Limit theorems for the Matsumoto zeta-function

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

NTANAS

L

AURIN ˇ

CIKAS

Limit theorems for the Matsumoto zeta-function

Journal de Théorie des Nombres de Bordeaux, tome

8, n

o

_{1 (1996),}

p. 143-158

<http://www.numdam.org/item?id=JTNB_1996__8_1_143_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 theorems for the Matsumoto Zeta-function

par ANTANAS

LAURIN010CIKAS

RÉSUMÉ. On démontre deux théorèmes limites fonctionnels _pondérés pour la fonction introduite par K. Matsumoto.

ABSTRACT. In this paper two weighted functional limit theorems for the

function introduced by K. Matsumoto are _proved.

Let N denote the set of all natural numbers. For _any

_{m e N,}

we define a

_positive

_integer

g(m).

Let be

_complex

numbers and

_{f ( j, m),1}

_j

_:5

g(m),

m e N,

be natural numbers. Now we can _{define the}

_polynomials

of

_degree

_{f (1, m)}

+ ~ ~ ~ +

f (g(m), m~ .

Let s be a

_{complex variable,}

and let _pn denote the n th

_prime

number. In

_[5]

K. Matsumoto introduced the

_following

zeta-function

and under some

_hypotheses

on

g(m),

am

and

cp(s)

he

_proved

the limit theorems for

_log

in the

_complex

_plane

C.

Let B denote a number

_(not

_always

the

_same)

bounded

_by

a constant.

Suppose

that

Manuscrit reçu Ie 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&#x26;T} Cooperation with Central and Eastern _European Countries" .

with some

non-negative

constats a

_{and ,Q.}

Let meas denote the

Lebes-gue measure of the set

A,

and,

for T &#x3E;

_0,

let

where in

_place

of dots we indicate a condition satisfied

_by

t. Under the condition

₍₁₎

is a

holomorphic

function in the

half-plane

u &#x3E; a

+,8

+1 with no zeros.

Let,

for u &#x3E;

_,Q,

and let R denote a closed

_rectangle

in the

_{complex plane}

with the

_edges

parallel

to the axes.

THEOREM A.

_(Matsumoto

_[5]).

_Suppose

_{that ao}

&#x3E;

_{a +,Q +l.}

Then there

exists the limit

-L , ...

If we assume that

~p(s)

can be

_{meromorphically}

continued to the

_region

~ &#x3E;_

Po, Po

a + (3 + 1,

then the theorem

analogue

to theorem A in this

region

was also

_proved

in

_[5].

More

_precisely,

let

_V(s)

_{by meromorphic}

in the

_poles

in this

_region

be included in a

compact

set,

and,

po,

with some

_positive

6.

Moreover,

let

We

_put

where s’

= a’ +

i~’ runs all

_possible

zeros and

_poles

of

_~p(s)

in the

_strip

a

+ (3 +1.

Define +

_ito)

for cro +

ito

E G

_by

the

_analytic

continuation

_along

the

_{path s}

+

_{ito, (I}

&#x3E; ~o.

THEOREM B.

_(Matsumoto,

_[5]).

_Suppose

that po then there exists

the limit

In fact Theorems A and B are the limit theorems in sense of weak con-vergence of

probability

mesures on

(C, B(C))

where

by

B(S)

we denote the class of Borel sets of the _space S. Our aim is to

_give

a

_{generalization}

of Theorems A and B in sense of

_[1]

and

_[4],

i.e. to _prove the

_weighted

functional limit theorems for the function

_cp(s).

Let

Coo =

C U

_{oo}

be the Riemann

_sphere

and let

_{d(sl, s2)}

be a metric on

Coo

_{given by}

the formulae

Here s,

81, S2 E C. This metric is

_compatible

with the

_topology

of Let

D =

Is

_{E &#x3E;}

a+,Q+1}.

Denote

by

H(D)

the _space of

analytic

on D functions

_{f :}

D -&#x3E;

_{(Coo, d)}

_equiped

with the

_topology

of uniform convergence on

_compacta.

In this

topology

a _sequence

_{f fn, f n E H(D)}}

converges to the function

f

E

_H(D)

if

as n -~ oo

uniformly

in s on

_compact

subsets of D. On the other hand let

To

be a fixed

_positive

_number,

and let

_{w (T)}

be a

_positive

function of bounded variation on Let us

_put

Suppose

that lim

_{U (T,}

_w)

= oo and consider the

probability

_measure

T --+ 00

THEOREM 1. There exists a

_probability

measure

Pw

on

(H(D),

B(H(D)))

such that the measure

_{PT,w converges}

_weakly

to

_Pw

as T - oo.

Denote

_by

_M(D1)

the space of

meromorphic

on

_Di

_{functions f :}

_Di

-equiped

with the

_topology

_{of uniform convergence} on

_compacta.

Suppose

that for the functions

_w(T)

and

_cp(s)

the estimate

is satisfied for all a &#x3E; _po and all t E R. Consider the

_probability

measure

THEOREM 2. Let the function

_~p(s)

be

_meromorphic

in the

_{half-plane u}

&#x3E;

Suppose

that aII

_poles

in this

_region

are included in a

_compact

_set, and that the estimates

₍₂₎

and

₍₃₎

hold. Then there exists a

_{probabih’ty}

measure

Qw

on such that the measure

_QT,w

_converges

_weakly

to

_Qw

as T -3 00.

For the

_proof

of Theorems 1 and 2 we will use the

_{following auxiliary}

results.

Let

81

and

_S2

be two metric _spaces. Let h :

_S2

be a measurable function. Then _every

_probability

measure P on

(81,B(81))

induces on

(92)B(?2))

the

_unique

_probability

measure

Ph-1

defined

_by

the

_equality

=

P(h-1 A),

A _E

8(82).

Let

P, Pn

be the

probability

_measures

on

LEMMA 1. Let h :

_S,

--~

S2

be a continuous function. Then the weak

convergence

of Pn

to P

implies

that

of Pnh’1

to as n - oo.

Proof.

This lemma is a

_particular

case of Theorem 5.1 from

[2].

Now let S be a

separable

metric _space with a metric p, and let

_Yn,

_Xln,

X2n, ...

be the S-valued random elements defined on Then the

following

assertion is true

_(Theorem

4.2 from

_[2]).

LEMMA 2.

_Suppose

that

Xkn

_£gXk

for each k and also If for &#x3E; 0

then Yn

Let 1

denote the unite circle on

_{complex plane}

that is ~y =

Is

_E

=

1 ~,

and let P be a

_probability

measure on

(~r, B(7’)),

r E N. The Fourier transform ... ,

kr)

of the measure P is defined

by

equality

where ki

E

Z,

xj

E 1, j

=

_{1, ...}

_r.

LEMMA 3. Let

_~Pn}

be a _sequence of

_probability

measures on

(,r, B(,r))

and let ... ,

kr) }

be a sequence of

corresponding

Fourier transforms.

Suppose

that for every set

_(ki,

... ,

kr)

of integers

the limit

g (ki,

... , =

lim

_{... , l~r)}

exists. Then there exists a

_probability

measure P on

n-~oo

(¡r, B(,r))

such that

Pn

converges

weakly

to P as n - oo.

_Moreover,

o (k1, ... , kr)

is the Fourier transform of P .

Proof.

Lemma is a

_special

case of the

_continuity

theorem for

_probability

measures on

_compact

Abelian groups, see, for

example,

[3].

The

_family

of

_probability

measures on

(8,B(8))

is

relatively

com-pact

if every sequence of elements of

{P}

contains a

_weakly

_convergent

subsequence.

The

_family

_~P}

is

_tight

if for

_arbitrary

e &#x3E; 0 there exists a

_compact

set

K such that

_P(K)

&#x3E; 1 - c for all P from

_{P}.

LEMMA 4.

_(The

Prokhorov

_theorem).

If the

_{family ofprobability}

measures

~P}

is

_tight,

then it is

_relatively

_compact.

Proof

of the lemma is contained in

_[2].

Let G be a

_region

in C. The

_family

of functions

_regular

on G is said to

be

_compact

on G if _{every sequence} of this

_family

contains a

_subsequence

which converges

uniformly

on _every

_compact

subset K C G.

LEMMA 5. If the

_family

of functions

_regular

on G is

_uniformly

bounded on

every

compact

subset

_{K C G,}

then it is

compact

on G. Proof can be

_found,

for

_example,

in

[6].

Proof of

Theorems 1. As it was noted in

_[5],

the function

_cp(s)

in the

half-plane a

&#x3E; a

+ {3

+1 is

_{presented by absolutely}

_convergent

Dirichlet series

First we will prove a limit theorem in the space

H(D)

for the Dirichlet

polynomial

Let

We will _prove that there exists a

_probability

measure

_PCPn,W

on

(H(D) , B (H (D)))

such that

_converges

_weakly

to as T - oo. Let _pl, ... , pr be the distinct

primes

which divide the

product

and let

where

_-yp,

is the unite circle on C for

_{all j}

=

1,

... , r. Let us define the

function

hn : Qr -

H(D~ by

the formula

for

_(xl,

_{... ,}

_x,)

E

Qr.

Here means that but

_Clearly,

Now we define on the

probability

measure

Then the Fourier

transform

is

_{given by}

the formula

Hence,

integrating

by

parts

and

_using

the

_properties

of the function

_{w (T),}

we

fing

that

Since the

_logarithms

of

_prime

numbers are

_{linearly independent}

over the field of rational

_numbers,

whence we deduce that

as T - oo.

Therefore, by

Lemma 3 the measure _{pT,w converges}

weakly

to the Haar measure mr on

(Hr)

B(Qr))

as T 2013~ oo.

Taking

into account

the

_continuity

of the function

hn

and the formula

_(4),

we obtain in view of Lemma 1 that the measure _converges

_weakly

to the measure

as T - oo.

Let K be a

_compact

subset of the

half-plane

D. Since the Dirichlet series for

_{p (s)}

is

_absolutely

and hence

_uniformly

_convergent

for a &#x3E; a

+ (3

_+1,

we have that

Let 8 be a random variable defined on the

_probability

space

,A,

_B(A),

P)

We take

Then from weak convergence of the measure we have that

where

_{X n}

denote an

H(D)-valued

random element with the distribution

It is well known that there is a _sequence of

_compact

subsets of D

such that ~

and the sets

_Kn

can be choosen to

satisfy

the

following

conditions:

a)

Kn

C

b)

if K is a

_compact

_{and K C D,}

then K C

_Kn

for some n. For

_{f , g}

E

_H(D)

let

where

Then from the remark made above we have that _p is a metric on

_H(D)

which induces its

_topology.

Let 1 E N. Then

_{by Chebyshev’s inequality}

Since the series for

_cp(s)

_converges

_absolutely

on

D,

we have that

Let c be an

arbitrary positive number,

and let

Then

₍₇₎

and

₍₈₎

_give

for all I E N. Let the function h : defined

_by

the formula

Then,

cleaxly, h

is

_continuous,

and

according

to

_(6),

_continuity

of h and Lemma 1.

_{Hence, using}

the

_inequality

(9),

we find that

for all l E N. Let us define

Then the

_family

of functions

H,

is

_uniformly

bounded on _every

_compact

K C D. But

_by

Lemma 5 then it is

_compact

on D. In view of

(10)

for all n &#x3E; 1. This

_gives

the

_tightness

of the

_family

of

_probability

measures Hence

_by

Lemma 4 it is

_relatively

_compact.

Let be a

subsequence

of such that

_{Pn,,w converges}

_weakly

to

Pw

as n --~ oo.

Then, clearly,

The convergence of the series for

cp(s)

is uniform on the

_{half-plane 7 &#x3E;}

a +

₍₃

+1

+ 6

for

_{every 6}

&#x3E; 0. Hence we find that

Let us

_put

Then from

₍₁₂₎

we

_get

for _{every -} &#x3E; 0.

_Thus,

_by

Lemma 2 and

_{(6), (11), (13)}

it follows that

what proves the theorem.

Proof of

Theorem 2. Part 1. First we _suppose that is

_analytic

in

_DI.

Then we will prove that the

probability

measure

Let ai

&#x3E; ~

and n E N. We define the function

in the

_strip

_{-1 1.}

_Here,

as

usual,

denote the Euler gamma-function.

_Moreover,

_let,

for 0" &#x3E; _p,

Since

uniformly

in _(7,

a’

_(7",

as

t ~ I

- _oo, it follows from

(2)

that the

integral

for

_{pi n (s)}

exists. Let

Consider the weak convergence of the measure

_{QT,n,w .}

Since 0’ &#x3E; a

_{+ /3}

+ 2

and _~1

&#x3E; 2 ,

then a + ~1 &#x3E; a

+ /?

_+1,

and the function

p(s

+

_z),

for Rez = is

presented

by absolutely

convergent

series

Let us consider the series

where

the series

_{(14) converges}

absolutely

for a &#x3E; a

+ {3 +!.

Thus,

_{interchanging}

sum and

_integral

in the definition of

cpln(s),

we find

It is well known

_that,

for

_positive

numbers a and

_b,

Therefore it _{is easy} to calculate that

Consequently,

the

_equality

₍₁₅₎

_may be written as

the series

_{being absolutely}

_convergent

for 7 &#x3E; _po.

_{Thus, repeating}

the

proof

of Theorem

_1,

we deduce that there exists a

_probability

measure

Qn,w

on such that

QT,n,w

_converges

_weakly

to

_Qn,w

as T -~ oo.

Let .K be a

_compact

subset of

_DI.

We will prove that

We

_begin

_by

_changing

the contour in the

_integral

for pi

n (s).

The

integrand

has a

_simple

_pole

at z = 0. Let _{po + e, e &#x3E;}

0,

whens _E

K,

and we

_put

~2 = _po

Denote

_by

L a

simple

closed contour

_lying

in

Dl

and

_enclosing

the set

_K,

and let 6 denote the distance of L from the set K. Then

_by

the

_Cauchy

formula we have

Therefore

Here

_by

_{I L we}

denote the

_lenght

of L.

_By

the formula

₍₁₇₎

Consequently,

since v is bounded

_by

a

_constant,

as n -~ oo. The contour L can be choosen so

that,

for s E

_L,

the

_inequalities

o &#x3E; po

_{+ 3:}

hold.

_Thus,

the relation

₍₁₆₎

is a _consequence of

(18)

and

_(19).

Now we

_already

can _prove a limit theorem for the measure Let

where the random variable 0 is defined in the

_proof

of Theorem 1. Since

QT,n,w converges

weakly

to

_Q~,w

as T - _oo, hence we have that

where

_{X n,w}

is an

H(Di)-

valued random element with the distribution

Qn,w.

· Since the series for is

absolutely

convergent,

we obtain

sim-ilarly

as in the

proof

of Theorem 1 that the

family

of measures is

tight,

and therefore

_by

Lemma 4 it is

_relatively

_compact.

_Applying

the

Chebyshev inequality,

we deduce from

(16)

that,

for _{every c} &#x3E;

_0,

Here _p is a metric on

_{H (D1)}

defined

by

the same manner as in the

_proof

of Theorem 1. Let us take

Let

_(Qn,,w)

be a

_subsequence

of _{which converges}

_weakly

to the measure _say,

as n’

---~ oo. Then we have that

From

_this,

_using

Lemma 2

_again,

we obtain in view of

(20)

and

(22)

that

The latter relation is an

_equivalent

of the weak convergence of

_QT,w

to

as T 2013~ oo .

Part 2.

_{Let o and p}

be two functions

_satisfying

the

_hypotheses

of Part

1. On

(H2(Dl)’ ,~ (HZ ~D1 ~~

we define the

_probability

measure

Then,

reasoning similarly

as in Part

_1,

we can _prove that there exists a

probability

measure

Q§/~

on such that

con-verges

weakly

to

Q~

as T 2013~ oo.

Part 3. Now let

_~p(s)

be as in Theorem 2. Since all

_poles

of

_p(s)

are included in a

_{compact set,}

the number r of these

_poles

in finite. Denote

they by

81, ... , sr, and let

Then the function is

_analytic

in

_{D1, and,}

for u &#x3E; 0:

_{+ /3}

_+1,

it is

_given

by

an

absolutely

_convergent

Dirichlet series. The estimate

(3)

for

0(s)

is

satisfied,

too.

_{Consequently,}

_by

Part 2 the measure

Q".

_converges

_weakly

to some measure

Q’

as T 2013~ oo.

Let the function h :

_{H2 (D1) --~}

_M(D1)

be

_{given by}

the formula

the function h is continuous.

_Therefore,

from Lemma 1 we find that the measure

QT,w

_converges

weakly

to as T - oo. Theorem 2 is

_proved.

Acknowledgement.

The author expresses his

deep

gratitude

to Prof. M.

_{Mendbs-France,}

Dr. M. Balazard and their

_colleagues

for constant

at-tention to his

_stay

at the

_University

Bordeaux I. He also thanks Prof.

B.

_Bagchi

for the

_possibility

to

_{get acquainted}

with the work

_[1].

REFERENCES

[1]

B. _Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet _series, Ph. D. _Thesis, Indian Statist. _Inst., Calcutta, 1982.

[2]

P. _Billingsley, Convergence of Probability Measures, John _{Wiley, 1968.}

[3]

H. _{Heyer, Probability} measures on locally _{compact groups, Springer-Verlag,} Berlin-Heidelberg-New York, 1977.

[4] A. _{Laurin010Dikas,} A _weighted limit theorem _for the Riemann _{zeta-function,} Liet. matem. rink. _{32(3) (1992),} 369-376. _(Russian)

[5] K. _Matsumoto, Value-distribution of zeta-functions, Lecture Notes in Math. 1434

(1990),178-187.

[6]

B. V. _Shabat, Introduction to _{complex analysis, Moscow, 1969. (Russian)}

Antanas LAURIN61KAS

Department of Mathematics Vilnius University

Naugarduko 24