Limit theorems for the Estermann zeta function i i i Текст научной статьи по специальности «Философия, этика, религиоведение»

ЧЕБЫШЕВСКИЙ СБОРНИК Том 12 Выпуск 4 (2011)

UDK 519.14

LIMIT THEOREMS FOR THE ESTERMANN ZETA FUNCTION. Ill

A joint limit theorem in the sense of the weak convergence of probability measures on the complex plane for Estermann zeta-functions is obtained.

1 Introduction

This paper is a continuation of [2] and [3], therefore first of all we recall the results obtained in the above papers.

Let, for a E C,

denote the generalized divisor function. The Estermann zeta-function E(s; k/l,a), s = a + it, with parameters k/l, (k,l) = 1, and a is defined, for a > max(1,1 + Ra)

and by analytic continuation elsewhere. It has two simple poles s = 1 and s = 1 + a if a = 0, and one double pole s = 1 if a = 0.

In [2] we started with the characterization of the function E(s; k/l, a) by limit theorems in the sense of the weak convergence of probability measures. Denote by meas{A} the Lebesgue measure of a measurable set A C R, and define, for T > 0,

A. LAURINCIKAS (Vilnius, Siauliai, Lithuania) D. SIAUCIUNAS (Siauliai, Lithuania)

Аннотация

by

ufT (...) = ~mea s{t £ [0,T ] : ... },

tt in vT only shows that the measure is taken over t E [0,T ] .Let B(S) be the class

S

Y = {s E C : \s\ = 1}■ Define

Q = II ^^, p

where yp = Y f°r all primes p. Then with the product topology and pointwise multiplication the infinite-dimensional torus Q is a compact topological Abelian group. Therefore, on (Q, B(Q)) the probability Haar measure mH can be defined, and we obtain a probability space (Q, B(Q),mH) ■ Denote by u(p) the projection of u E Q to the coordinate space yp ■> and define, for m E N,

u(m) = ]^[ ur (p),

pr\\m

where pr || m means that pr \ m but pr+1 \ m. On the probability space (Q, B(Q), mH) define, for a > 1/2, a complex-valued random element E(a; k/l,a; u) by the formula

ev z /; ^ v^a«(m)u(m) fo • kl

E(a; k/l,a; u) = -----#----exp < 2nimj >.

m=1 ^ '

Since E(s; k/l,a) = E(s — a; k/l, -a), we suppose without loss of generality throughout the paper, as in [2] and [3], that Ra < 0.

Theorem A. [2]. Supose that a > 1/2 and Ra < 0. Then the probability measure

vfT(E(a + it; k/l, a) E A), A E B(C),

converges weakly to the distribution of the random, element E(a; k/l, a; u) as T ^ to

In [3], Theorem A has been generalized to the space of meromorphic functions. For a region G on the complex plane, denote by M(G) the space of meromorphic on G functions equipped with the topology of uniform convergence on compacta. The space H (G) of analytic on G functions forms a subspace of M (G) .Let D = {s E C : a > 1/2}, and on the probability space (Q, B(Q),mH) define an H(D)-valued random element E(s; k/l, a; u) by the formula

jp( j /1 \ ^ aa(m)u(m) (0 k\

E(s; k/l, a; u) = ------s-----exp < 2nmj >.

m=1 ^ *

Ra < 0

vT(E(s + ir; k/l, a) E A), A E B(M(D)),

converges weakly to the distribution of the random element E(s; k/l, a; u) as T ^ to.

We recall that the distribution of E(s; k/l,a; u) is the probability measure

mH(u E Q : E(s; k/l, a; u) E A), A E B(H(D)).

The aim of this paper is to obtain a joint limit theorem on the complex plane

for Estermann zeta-functions. Let, for a > max(1,1 + Raj),

^ ( )

E(s;kj/lj,aj) = °amT exp{2nmkj/lj}, j = 1,...,r.

m=1

Denote by

Cr = C x ■ ■ ■ x C

'-----v-----'

r

the Cartesian product, and define the probability measure Pt;al,...,ar = vT((E(a1 + it; k1/l1, a1),..., E(ar + it; kr/lr, ar)) E A), A E B(Cr).

Moreover, for min aj > 1/2 and u E Q, let

1<j<r

E (a1,... ,ar; u) = (E (a1; k1/l1, a1; u),... , E (ar; kr/lr, ar; u)),

where

aa. (m)u(m) r , ^

E(aj;kj/lj,aj;u) = —m.— exp{2nmkj/lj}, j = 1,...,r.

m.

m=1

Then E (a1,... ,ar; u) is a Cr-valued random element define on the probability space (Q, B(Q),mH)• The main result of this paper is the following statement.

Theorem 1. Let min aj > 1/2 and Raj < 0, j = 1,...,r. Then the

1<j<r

probability measure PT;ai,..,ar converges weakly to the distribution of the random element E (a1,... ,ar; u) as T ^ to.

The proof of Theorem 1 is based on Theorem A and Prokhorov’s theorems.

2 Application of Prokhorov’s theory

In the theory of the weak convergence of probability measures an important role is played by the relative compactness of families of probability measures. By the definition, a family of probability measures {P} on (S, B(S)) is relatively compact if every sequence of {P} contains a weakly convergent subsequence. Clearly, if the probability measure Pn converges weakly to some measure P as n ^ to, then the family {Pn} is relatively compact. However, the relative compactness of {Pn}

does not imply in general the weak convergence of Pn, but, under some additional condition, does. Therefore, it is important to know how to obtain the relative compactness of a given family. Yu. V. Prokhorov observed [5] that for this a notion of the tightness can be used. By the definition, the family of probability measures {P} is tight if, for arbitrary e > 0, there exists a compact subset K C S such that P(K) > 1 - e for all P E {P}.

Lemma 1. (First Prokhorov’s theorem) If the family of probability measures {P} is tight, then it is relatively compact.

S

complete metric space. If the family of probability measure {P} on (S, B(S)) is relatively compact, then it is tight.

Proof of Prokhorov’s theorems can be found, for example, in [1].

Lemma 3. The family of probability measures {Pt;ai,..,ar} is tight.

Proof. Let, for aj > 1/2 and Raj < 0,

Pt;aj (A) = vT ((E (aj + it; kj/lj ,aj)) E A), A E B(C), j = 1,...,r.

Then, by Theorem A, we have that the measure PT;. converges weakly to the distribution of the random element

~ aa. (m)u(m) r , ^

—mF.—exp{2nimk/lj}

m=1

as T ^ to, j = 1,..., r. Hence, the family of probability measures {PT]a } is relatively compact, j = 1,... ,r. Sinee C is a separable complete metric space, in view of Lemma 2, the family {Pt;aj} is right, j = 1,... ,r. Therefore, for arbitrary e > 0, there exists a compact subset Kj C C such that

e

Ptf(Kj) > 1 -j = 1..............r, (1)

T > 0

Suppose that the random variable 9 is defined on a certain probability space (Q, B(Q), P) and uniformly distributed on [0,1]. Let, for aj > 1/2,

ET,j (aj) = E(aj + iT9; kj/lj, aj), j = 1,...,r,

and

Et (a1,... ,ar) = (ET,1(a1),... ,Et,r (ar)).

We put K = K1 x ■ ■ ■ x Kr. Then, clearly, K is a compact subset of the space Cr. Taking into account inequality (1), we obtain that

Pt ;au..Fr (Cr \ K) = P(Et (a1,...,ar) E Cr \ K ) =

p( [JiETj(a,) £ C \ K,)) <

< E riETj (a,) £ C \ Kj ) = £ PT--j (C \ K,) < £

j=1 j=1

for all T > 0. Therefore, PT■;a1,...,ar (K) > 1 — £, for all T > 0, i. e., the family {pt;*i,...,„r} is tight. □

Corollary 1. The family of probability measures {PT;ai,...,ar} is relatively compact.

□

3 Limit theorem for a linear combination

In this section we consider the linear combination

r

E (s) = Y1u,E (s + a°,;k,/l,, a,)

,=i

where u, ,j = 1,..., r, are arbitrary complex numbers, a > max a1, = max (1/2 —

1<,<r 1<,<r

a0,), a0, > 1/2. So we have t hat a, < 0, j = 1,... ,r.

Let

r

E(a, u) = u,E(s + a0,; k,/1,, a,; u).

,=1

If a > max a,, then E(a,u) is a complex-valued random element defined on the

1<,<r

probability space ( £1,B(1 ),mH)■

Theorem 2. Let a > max a,. Then the probability measure

1<,<r

Pt;a(A) = vT(E(a + it) £ A), A £ B(C),

converges weakly to the distribution of the random element E(a,u) as T ^ to.

Proof. For the proof, we will use a way similar to that of the proof of Theorem A. Therefore, we will give only sketches of the proofs of statements.

Let, for positive integers N and n, and a fixed a0 > 1/2,

EN,n(a +lt;k,/l,,a,Hj] exp{ — (}> j = 1,...,r,

and

N

’N,n(a + it; k,/i,, a, ' ^

EN,n(a +it;k,/l,,a,;u) = m+r exp{2ni^} exP { — (m) }’

j = 1,... ,r, u £ 1,

and define

and

EN,n(a + it) — ^ ^u,EN,n(a + —o, + it; k,/1,,a,), ,=1

EN,n(a + it; u) = ^ u,EN,n(a + ao, + it; k,/1,, a,; u).

,=i

Consider the weak convergence of two probability measures

PT,N,n;a (A) = vT (EN,n(- + it) £ A), A £ B(C),

and

PDT,N,n;a (A) = Vy (EN,n(- + it; u) £ A), A £ B(C).

□

Lemma 4. The probability measures PT,N,n;a and PT,Nn;a both converge weakly to the same probability measure on (C, B(C)) as T ^ to.

Proof. Since logarithms of prime numbers are linearly independent over the field of rational numbers, we easily find that the probability measure

QT(A) = vT((p-it : p is prime) £ A), A £ B( 1 ),

converges weakly to the Haar measure mH as T ^ to. Moreover, there exist a continuous functions ^d h : 1 ^ C such that

h((p-it : p is pri me)) = EN,n(a + it)

and

h((p-it : p is pri me)) = EN,n(a + it; u).

Therefore, we obtain that PT,N,n;a and PT,N,n;a converge weakly to mHh-1 and mHh-1, respectively, as T ^ to . However, the invariance of the Haar measure mH show that mHh-1 = mHh-1.

Now let

^ -aj (m) f . k,' f (™ )a°'

En(a + it; k,/l,,a, ) = a^mi)exp{2nim^} ex^ — ( m) }, j = 1^..

and

En{a + it; k3 /lj a; u) = ^ exP { 2n im^ } exP { — ( m) } >

m=1 j ' \ \ / J

j = 1,... ,r, u E Q.

Then, for each j = 1,r, the series for En(-+it; kj/lj, aj) and En(-+it; kj/lj ,aj; u) absolutely converge for - > 1/2. This was observed in [2]. Next, define the functions

r

En(- + it) = ^ Uj En(- + -oj + it; kj/lj, aj), j = 1,...

j=i

and

En(- + it; u) = Uj En(- + -0j + it; kj/lj, aj; u), j = 1,... ,r, u E Q,

j=i

and the probability measures

Pr,n;o-(A) = vT(En(- + it) E A), A E B(C),

and

Pt , n; a (A) = Vy (En(- + it) E A), A E B(C).

□

Lemma 5. Let - > max -1j. Then the probability measures PT,n;a and PT,n;a

1<j<r ’ ’ ’ ’

both converge weakly to the same probability measure on C, B(C) as T ^ to.

Proof A scheme of the proof is the same as that of Lemma 2 in [2]. By Lemma 4 we have that the measures PT,N,n;a and pT,N,n;a both converge weakly to some measure PN,n;a as T ^ to. The first step is to prove that the family of probability measures {PN,n;a} is tight for fixed n. Let 9 be the random variable defined in Section 2, and let

Xt , n , n(-) = en, n (- + iT9).

Then, in view of Lemma 4,

Xt ,n ,n(-) —Xn,n(-), (2)

T

T)

where means the convergence in distribution, and XNn(-) is a complex-valued random element with the distribution PN,n;a. By the definition of EN,n(- + it) we find that

T

suplimsup— \EN,n(- + it)|dt < R < to.

N>1 TT J

o

r

Hence, taking M = R/e, denoting Ks = {s E C : \s\ < M} and using (2), we obtain that

P(XN,n(-) E K£) ^ 1 — e

for all N E N. This gives the tightness of the family {PN,n}.

Since EN,n(- + it; kj/lj,aj) is a partial sum of En(- + it; kj/lj, aj), we have, for -> 1/2,

T

lim limsup — \ENn(- + it; kj/lj ,aj) — En(- + it; kj/lj ,aj)\dt = 0, j = 1,...,r.

N ^ t^ T J o

Hence, obviously, for - > 1/2,

T

lim limsup— I \Enn(- + it) — En(- + it)\dt = 0. (3)

N ^ t^ T J

o

Let

XT,n(-) = En(- + iT9).

e > 0

lim limsupP(\Xtnn(-) — Xtn(-)\ > e) = 0. (4)

Since the family {PN,n} is tight, by Lemma 1 it is relatively compact. Let {PNl,n;a} C {PN,n;a} be suchthat PNl,n;a converges weakly to Pn;a, say, as N1 > to,

T>

i. e., XNl,n(-) ------> Pn;a. This, (2), (4) and Theorem 4.2 from [1] now yield the

relation

XT,n(-) —— Pn;a. (5)

N1^^

Thus, we proved that PT,n;a converges weak ly to Pn;a as T > to. Moreover, (5)

shows that the measure Pn;a is independent of {PNl,n;a}. Hence,

XN,n(-) — > Pn;a. (6)

N

Now let

XT,N,n(-) = EN,n(- + iT9; u)

and

XT,n(-) = En(- + iT9; u).

Then the above arguments with (6) show that the measure PTna also converge weakly to Pn;a as T > to. □

Proof. [Proof of Theorem 2] Define once one probability measure

PT,a(A) = vT(E(- + it;u) E A), A E B(C).

First we will prove that, for a > max a1j, the probability measures PT;a and Рт;а

1<j<r ’ ’

both converge weakly to the same probability measure Pa on (C, B(C)) as T ^ to. This can be done in the same way as in the proof of Lemma 5, using Lemma 5 and the relations

т

lim limsup — \E(a + it) — En(a + it)\dt = 0

т^ж T J 0

and, for almost all ш Є П,

т

lim limsup— \E (a + гЬ,ш) — En(a + it; w)|dt = 0,

n^<x т^ж T J

0

where a > max a\j. The later relations follow from the analogical relations for

1 < j< r

the functions E(a + it; kj/lj,aj), En(a + it; kj/lj,aj^d E(a + it; kj/lj, aj; ш), En (a + it; kj/lj ,aj; ш), for a > 1/2, obtained in [2], j = 1,... ,r, and the definitions of the functions E(a + it), En(a + it), E(a + it; ш) and En(a + it; ш). Therefore, it remains to identify the limit measure Pa.

Let, for t Є R,ot = [p-u : p is prime} and let ф*(ш) = о^ш,ш Є П. Then {фі : t Є R} is an ergodic one-parameter group of measurable transformations on П [4].

If A Є B(C) is a continuity set of the measure P^, then, for a > max a1j,

1<j<r

lim vT(E(a + it; ш) Є A) = Pa (A) (7)

т ^ж

for almost all ш Є П. Suppose that the set A is fixed, and define a random variable П от the probability space (П, В(П),тн) by the formula

{■

1 E(a, ш) Є A,

п(ш) = <

V ; ' 0 if E(a, ш) = A.

Then we have that the expectation

E(n) = ndmH = mH(ш Є П : E(a^) Є A) d= PE;a(A), (8)

where PE;a is the distribution of the random element E(a,u). In view of the ergodicity of : t E R}, the process n(pt(u)) is ergodic. Thus, the Birkhoff-Khintchine theorem yields

T

lim 1 f n(^t(^))dt = En (9)

T^<x 1 J 0

for almost all u E Q. However,

T

v(<pt(u))dt = vT(E(a + it,u) E A).

0

Hence, and from (8) and (9) we deduce that

lim vT(E(a + it,u) E A) = PEo(A),

T^x ’

for almost all u E Q. This together with (7) shows that Po(A) = PE;o(A) for all continuity sets A of Po. Hence the theorem follows. □

4 Proof of Theorem 1

In view of Corollary 1, there exists a sequence Ti — to such that PTl;ai,...,ar converges weakly to some probability measure on (Cr, B(Cr)) as Ti — to.

Hence, there exists a Cr-valued random element

E = E(ai,... ,ar) = (ei(ai),... ,er(ar))

defined on a certain probability space, and P0l,..,0r is its distribution. In other words,

Eti (ai,... ,ar) V > E (ai,... a). (10)

T1^x

Let aj = a + a0j, where a > max a\j. Then (10) and the continuity of the function

i<j<r

u : Cr — C given by

r

u(si, . . . , Sr) = UjSj

j=i

u(Eti(ai,... ,ar)) V > u(E(ai,... ,ar)). (11)

Ti^x

show that

Here

and

r

u(Eti (ai,.. .,ar)) = ^2 ujEtx j (a + aoj)

j=i

u(E (ai,... ,ar)) = ^2 uj ej (a + aoj). j=i

In view of the definition of ET(ai,..., ar) and E(s), we rewrite (11) in the from

E(a + iTi9) ———> u(E (ai,...,ar)). (12)

T1^x

On the other hand, by Theorem 2

E(a + iT19) —— E(a,u).

Now this and (12) imply

Taking in this relation a = 0 (this is possible be cause a1j < 0, j = 1,... ,r) we find that

u1 , . . . , ur

It is known [1] that the family of R2r generated by all hyperplanes forms a determining class, hence also in Cr. Therefore, in virtue of (13) we have that the Cr-valued random elements (E (a01; k1/l1,a1; u),...,E (a0r; kr/lr ,ar; u)) and (ei(a01),

...,er(a0r)) have the same distribution. However, for j = 1,...,r, a0j > 1/2. Therefore, remembering the definition of E (a1 ,...,ar; u), we obtain that, for min aj > 1/2,

In the latter relation, the random element E(a1,... ,ar; u) is independent of the sequence T1. Since the family of probability measures {PT;ai,...,ar} is relatively compact, hence the theorem follows.

5 Concluding remarks

The proof of Theorem 1 is based on the modified Cramer-Wald criterion. An

r

for Dirichlet polynomials on the weak convergence of the probability measures vT((EN,n(a1 + it; k1/l1,ai),..., EN>n(ar + it; kr/lr,®r)) ^ A), A e B(Cr),

VT((ENn(a1 + it; k1/l1,a1; u),..., ENn(ar + it; kr/lr,ar; u)) e A), A e B(Cr),

(13)

1<j<r

E(a1;... ,ar; u) = E(a1;... ,ar).

Hence, in view of (10)

ETl (a1;... , ar) ———> E (a1;... ,ar; u).

and

to the same measure as T ^ to must be proved. Using the latter theorem, a r-dimensional limit theorem for absolutely convergent Dirichlet series on the weak convergence of the probability measures

vT((En(ui + it; ,..., En(ar + it; kr/lr, ar)) £ A), A £ B(Cr),

and

vT((En(al + it; kl/ll, al; w),..., En(ar + it; kr/lr, ar; w)) £ A), A £ B(Cr),

to the same measure as T ^ to must be obtained. This theorem implies the weak convergence of the probability measures ,...,ar and

vT((E(al + it; kl/ll,al; w),..., E(ar + it; kr/lr, ar; w)) £ A), A £ B(Cr),

to the same measure as T ^ to. Now, an application of elements of the ergodic theory gives the assertion of Theorem 1. Note that this direct way of the proof is more clear, however, it uses more complicated formulas.

СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

[1] Billingsley P. Convergence of Probability Measures. New York: Wiley, 1968.

[2] Laurincikas A. Limit Theorems for the Riemann Zeta-Function. Dordrecht, Boston, London: Kluwer Academic Publishers, 1996.

[3] Laurincikas A. Limit theorems for the Estermann zeta-function. I// Statist. Probab. Letters. 2005. V. 72 No. 3 P. 227-235.

[4] Laurincikas A. Limit theorems for the Estermann zeta-function. II// CEJM. 2005. V 3 No. 4 P. 580-590.

[5] Prokhorov Yu. V. Convergence of random processes and limit theorems of probability theory// Probab. Theory and Appl. 1956. V. 1 No. 2, P. 177-238 (in Russian).

Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, LT-03225 Vilnius, Lithuania.

Faculty of Mathematics and Informatics, Siauliai University, P. Visinskio 19, LT-77156, Siauliai, Lithuania.

E-mail: antanas.laurincikas@mif.vu.lt, siauciunas@fm.su.lt Поступило 21.12.2011