22
s = σ + it m=1 a m e λms , {a m } {λ m } ζ (s) σ> 1

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iauliai Math. Semin.,7 (15), 2012, 1940

LIMIT THEOREMS FOR ZETA-FUNCTIONS

WITH APPLICATION IN UNIVERSALITY

Roma KAINSKAIT EFaculty of Mathematics and Informatics, iauliai University,

P. Vi²inskio str. 19, LT-77156 iauliai, Lithuania;e-mail: [email protected]

Abstract. The zeta-functions are very interesting objects in analytic num-ber theory. In the paper, the main attention is focused to the Riemannzeta-function, from the rst results on the denseness until limit theoremswith application in the investigation of universality in Voronin's sense. Aspecial attention is devoted to the Matsumoto zeta-function. The paper isbased on the content of a lecture to the master and doctoral students at theGraduate School of Mathematics of Nagoya University (Japan).

Key words and phrases: density, discrete limit theorems, Matsumoto zeta-function, Riemann zeta-function, probability measure, universality, value-distribution, weak convergence.

2010 Mathematics Subject Classication: 11M41, 11M35, 11M06.

1. Introduction

So called zeta-functions are important objects nowadays in number theory.We recall that the zeta-functions are functions of complex variable s = σ+ itwhich in some half-plane are dened by the Dirichlet series

∞∑m=1

ame−λms,

where am is a sequence of complex numbers having some arithmeticalsense, λm is an increasing sequence of positive numbers.

The zeta-functions are generalizations of the Riemann zeta-function ζ(s).On the half-plane σ > 1, it is dened by ordinary Dirichlet series (when

20 R. Ka£inskaite

λm = logm) or by Euler product over prime numbers

ζ(s) =∞∑

m=1

1

ms=

∏p∈P

(1− 1

ps

)−1

(here P denotes the set of all prime numbers). It is well-known that it can beanalytically continued over the whole complex plane C, except for a simplepole at the point s = 1 with residue 1.

The function ζ(s) as a function of a real argument s was studied byL. Euler in 1737 [17]. B. Riemann [41] was the rst who investigated (andpresented the notation of ζ(s)) the function ζ(s) as a function of complexvariable.

2. Bohr's denseness theorem

In 1910, H. Bohr initiated the study of value-distribution of ζ(s) using dio-phantine, geometric and probabilistic methods.

In the half-plane of absolute convergence σ > 1, we have

0 < |ζ(s)| 6 ζ(σ).

Then the values of ζ(s), in half-plane σ > σ0 > 1, are lying in the disc ofthe radius ζ(σ0) centered in the origin. It is possible to show that ζ(s) takesquite many complex values inside the disc with t ∈ R, i.e., the orbit ζ(s+ it)is included in a compact set of C. In 1911, H. Bohr [4] proved the followingstatement.

Theorem 1 ([4]). In every strip 1 < σ < 1 + ε, ζ(s) takes non-zero values

innitely often.

When σ 6 1, we have the following situation.

Theorem 2 ([6]). For every σ, 12 < σ 6 1, the set t ∈ R : ζ(σ + it) is

dense in C.

Theorem 3 ([5]). For every σ, 12 < σ 6 1, the set t ∈ R : log ζ(σ + it) is

dense in C.Here, for σ > 1,

log ζ(s) = −∑p∈P

Log

(1− 1

ps

),

and Log denotes the branch of the logarithm which is real on the positivereal axis.

Limit theorems for zeta-functions... 21

In 1972, S. M. Voronin obtained [45] the multidimensional generalizationof Bohr's denseness result.

Theorem 4 ([45]). For every xed distinct numbers s1, . . . , sn,12 < Re si <

1, 1 6 i 6 n, and sk = sl for k = l, the set

(ζ(s1 + it), . . . , ζ(sn + it)) : t ∈ R ,

is dense in Cn, n ∈ N. Moreover, for every xed number s, 12 < σ < 1, the

set (ζ(s+ it), ζ ′(s+ it), . . . , ζ(n−1)(s+ it)

): t ∈ R

is dense in Cn.

3. Bohr's probabilistic approach

In the second decade of XXth century, H. Bohr developed the studies of valuedistribution of the Riemann zeta-function using probabilistic approach. Itis dicult to say anything about concrete values of zeta-functions, but it ismuch easier to study how often the values get into a given set. H. Bohr foundthat these frequencies follow by strict mathematical laws. The rst result inthis direction belongs to H. Bohr and B. Jessen, and was obtained in [8].

Let R be a closed rectangle in the complex plane with the edges parallelto the coordinate axes, and L(T,R) denotes the Jordan measure of the sett ∈ [0, T ] : log ζ(σ + it) ∈ R.

Theorem 5 ([8]). For σ > 1, there exists the limit

limT→∞

L(T,R)

T= W (R, σ)

which depends only on σ and R.

In 1932, H. Bohr and B. Jessen obtained [9] a similar result in the caseσ > 1

2 . Let G be the half-plane σ > 12 , where all points are removed which

have the same imaginary part as, and smaller real part than one of possiblezeros of ζ(s) in this region. Let L1(T,R) denote the Jordan measure of theset t ∈ [0, T ] : σ + it ∈ G, log ζ(σ + it) ∈ R.

Theorem 6 ([9]). There exists the limit

limT→∞

L1(T,R)

T= W1(R, σ)

which depends only on σ and R.

22 R. Ka£inskaite

For the proof, the theory of sums of convex curves developed by H. Bohrand B. Jessen [7] was applied.

In 1987, K. Matsumoto obtained [34] some quantitative version of Bohr-Jessen's results. Let m(R) denote the Jordan measure of the rectangle R.

Theorem 7 ([34]). For every ε > 0 and σ > 1,

L(T,R) = W (R)T +BT (m(R) + ε)(log log T )−σ−17

+ε.

Here and latter B is a quantity which is bounded by a constant, and it depends

only on σ and ε.

The case σ > 12 is contained in the following theorem.

Theorem 8 ([35]). For any xed σ ∈(12 , 1

),

L1(T,R) = W1(R)T +BT (m(R) + ε)(log log T )−2σ−115

+ε,

where B depends only on σ and ε.

In 1999, T. Hattori and K. Matsumoto identied [19] this limit distribu-tion. The limit value in Theorem 8 could be understood as the probabilityhow many values of log ζ(σ + it) belong to the rectangle R.

In 1935, B. Jessen and A. Wintner proved [21] similar theorems to thoseof Bohr-Jessen using another method from the probability theory as inniteconvolutions of probability measures. Also, similar type result was obtainedby V. Borchsenius and B. Jessen in 1948 [10].

The above results were improved and generalized by many mathemati-cians. Among them B. Bagchi, P. D. T. A. Elliott, R. Garunk²tis, A. Ghosh,D. Joyner, A. Laurin£ikas, K. Matsumoto, H. Mishou, T. Nakamura, A. Sel-berg, E. Stankus, J. Steuding, and so on.

It is more convenient to formulate the above type limit theorems in theterminology of modern probability theory, i.e., in terms of the weak conver-gence of probability measures.

4. The theory of the weak convergence of probability

measures with some denitions

In the fourth decade of the XXth century, the theory of the weak convergenceof the probability measures was created. The main principles were given byA. N. Kolmogorov [26], P. Erdös and M. Kac [15][16], J. L. Doob [14] and

Limit theorems for zeta-functions... 23

M. Donsker [12][13]. The theory of weak convergence was developed byJ. V. Prokhorov [39][40], A. V. Skorokhod [42], L. LeCam [32], V. S. Vara-darajan [43] and other mathematicians. A modern theory of the weak con-vergence of probability measures is presented by P. Billingsley in [3].

Now we recall some basic notations from probability theory and weakconvergence. Let S be a non-empty set. The triple (S,F , P ) is called aprobability space, where F is a σ-eld consisting of some subsets of S, and Pis the probability measure on F . We investigate the case when S is a metricspace and F = B(S) is the family of all Borel subsets of S (when σ-eld isgenerated by all open subsets).

Let Pn and P be probability measures on the space (S,B(S)). We saythat Pn converges weakly to P as n tends to innity if, for every boundedcontinuous real function f on S,∫

S

fdPn →∫S

fdP as n → ∞.

Some equivalent statements of the weak convergence of probability mea-sures follow.

(1) For every bounded uniformly continuous real function f on S,

limn→∞

∫S

fdPn =

∫S

fdP.

(2) For every closed set F ,

lim supn→∞

Pn(F ) 6 P (F ).

(3) For every open set G,

lim infn→∞

Pn(G) > P (G).

(4) For every P -continuity set A (if P (∂A) = 0),

limn→∞

Pn(A) = P (A).

In the theory of weak convergence of probability measures, the Prokhorovtheorems play an important role: the relative compactness is connected withtightness of a family of probability measures.

24 R. Ka£inskaite

A family Pn of probability measures on (S,B(S)) is said to be relativelycompact if every sequence of elements of Pn contains a weakly convergentsubsequence. A family Pn is called tight if, for arbitrary ε > 0, there existsa compact set K such that P (K) > 1− ε for all P from Pn.

Lemma 1 ([3]). If a family of probability measures is tight, then it is relatively

compact.

Lemma 2 ([3]). Let the space S be separable and complete. If a family of

probability measures on (S,B(S)) is relatively compact, then it is tight.

5. Modern limit theorems for the Riemann zeta-function

Denote by measA the Lebesgue measure of a measurable set A ∈ R, andlet, for positive T ,

νT (...) =1

Tmeast ∈ [0, T ] : ...,

where in place of dots we write a condition satised by t.

Now we can state the analogue of Theorem 6 in modern terminology.Dene the probability measure

PT,σ(A) = νT (t ∈ [0, T ] : ζ(σ + it) ∈ A), A ∈ B(C).

Theorem 9 ([29]). For every xed σ > 12 , there exists a probability measure

Pσ on (C,B(C)) such that PT,σ weakly converges to Pσ as T → ∞.

Proof is obtained using the method of characteristic functions. Insteadof the measure PT,σ, we can consider the probability measure

νT ((Re ζ(σ + it), Im ζ(σ + it)) ∈ A) , A ∈ B(R2).

First it is proved that the probability measure

νT ((Re ζn(σ + it), Im ζn(σ + it)) ∈ A) , A ∈ B(R2),

weakly converges to some probability measure as T → ∞, where

ζn(s) =

n∑m=1

1

ms.

Limit theorems for zeta-functions... 25

Later the almost periodicity of ζ(σ + it) is applied. For each ε > 0, thereexists a number n0 = n0(ε, σ) such that, for n > n0,

lim supT→∞

T∫0

|ζ(σ + it)− ζn(σ + it)|2dt < ε.

Hence, the weak convergence of the measure PT,σ follows.

We can consider the limit theorems in various functional spaces.

Let H(G) be the set of all holomorphic on the region G functions withthe topology of uniform convergence on compacta. There exists a family ofcompact subsets Kj on G such that

G =∞∪j=1

Kj ,

and satisfying following properties:

(1) Kj ⊂ Kj+1 for every j ∈ N;

(2) if K is compact and K ⊂ G, then K ⊂ Kj for some j ∈ N.

Now, for every functions f, g ∈ H(G), let

ρj(f, g) = maxs∈Kj

|f(s)− g(s)|,

and dene

ρ(f, g) =

∞∑j=1

1

2jρj(f, g)

1 + ρj(f, g).

Then ρ is a metric on H(G) which induces its topology.

It is well-known that the space H(G) is a separable complete metric space[11]. Let D =

s ∈ C : 1

2 < σ < 1.

In 1981, B. Bagchi proved [1] the following statement for ζ(s). For everyA ∈ B(H(D)), dene

PHT (A) =

1

Tmeas τ ∈ [0, T ] : ζ(s+ iτ) ∈ A .

Theorem 10 ([1]). There exists a probability measure QH on (H(D),B(H(D))) such that PH

T weakly converges to QH as T → ∞.

26 R. Ka£inskaite

Theorem 10 is an analogue of Theorem 9 in the space of analytic func-tions. Moreover, B. Bagchi gave an explicit form of the limit measure QH .

Denote by γ the unit circle on the complex plane C, i.e., γ = s ∈ C :|s| = 1, and let

Ω =∏p∈P

γp,

where γp = γ for all p ∈ P. The torus Ω is a compact topological group, andhence, there exists the unique probability Haar measure mH on Ω. Then(Ω,B(Ω),mH) is a probability space. For ω ∈ Ω, denote by ω(p) the projec-tion of ω to the coordinate space γp. Dene

ζ(s, ω) =∏p∈P

(1− ω(p)

ps

)−1

. (1)

Let 12 < σ0 < 1. Then ζ(σ0, ω) converges almost surely (with respect to the

measure mH), and equals to∞∑n=1

ω(n)

nσ0, (2)

where ω(n) = ω(p1)α1 . . . ω(pr)

αr for n = pα11 . . . pαr

r . From Dirichlet seriestheory, we nd that, in view of convergence (2), it follows that the series

∞∑n=1

ω(n)

ns(3)

converges uniformly on every compact subset of the half-plane σ > σ0. Theexpression (3) equals to the right-side of (1). Then the product in (1) con-verges almost surely on every compact subset of D. Therefore, ζ(s, ω) is anH(D)-valued random element.

Let Qζ be the distribution of the random element ζ(s, ω), i.e.,

Qζ(A) = mH(ω ∈ Ω : ζ(s, ω) ∈ A), A ∈ B(H(D)).

Theorem 11 ([1]). The relation Qζ = QH holds.

From the last theorem, we can obtain an explicit form of the measure Pσ

in Theorem 9. Let h : H(D) → C which is dened by formula h(f) = f(σ)for every f ∈ H(D). Then PT,σ = PH

T · h−1. Since h is continuous, fromTheorems 9 and 11 we nd that Pσ = QH · h−1 = Qζ · h−1, i.e.,

Pσ(A) = mH(ω ∈ Ω : ζ(σ, ω) ∈ A), A ∈ B(C).

Limit theorems for zeta-functions... 27

6. Proof of the limit theorem for ζ(s)

In this Section, we will give the main steps for the proof of a limit theoremfor the Riemann zeta-function ζ(s) with explicitly given limit measure (fordetails, see, [29]).

First it is proved a limit theorem on torus Ω, i.e., that the probabilitymeasure

νT((p−it : p is prime) ∈ A

), A ∈ B(Ω),

converges weakly to the Haar measure mH as T → ∞. For this purpose, thelinear independence of the set log p : p is prime over the eld of rationalnumbers Q is applied.

The next step consists of the proof a limit theorems for Dirichlet polyno-mials associated with ζ(s), i.e., for nite sum

pn(s) =n∑

m=1

1

mit.

On (C,B(C)), there exists a probability measure Pn such that the measure

νT (pn(σ + it) ∈ A) , A ∈ B(C),

converges weakly to Pn as T → ∞.Let ω ∈ Ω, and dene

pn(σ + it, ω) =

n∑m=1

ω(m)

mσ.

Then

νT (pn(σ + it, ω) ∈ A) , A ∈ B(C),

converges weakly to Pn as T → ∞, also.The third step are limit theorems for absolutely convergent Dirichlet se-

ries for σ > 12 . Limit theorems for the series

ζn(s) =∞∑

m=1

1

msexp

(−(mn

)σ1)

and

ζn(s, ω) =

∞∑m=1

ω(m)

msexp

(−(mn

)σ1),

28 R. Ka£inskaite

where σ1 > 12 is xed, are proved. For both functions, the limit measures

coincide.In order to pass from ζn(s) to ζ(s) and ζn(s, ω) to ζ(s, ω), the approxi-

mation by the mean is applied, respectively. It is proved that

limn→∞

lim supT→∞

1

T

T∫0

|ζ(σ + it)− ζn(σ + it)|dt = 0,

and, for almost all ω ∈ Ω,

limn→∞

lim supT→∞

1

T

T∫0

|ζ(σ + it, ω)− ζn(σ + it, ω)|dt = 0.

For the proof, it is shown that the mean square of ζ(s, ω) is bounded. Forthis aim, the elements from ergodic theory are applied.

In the last step of the proof, the explicit form of the measure Pσ isobtained, i.e., it is shown that the probability measure PT,σ weakly convergesto the distribution of the random element ζ(s, ω) as T → ∞.

7. Matsumoto zeta-function

Analogous results are true for other zeta-functions. A wide class of zeta-functions have the Euler product over primes. In 1990, K. Matsumoto in-troduced [36] the zeta-function dened by polynomial Euler product overprimes.

For any m ∈ N, we dene a positive integer g(m). Moreover, let a(j)m ∈ C,

and f(j,m), 1 6 j 6 g(m), be natural numbers. Dene the polynomials

Am(X) =

g(m)∏j=1

(1− a(j)m Xf(j,m)

)of degree f(1,m) + . . . + f(g(m),m), and by pm, as usual, denote the mthprime number. Then the following zeta-function

φ(s) =

∞∏m=1

A−1m (p−s

m ) (4)

is called the Matsumoto zeta-function. It is assumed the conditions

g(m) = Bpαm, |a(j)m | 6 pβm, (5)

Limit theorems for zeta-functions... 29

where α and β are non-negative constants. The innite product in (4) con-verges absolutely for σ > α+β+1, and denes there a holomorphic functionφ(s) without zeros.

It is not dicult to see that, in the half-plane σ > α + β + 1, the Mat-sumoto zeta-function is one of the functions dened by absolutely convergentDirichlet series

∞∑m=1

bmms

,

and, for every ε > 0, coecients bm are estimated, i.e., bm = Bmα+β+ε.The function φ(s) is a generalization of classical zeta-functions. We will givesome examples.

1. Dirichlet L-functions L(s, χ), for σ > 1, are given by

L(s, χ) =∞∏

m=1

(1− χ(pm)

psm

)−1

,

where χ(m) denotes a Dirichlet character modulo k. χ(m) is an arith-metic function which is not identically zero, it is completely multi-plicative and periodic with period k, and χ(m) = 0 if (m, k) > 1. Here

α = β = 0, a(j)m = χ(pm), g(m) = 1, f(j,m) = 1.

2. The Dedekind zeta-function of the algebraic number eld K is denedby the formula

ζK(s) =∏p

(1− 1

(Np)s

)−1

, σ > 1.

Here p runs over all prime divisors of the eld K, and Np denotes

the norm of p. Let p(1)m , . . . , p

(g(m))m be the set of all prime divisors of

pm in K. Then, for σ > 1,

ζK(s) =

∞∏m=1

g(m)∏j=1

(1− 1

pf(j,m)sm

)−1

,

where f(j,m) are dened by Np(j)m = p

f(j,m)m .

3. Let SL(2,Z) be the full modular group, i.e.,

SL(2,Z) =(

a bc d

): a, b, c, d ∈ Z, ad− bc = 1

.

30 R. Ka£inskaite

A function analytic in the half-plane Im z > 0 such that

f

(az + b

cz + d

)= (cz + d)kf(z)

for all

(a bc d

)∈ SL(2,Z) and limIm z→∞ f(z) = 0 is called a cusp

form of weight k with respect to SL(2,Z). The function f(z) has theFourier expansion

f(z) =

∞∑n=1

c(n)e2πinz.

Then a Dirichlet series

L(s, f) =

∞∑m=1

c(m)m−s

is called the zeta-function attached to the cusp form f . For σ > k+12 ,

it converges absolutely and, for some additional conditions,

L(s, f) =

∞∏m=1

(1− αm

psm

)−1(1− βm

psm

)−1

,

with |αm| 6 p(k−1)/2m and |βm| 6 p

(k−1)/2m . Here α = β = k−1

2 , andg(m) = 2.

8. Continuous limit theorems for the Matsumoto

zeta-function

In 1990, K. Matsumoto proved [36] two limit theorems for the function φ(s)analogous to Bohr-Jessen's results.

Let R be the same as in Section 3. The rst theorem from [36] considersthe case σ > α + β + 1. For σ0 > α + β + 1 dene logφ(σ0 + it) as thefollowing sum of principal values

logφ(σ0 + it) = −∞∑

m=1

g(m)∑j=1

Log(1− a(j)m p−f(j,m)s

m

).

Theorem 12 ([36]). Under the conditions (5), there exists the limit

limT→∞

1

Tmeas t ∈ [0, T ] : logφ(σ0 + it) ∈ R .

Limit theorems for zeta-functions... 31

The second theorem of [36] deals with logφ(σ0+it) where σ0 < α+β+1.Let ρ0 be a constant with α+β+ 1

2 < ρ0 6 α+β+1. Assume that φ(s) canbe meromorphically continued to the half-plane σ > ρ0, and all poles andpossible zeros of φ(s) are included in a compact set. Let

G =s ∈ C : σ > ρ0

\

∪s′=σ′+it′

s = σ + it′ : ρ0 < σ 6 σ′,

where s′ = σ′+ it′ runs over all possible zeros and poles of φ(s) in the stripρ0 6 σ 6 α + β + 1. For any s0 = σ0 + it0 ∈ G, the value of logφ(s0) isdened by the analytic continuation along the path s = σ + it0 : σ0 6 σ.Let, for ρ0 6 σ 6 α+ β + 1

2 ,

φ(σ + it) = B|t|δ (6)

with some constant δ > 0, and

T∫0

|φ(σ + it)|2dt = BT, T → ∞. (7)

Theorem 13 ([36]). Let the conditions (5)(7) be satised. Suppose that

σ0 > ρ0. Then there exists the limit

limT→∞

1

Tmeas

t ∈ [0, T ] : σ0 + it ∈ G, logφ(σ0 + it) ∈ R

.

K. Matsumoto himself [37], [38] and jointly with T. Hattori [18] treatedthe limit measure in Theorem 13 and proved precise upper and lower boundsfor it.

A. Laurin£ikas generalized [27] Theorems 12 and 13, and proved twofunctional limit theorems with a weight for the function φ(s). Let T0 bea xed positive number, and let w(t) be a positive function of boundedvariation on [T0,∞). Take

U = U(T,w) =

T∫T0

w(τ)dτ,

and suppose that limT→∞ U(T,w) = ∞. Let D1 = s ∈ C : σ > α+ β + 1,and dene the probability measure

PT,w(A) =1

U

T∫T0

w(τ)Iτ : φ(s+iτ)∈Adτ, A ∈ B(H(D1)),

32 R. Ka£inskaite

where IA denotes the indicator function of the set A.

Theorem 14 ([27]). Suppose that conditions (5)(7) hold. Then there exists

a probability measure Pw on (H(D1),B(H(D1))) such that the measure PT,w

converges weakly to Pw as T → ∞.

A. Laurin£ikas gave in [28], [30] the explicit form of the limit measure inTheorem 14.

9. Discrete limit theorems for the Matsumoto zeta-function

In all limit theorems stated above, probability measures are dened by termsof continuously varied translations of the imaginary part of a complex variablein the interval [0, T ], i.e., in these theorems the frequency of the sets

t ∈ [0, T ] : φ(σ + it) ∈ A or τ ∈ [0, T ] : φ(s+ iτ) ∈ A

was considered (here A denotes some set on the complex plane or in thespaces of analytic or meromorphic functions). Such kind of limit theoremscan be called continuous limit theorems.

We call limit theorems discrete if the frequency of the sets

0 6 m 6 N : φ(σ + ihm) ∈ A or 0 6 m 6 N : φ(s+ ihm) ∈ A

is considered, i.e., in this case the imaginary part of translations takes valuesin some arithmetical progression with a xed step h > 0.

The rst discrete results on the value distribution of Riemann zeta-function were obtained by S. M. Voronin [45] in 1972. In 1981, B. Bagchi [1]investigated discrete value distribution of some class of Dirichlet L-functions.There it is assumed that the step of arithmetic progression h > 0 is any xedreal number.

Doctoral dissertation of the author [23] is devoted to the investigationof discrete value-distribution of the Matsumoto zeta-function. In view of anapplication of limit theorems with explicitly given limit measure, there theclass of numbers h was limited, i.e., there it was assumed that exp2πk

h is anirrational number for all k = 0, and limit theorems in the sense of the weakconvergence of probability measures for φ(s) were proved on the complexplane as well as in the space of analytic functions. Also, some results forφ(s) on the complex plane and in the space of analytic functions with suchh for which exp2πk

h is rational for some k = 0 are obtained (see, [24], [25]).

Limit theorems for zeta-functions... 33

There exists integers k such that exp2πkh are rational. Denote by k0

the smallest of such k.

Let exp2πk0h = m0

n0with m0, n0 ∈ N, (m0, n0) = 1, and let Ωh = ω ∈

Ω : ω(m0) = ω(n0). Then Ωh is a closed subgroup of Ω, and it is a compacttopological Abelian group, also. Hence we obtain that, on (Ωh,B(Ωh)), theprobability Haar measure mhH exists, and this leads to a probability space(Ωh,B(Ωh),mhH).

For σ > α+β+ 12 , on (Ωh,B(Ωh),mhH) dene a complex-valued random

variable by the formula

φ(σ, ωh) =∞∑k=1

b(k)ωh(k)

kσ, ωh ∈ Ωh.

Let N ∈ N, and

µN (. . . ) =1

N + 1#0 6 m 6 N : . . . ,

where in place of dots a condition satised by m is to be written. Then thefollowing statement holds.

Theorem 15 ([24]). Suppose that the number h satises the above conditions,

and σ > α+ β + 12 . Then the probability measure

µN (φ(σ + imh) ∈ A), A ∈ B(C),

converges weakly to the distribution of the random variable φ(σ, ωh) as N →∞.

Discrete limit theorems for various zeta-functions were investigated in afew Doctoral thesis of Lithuanian Number Theory school by supervision ofA. Laurin£ikas. Namely, J. Ignatavi£iute considered [20] the value distribu-tion of the Lerch zeta-function, R. Macaitiene [33] obtained limit theoremsfor general Dirichlet series, V. Balinskaite [2] investigated value distributionfor the Mellin transforms of the Riemann zeta-function.

10. Universality of the Riemann zeta-function

B. Bagchi applied functional limit theorems for the proof of universality [1].In 1975, this property was discovered by S. M. Voronin [44].

34 R. Ka£inskaite

Theorem 16 ([44]). Let 0 < r < 14 , and f(s) be any non-vanishing conti-

nuous function on the disc |s| 6 r which is analytic in the interior of this

disc. Then, for every ε > 0, there exists a real number τ = τ(ε) such that

max|s|6r

∣∣∣∣f(s)− ζ(s+

3

4+ iτ

)∣∣∣∣ < ε.

From the limit theorem for ζ(s) in the space of analytic functions (Theo-rem 11), we have the following more general result.

Theorem 17 ([29]). Let K be a compact subset of the strip D = s ∈ C : 12 <

s < 1 with connected complement. Let f(s) be a non-vanishing continuous

function on K which is analytic in the interior of K. Then, for every ε > 0,

lim infT→∞

νT

(sups∈K

|f(s)− ζ(s+ iτ)| < ε

)> 0.

11. Proof of universality for ζ(s)

In this Section, we will give the main steps to the proof of universality inVoronin's sense.

Theorem 10 is used with explicitly given support of the limit measure.

We recall that the support of the probability measure P on (S,B(S)),S is seprable, is a minimal closed set A ⊆ S such that P (A) = 1. Theset A consists of all x ∈ S such that, for every open neighbourhood G ofx, P (G) > 0. The support of S-valued random element is a support of itsdistribution.

Lemma 3 (Mergelyan theorem, [46]). Let K be a compact subset of C whose

complement is connected. Then every continuous function f(s) on K which

is analytic in the interior of K is approximable uniformly on K by the poly-

nomials of s.

Now we will give a sketch of the proof of Theorem 17.

Let

S = f ∈ H(D) : f(s) = 0 or f(s) ≡ 0.

Using the independence of random variables ω(p) and applying some proper-ties of the Hilbert spaces, it is proved that the support of the measure Qζ isthe set S.

Limit theorems for zeta-functions... 35

Let the function f(s) be non-vanishing analytic continued to H(D). De-note by G the set of functions g ∈ H(D) such that

sups∈K

|g(s)− f(s)| < ε.

Clearly, the function f(s) ∈ S. The set G is open. Since the probabilitymeasure PH

T weakly converges to the measure Qζ as T → ∞, applying theproperties of weak convergence and of the support, it is obtained that

lim infT→∞

νT

(sups∈K

|ζ(s+ iτ)− f(s)| < ε

)> Qζ(G) > 0.

Now let f(s) be the same as in the statement of Theorem 17. Then byLemma 3 there exists a sequence of polynomials pn(s) such that pn → f(s)uniformly on K as n → ∞. Since f(s) = 0 on K, pn0(s) = 0 on K for asuciently large n0, and

sups∈K

|f(s)− pn0(s)| <ε

4. (8)

It is well-known that the polynomial pn0(s) has only nitely many zeros.Thus, it can be choose the region G whose complement is connected suchthat K ⊂ G and pn0 = 0 on G. Then there a continuous version of log pn0(s)exists. Clearly, log pn0(s) is analytic in the interior of G. Again applyingMergelyan theorem, it is nd a sequence of polynomials qn(s) such thatqn(s) → log pn0(s) uniformly on K as n → ∞. Consequently, for sucientlylarge n1,

sups∈K

∣∣pn0 − eqn1 (s)∣∣ < ε

4.

The last inequality together with (8) yields

sups∈K

|f(s)− eqn1 (s)| < ε

2. (9)

But eqn1 (s) = 0 for all s. Therefore, in view of the rst part of this proof,

lim infT→∞

νT

(sups∈K

|ζ(s+ iτ)− eqn1 (s)| < ε

2

)> 0.

From the inequality (9), it follows

lim infT→∞

νT

(sups∈K

|ζ(s+ iτ)− f(s)| < ε

)> 0.

This completes the proof of the theorem.

Obviously, Theorem 17 is stronger than Theorem 16.

36 R. Ka£inskaite

12. Universality of the Matsumoto zeta-function

In 1998, the weighted universality in the Voronin sense for the Matsumotozeta-function was proved by A. Laurin£ikas [31]. Suppose that the functionφ(s) is analytic in the strip D3 = s ∈ C : ρ0 < σ < α+β+1, α+β+ 1

2 6ρ0 < α+ β + 1. Assume that

M(m) :=

∣∣∣∣∣∣∣∣g(m)∑j=1

f(j,m)=1

a(j)m

∣∣∣∣∣∣∣∣1

pα+β> c > 0 (10)

for all m > 1.

Theorem 18 ([31]). Let the conditions of Theorem 14 and (10) be satised.

Let K be a compact subset of the strip D3 with connected complement. Let

f(s) be a non-vanishing continuous function on K which is analytic in the

interior of K. Then, for every ε > 0,

lim infT→∞

1

U

T∫T0

w(τ)Iτ : sups∈K

|φ(s+iτ)−f(s)|<εdτ > 0.

Discrete universality theorem for the Matsumoto zeta-function φ(s) wasobtained by the author in [22].

Theorem 19 ([22]). Let h > 0 be a xed number, and exp2πkh

be an

irrational number for all integers k = 0. Suppose that the conditions (4)(7)are satised, and M(m), K, f(s) are the same as in Theorem 18. Then, forevery ε > 0,

lim infN→∞

1

N + 1#

0 6 m 6 N : sup

s∈K|φ(s+ imh)− f(s)| < ε

> 0.

Acknowledgement. I would like to thank Professor Kohji Matsumoto forthe invitation to Graduate School of Mathematics of Nagoya University andhospitality. Also, I thank all participants of the Seminar on Analytic NumberTheory at 27th of October, 2011, for their interest on my lecture.

Limit theorems for zeta-functions... 37

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Received

10 February 2012