{paper is} to _prove the limit theorem in _{the space of continuous} functions

for

_{Su (s) .}

This theorem will be the first

_step

to

_study

_{the weak convergence}

of the

_probability

measure

(1).

Now let ₂

_{log T,}

aT

= 1/2

+ and let lT

Moreover we _suppose that

for all U &#x3E; 0 as T --j oo. Here B denotes a number

(not

_always

the

_same)

which is bounded

_by

a constant.

THEOREM There exists a

probability

measure P on

(C(lI8), t3(C(It8)))

such

that

_PT,ST

_converges

_weakly

to P as T - oo.

Proof of the theorem is based on the

following

probability

result. Let

Sl

and be two metric spaces, and let h :

_Si

-~

S2

be a measurable

function. Then _every

_probability

measure P on

(81, B(8I))

induces on

(S2, Zi(S2))

the

_unique

_probability

measure defined

by

the

equality

Ph-’(A)

=

P(h-’A),

A _E

B(82).

Now let h and

hn

be the measurable functions from

_Sl

into

_S2

and

LEMMAl. Let P and

Pn

be the

_probability

measures on

(81,8(81)).

Sup-pose that

Pn

converges

weakly

to P as n - oo and that

P(E)

= 0. Then

the measure

P,,hn 1

_converges

_weakly

to

Ph-’

as n - oo.

Proof. This lemma is Theorem 5.5 from

_[13].

Let _~y denote the unit circle on

complex

plane,

that

_{is 1}

=

Is

_E C

:1

s

1=

1 }.We

put

where 7P = 1 for each

prime

p. With the

product topology

and

point-wise

_{multiplication}

the infinite-dimentional torus 52 is a

_compact

Abelian

The Fourier transform

_g(k)

of the measure P is defined

by

the formula

Here k =

(k2,

k3,

...)

where

only

a finite number of

_integers

_kp

are distinct

of _zero, and _{xp E -y.}

LEMMA 2. Let be a _{sequence of}

probability

measures on

(Q,

Ii(S2))

and let

_{(k) }}

be a _sequence of

_{corresponding}

Fourier transforms.

_Suppose

that for _every vector k the limit

_g(k)

= lim

gn(k)

Then there exists -

n-oo

a

probability

measure P on

(S2, j3(S2))

such that

Pn

_converges

_weakly

to P

as n ~ oo.

Moreover,

g(k)

is the Fourier transform of P.

Proof. The lemma is the

_special

case of the

_continuity

theorem for

_compact

Abelian group, see, for

example,

[14].

Let

LEMMA 3. The

_probability

measure

_QT

_converges

_weakly

to the Haar

measure Tn on

(S2,

B(S2))

as T 2013~ oo.

Prooi The Fourier transform

_9T(k)

of the measure

_QT

is

_given

_by

Here

_{xp E}

_{q, k}

=

(ki ,

k2,

...) .

_By

definition of the Fourier transform of

probability

measure on

_~5~,13(S~~),

_only

a finite number

_{of kj}

are distinct

from zero. Since the

logarithms

of

_prime

numbers are

_linearly

_independent

over the field of rational

numbers,

we find that

We define the function

_{hT :}

S2 ~

_C(R)

_by

the formula

Here

_pa

₁₁

J~ means that

_pO:

h

but

pc’+’

Then,

clearly,

Let,

for

_brevity,

and let

Let K be a

_compact

subset of R. For _every 6 &#x3E; 0 we define the set

_A~~

_by

and we

_put

LEMMA 4.

_m(Ak(K))

= 0 for _every E &#x3E;

_{0, K,}

and k G N.

Proof.

_By

the

Chebyshev

inequality

Using

the

_{Cauchy formula,}

we have that

where L denotes the

_restangle,

_enclosing

the set iK =

lia, a

_E

KI,

with

to the real axis.

_Moreover,

we _suppose that the distance of L from the set

iK is &#x3E; From this

_equality

it follows that

Hence, having

in mind the

_inequality

_(5),

we obtain that

where !

I L

is

the

_length

of L. From the definitions of and

_{Sn ( Un +}

z +

_iT)

we have

that,

for z = u +

_iv,

Since

hence we find that

By

a similar manner we find that

From the definition of the contour L it follows that

for z = u + iv _E L.Then

(8)

together

with

(2)

and the well-known estimate

where _{/0 is} the Euler

_constant,

_yields

for n &#x3E; no. Here we have used the

_inequality

0

_{1, n &#x3E;}

_no,

which follows

_trivially

from the

_{multiplicativity}

of

_(rrt)

and from the

inequality

0

_{l, n &#x3E;}

_no,

_{implied by}

the formula

_{[11], [12]}

Thus,

Consequently,

in view of

₍

Hence,

for m

_n,

Repeating

the

_proof

of Lemma 3 from

_[10)

and

_taking

into account

_(10),

1

we see that , , ,

Consequently,

this and

₍₁₅₎

_give

the estimate

From

_this,

_(6),

₍₇₎

and

₍₁₁₎

we find that

for every E &#x3E; 0 and k G N. Thus it follows from the definition of the set

that

The lemma is

_proved.

Proof of Theorem. We will deduce the theorem from lemmas

_1,

3 and 4.

Let

converges to

as T 2013~ _oo, and let E denote the

e"2,

...) I

of elements of Q such

that

-does not _converge to some function

h(t; eiTl, eiT2 ,

_...)

as T --+ 00. In order to prove the theorem we must show that

m(E)

= O-Since Q is

compact,

it

is

_{separable. Consequently}

_[13],

E E

_B(Q).

does not _converge to some function

h(t; ei’Tl, ei’T2, ...)

as T 2013~ oc. We will

prove that

m(EI)

= 0. First _we consider the

sequence

hn(t; ei’Tl, ei’T2, ...).

Note that there exists a _sequence of

_compact

subsets of R such

that

--Kj

C and if .K is as

_compact

of R then K

_{C Kj}

for

_{some j .}

Let

Then

is a metric in

C(R) .

Since

_C(R)

is a

_complete

metric space, we have that _every fundamental

sequence is

convergent.

Thus it follows from the definition of the funda-mental sequence that

Thus, by

Lemma

_4,

From the definition of the function

_{hT, using}

the estimates of

_types

₍₁₃₎

and

_(14),

we find that

uniformly

E R and in

_{ei’T2 , ...)}

E 9.

_Therefore,

in view of

_{( 16) ,}

We have shown that there exists a function h such that for almost all

. , . ,

uniformly

in t on

_compact

subsets of R.

_Similarly

as above in the case of

the variable t it can be

proved

_using

the

Cauchy

formula that for almost

all

_(ei7"l,

_{... )}

E S2 the relation

(17)

is valid

_uniformly

in Ti on

_compact

subsets of

_R,

_uniformly

in 72 on

_compact

subsets of

_{I~, ....}

Since the

_family

of sets of m-measure one is closed under countable

_{intersection,}

hence we

have that

₍₁₇₎

is true for almost all

_{(ei7"l, eiT2, ... )}

E Q

_uniformly

in t on

compact

subsets of

_R,

the convergence

being

uniform in _7j on

_compact

subsets of R, j = 1, 2, ....

Since,

for _every M &#x3E;

_0,

in view of the estimate

we have that

_h(t;

ei’T2,

_{...) ~}

oc for almost all

ei’T2,

...)

E 9. The relation

₍₁₇₎

and the uniform convergence

imply

that for almost all

(ei’TI,

,.. . )

E 9

uniformly

in t on

_compact

subsets of R. This

_yields

_rri(E)

= O.The latter

equality together

with Lemmas 1 and 3 _proves the theorem.

!!:I..

COROLLARY. There exists a

_probability

measure P on

(C(R),

13

(C(R)))

such that

_PT,SnT

_converges

_weakly

to P 00.

Proof Let K be a

_compact

subset of R. Denote

_by

ZT(it+iT)

the diference

Let CT =

(log

1T) -’.Then

In view of the

_Cauchy

formula

where L is the contour similar to that in the

_proof

of Lemma 4. Hence we

find

_by

the

_{Montgomery-Vaughan}

theorem for

_{trigonometrical polynomials}

[15], [12]

that

From this and from

₍₁₈₎

we deduce that

By

(19)

the second

_integral

in the latter formula is

_o(T)

as T 2013~ _oo, and

the first

_{integral trivially}

is

BET

T. Hence and from

₍₂₀₎

as oo for every E &#x3E; 0.

_Thus,

the

_corollary

follows from Theorem and Theorem 4.1 from

_[13]:

Let

_{(S, p)}

be a

_separable

_space and

_Xn

and

D P

Yn

he S-valued random elements. If and

_p(Xn,

_{Yn) ---+}

_0,

then

D

Acknowledgement.

For the warm

_hospitality

_during

our

_stay

at the

University

Bordeaux I we _express our

deep

gratitude

to Prof. M.

MENDTS

FRANCE,

Prof. M. BALAZARD and to their

_colleagues.

We also would like to thank Prof. J.-P. ALLOUCHE and the referee for several

_helpful

remarks.

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[1]

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_(1930),

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[2]

H. Bohr and B. _{Jessen, Über} die _{Wertverteilung} der Riemannschen Zeta _funktion,

Zweite _Mitteilung, Acta Math. 58

_(1932),1-55.

[3]

B. Jessen and A. _Wintner, Distribution _functions and the Riemann zeta-function,

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[8]

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(1974), 73-82. Antanas LAURINCIKAS Department of Mathematics Vilnius University Naugarduko 24 2006 Vilnius, Lithuania