Extention of the Laurincikas - Matsumoto theorem Текст научной статьи по специальности «Математика»

ЧЕБЫШЕВСКИЙ СБОРНИК

Том 20. Выпуск 1.

УДК 511 DOI 10.22405/2226-8383-2019-20-1-82-93

Расширение теоремы Лауринчикаса — Матсумото

А. Вайгините

Вайгините Адель — докторант кафедры теории вероятностей и теории чисел, Вильнюсский

университет, Литва.

e-mail: [email protected]

Аннотация

В 1975 г. С.М. Воронин открыл замечательное свойство универсальности дзета функции Римана С(s). Он показал, что широкого класса аналитические функции могут быть приближены с желаемой точностью сдвигами £(s + гт), т G R, одной и той же функции £(s). Открытие Воронина вдохновило продолжить исследования в этом направлении. Оказалось, что универсальность является свойством многих других дзета и L-функций, а также некоторых классов рядов Дирихле. Среди них L-функции Дирихле, дзета функции Дедекинда, Гурвица и Лерха. В 2001 г. А. Лауринчикас и К. Матсумото получили универсальность дзета-функций £(s, F), связанных с некоторыми параболическими формами F. В статье получено расширение теоремы Лауринчикаса-Матсумото с использованием для приближения аналитических функций сдвигов £(s + i^>(r), F). Здесь tp(т) - дифференцируемая функция, при т ^ то, имеющая непрерывную монотонную положительную производную (р'(т), удовлетворяющую при т ^ то оценкам у,(т) = о(т) и <^>(2т) maxT^t<2r ^ т. Более точно, в статье док^ано, что е ели к - вес параболической формы F,K компактное множество полосы {s G C : < а < , обладающее связным дополнением, и f (s) -непрерывная, неимеющая пулей в К и аналитическая внутри К функция, то для всякого £ > 0 множество [т G R : supse_^ (s + ), F) — f (s)| < e} имеет положительную нижнюю плотность.

Ключевые слова: дзета-функция параболической формы, параболическая форма Гекке, универсальность.

Библиография: 15 названий. Для цитирования:

А. Вайгините. Расширение теоремы Лауринчикаса — Матсумото // Чебышевский сборник, 2019, т. 20, вып. 1, с. 82-93.

CHEBYSHEVSKII SBORNIK Vol. 20. No. 1.

UDC 511 DOI 10.22405/2226-8383-2019-20-1-82-93

Extention of the LaurinCikas — Matsumoto theorem

A. Vaiginyte

Vaiginyte Adele — doctoral student of the Department of Probability Theory and Number Theory, Vilnius University, Lithuania. e-mail: [email protected]

Abstract

In 1975, S. M. Voronin discovered the remarkable universality property of the Riemann zeta-function ((s). He proved that analytic functions from a wide class can be approximated with a given accuracy by shifts ((s + ir), t g R, of one and the same function ((s). The Voronin discovery inspired to continue investigations in the field. It turned out that some other zeta and L—functions as well as certain classes of Dirichlet series are universal in the Voronin sense. Among them, Dirichlet L-functions, Dedekind, Hurwitz and Lerch zeta-functions. In 2001, A. Laurincikas and K. Matsumoto obtained the universality of zeta-functions ((s,F) attached to certain cusp forms F. In the paper, the extention of the Laurincikas-Matsumoto theorem is given by using the shifts ((s + i<p(t), F) for the approximation of analytic functions. Here f(r) is a differentiate real-valued positive increasing function, having, for t > t0, the monotonic continuous positive derivative, satisfying, for t ^ <x>, the traditions ^fy = o(t) and y(2r) maxT^¿y C t. More precisely, in the paper it is proved that, if k is the weight of the cusp form F, K is the compact subset of th e strip {s G C : | < a < -^j1} with connected complement, and f(s) is a continuous non-vanishing function on K which is analytic in the interior of K, then , for every e > 0, the set [t G R : supseK |Q(s + i<p(r), F) — /(s)| < e} has a positive lower density.

Keywords: zeta-function of cusp forms, Hecke-eigen cusp form, universality.

Bibliography: 15 titles.

For citation:

A. Vaiginvte, 2019, "Extention of the Laurincikas — Matsumoto theorem" , Chebyshevskii sbornik, vol. 20, no. 1, pp. 82-93.

Dedicated to Professor Antanas LAURINCIKAS on the occasion of his 70th birthday

1. Introduction

In 1975 the remarkable property of universality was discovered by Voronin [25]. By analyzing the Riemann zeta-function, he noticed that with certain shifts of one and the same function a whole class of analytic functions can be approximated. This fact inspired further research of functions with similar properties and became a subject of interest for number theory specialists, among them Reich [14], Gonek [4], Good [6], Bagchi [1], Laurincikas [8], [9] and others. The aim of this paper is certain extended results on the universality for zeta-functions attached to certain cusp forms. Denote by s = a + it a complex variable. Let

SL(2, Z) := = ^ ^ :a,b,c,d G Z,ad — be = 1 j

be the full modular group. We say that the function F(z), z G C, is a holomorphic cusp form of weight k for SL(2, Z) if it is holomorphic for Im(z) > 0, for all 7 G SL(2, Z) satisfies the functional equation

F{Wbi)=(a+*)"F (i>

and at infinity has the Fourier series expansion

F(z) = ^ c(m)e2

Jl-Kimz

c(m )(

m= 1

We assume additionally that F(z) is an eigen form of all Hecke operators

TmF(Z) = m*-1 £ 1 £ F , m € N.

a,d>0 b (mod d) V /

ad=m

Then c(m) = 0, and, therefore, F(z) can be normalized to have the Fourier coefficient c(l) = 1.

Having all the aforementioned assumptions, the zeta-function ((s, F) associated with the cusp form F(2) of weight k is defined, for a > , by absolutely convergent Dirichlet series

C(s,F) = V ^.

sv ' 7 ^ ms

m=1

It is proved [5] that ((s,F) is analytically continued to an entire function. Moreover, for a > the function ((s, F) has the Euler product expansion over primes, i.e.,

-1 / \\-i

2(1 - - (1 -m

where P is the set of all prime numbers, and a(p) and @(p) are complex conjugate numbers satisfying a(p)+P (p) = c(p).

The first result on the universality of ((s, F) was obtained by the Laurincikas and Matsumoto in 2001 [10]. For the formulation of the theorem, we need some notation.

Let D = Dp = {s € C : | < a < 1C = be the class of compact subsets in the strip D

with connected complements, and H0(K), K € K, stand for the class of continuous non-vanishing functions on K that are analytic in the interior of K. The Lebesgue measure of a measurable set A C R is denoted by measA. Then the Laurincikas-Matsumoto universality theorem for ((s, F) can be formulated as follows.

Theorem 1 ([10]). Suppose that K € K. and f(s) € H0(K). Then, for every e > 0, the following inequality

liminf ^meas It € [0,T] : sup |((s + it,F) - f(s)l < el > 0 T^ 1 I seK )

holds.

shifts can be considered. The aim of this paper is taking shifts for the universality theorem from a certain class of functions U(r0). We say that a function ^(t) € U(r0), r0 > 0, if the following conditions are satisfied:

1. <^(t) is a differentiate real-valued positive increasing function on [t0, to);

2. (p'(t) is monotonic, continuous, positive on [t0, to) satisfying ^p) = o(t), t ^ to;

3. ip(2r) maxr^^t ^¿y ^ to. Then the following result is true.

Theorem 2. Suppose that <p(t) € U (r0), K €K, f(s) € H0(K). Then, for ev ery e> 0,

liminf—-—meas < t € [t0,T] : sup | ((s + i<p(r),F) - f(s)l < el > 0. T^^ T - T0 [ sEK )

It is known [11], [12] that universality theorems can be stated in a slightly different form. Theorem 2 has the following modification which will be proved in the paper.

Theorem 3. Suppose that <p(t) G U(to), K gK,, f(s) G H0(K). Then the limit

lim meas jt G [to,T] : sup |((s + î<p(t), F) — f(s)l <e \ > 0

1

T^œT — to I L° seK

exists for all but at most countably many e > 0.

In the following section, some lemmas necessary for the proof of the above mentioned theorems will be introduced.

2. Auxiliary results

Denote by B(X) the Borel a-field of the space X, and by 7 the unit circle on the complex plane. Define

<xl

Q = n7P,

pe p

where 7P = 7 for all primes p G P. By the Tikhonov theorem, with product topology and pointwise multiplication, the infinite-dimensional torus Q is a compact topological Abelian group. Therefore, the probability Haar measure mg on (Q, B(Q)) can be defined, and so we have a probability space (Q, B(Q),mn) .Denote by the projection of an element w G Q to the coordinate space 7P, p G P, by H(D) the space of analytic functions on D endowed with the topology of uniform convergence on compacta, and on probability space (Q, B(Q),mn) define the H(D)-valued random element ((s,w,F) by the formula

« >,»,F ) = n(l — ^ )-1 (l — ^ )-1

peP

Denote by , the distribution of ((s,w,F), i.e.,

PCF(A) = mH[w G Q : ((s, w, F) G A}, A G B(H(D)), Proof of the universality theorem is based on the weak convergence, as T ^ <x>, for

Pt,f(A) = —^meas [t g [to, T] : ((s + i<p(r), F) G A} , A G B(H(D)). T — to

Theorem 4. Suppose that <p(t) g U(t0). Then Pt,f converges weakly to P^,f as T ^ m. Moreover, the support of is the set Sf = [g G H(D) : g(s) = 0 or g(s) = 0}.

We divide the proof of Theorem 4 into several lemmas. The first of them is a limit theorem on the torus Q. For the proof of this lemma, properties of the function (p(t) are needed. For A G B(Q), define

QT(A) = —1—measiT G [To, T] : (p: p g P) G A} . T — To l J

Lemma 1. Suppose that <p(t) G U(to). Then Qt converges weakly to the Haar measure mn as T ->• 00.

Доказательство. [Proof] For the proof, we will apply the Fourier transform method. Let дт(к), к = (kp : kp <E Z, p <E P), be the Fourier transform of QT, i.e.,

дт(k)= i ( (p))dQT,

Jn \ p€p I

where "'" means that only a finite number of kp are distinct from zero. Thus, from the definition of Qt , we have

9T (k) = ^ fT ( n' P-tkMT) ) dT = f exp \ -ip(r) ^'kp logp \ dr, (1)

1 T0 JT° \PeP J 1 T0 JT0 ( PeP )

Obviously,

9T (0) = 1. (2)

Since the set {logp : p € P} is linearly independent over the field of rational numbers Q, we have that

a := kp logp = 0

p€P

for all k = 0.

гТ rT ГТ

Clearly,

r f1

exp {-iap(r)}dr = / cos(ap(r))dr -i / sin(ap(r))dr. (3)

jto Jto Jto

Suppose that p(t) is decreasing. Then, ^ly is increasing, and therefore, by the mean value theorem,

[ cos(ap(r))dr = 1 [ ap (T) dr = —[ ap'(t) cos(ap(r))dr

Jto a J tO P (t) aP (I ) Jt

iто

i F

P (j) ap' (T)Jz

d sm(ap(r)) = o(r)

ap' (T)Jz

as T ^ to, where r0 ^ £ ^ T. The same is also true for the second integral in (3). Thus, by (3),

fT

/ exp {-iap(r)} dr = o(t), T ^ to. (4)

Jto

Similarly, if p(t) is increasing, then

fT 1

exp{-iap(r)}dr < . (5)

JTo Mp^

From (4) and (5) together with (1), we get that

lim 9T(k) = °

T ^^

whenever k = 0.. Therefore, in view of (2),

lim qt(k) = < 1 ^ k 0 v"7 [0 if k = 0.

The right-hand side of the latter equality is the Fourier transform of the Haar measure mn. Therefore, the lemma follows from the continuity theorem for probability measures on compact groups. □

Now, some absolutely convergent Dirichlet series will be analysed. Let d > 2 be a fixed number, and m,n £ N. We define series

,F) = £ C(m)V-(m)

and

т*

т=1

Л, ,, рч ^ с(т)ш(т)vn(m)

,ш,ь) = -m*-'

m=l

where

vn(m)=expi^ — ^| and ш(т) = ^ ш1(p), m e N.

pl \m pl+1\m

The latter series are absolutely convergent for a > f [10]. Define the function un,F ■ Q ^ H(D) bv the formula un,F(ш) = (n(s, ш, F). Due to absolute convergence of (n(s, ш, F), we have that the function un,F(ш) is continuous, hence (B(Q), B(H(D)))-measurable. Therefore, the Haar measure тн on (Q, B(Q)) induces the unique probability measure Pn,F on (H(D), B(H(D))) defined by

Pn,F(A) = тни-F(A) = тн(u-pA), A e B(H(D))). Lemma 2. Suppose that <(t) e U(To). Then

PT,n,F (A) ^—^meas [т e [tq,T] ■ Cn(s + i<p(r),F) e A} , A e B(H (D))),

1 - To

converges weakly to Pn,F as Т ^ ж.

Доказательство. [Proof] The lemma is derived by standard arguments from Lemma 1 and the continuity of the function un>F. □

Our aim is to prove that PT,F converges weakly to the limit measure PF of the measure Pn,F as n ^ ж. For the proof of Theorem 4, approximation in the mean of ((s, F) by (n(s, F) is used. Thus, the following estimate of the mean square is needed.

Lemma 3. Suppose that <р(т) e U(т0), and a, f < a < Щ, is fixed. Then, for all t e R,

f \((a + it + i<p(r),F)\2dT < Т(1 + \t\).

Jtо

Доказательство. [Proof] It is known that, for fixed a, f < a < Щ,

T

[ \((a + it,F)\2dt < Т. (6)

о

For X > т0, we get

i'2X i'2X i

\ a a + i t + i <р(т), F )\2(dT= -— \ C(a + it + i<p(r),F )\2d<p(T) Jx Jx < (V

IX Jx < (J)

1 r 2X / г Щ+<р(т) \

^ max . d / \((a + iu,F)\ du\

^—'0'(t)JX \Jo )

1 ( rM+V(T) \ 2X

max ,, N \C(a + iu, F)\ du\

x(T)\ Jo ) x

Consequently, by (6),

AA+v^t) 2X

/ |((a + iu,F)I du |i| + p(2X),

lo X

and thus,

f2X 1

I((a + it + ip(r),F)\2(1t < (\t\ + p(2X)) max —— JX X ^T^2Xp'(r)

^X + \i\ max —^ <X(1 + \i\). Xp(t)

Taking X = 2-k- 1T and summing over k = 0,1,... prove the lemma. □

Now, we can approximate ((s,F) by (n(s,F) in the mean. For g1, g2 € H(D), take

p(gi, g2) = ± 2- - °2(S)I

=1 1 + sup,eKl 1 gi(s) - g2(s)|! where [Ki : I £ N} С D is a sequence of compact subsets such that

D = Uk ,

1=1

Ki C Kl+1ioi all I € N, and if K C D is a compact subset, then K C K for some I € N. Then p H( D)

Lemma 4. Suppose that p(r) £ U(т0). Then

1 fT

lim lim sup —- p (((s + ip(r),F), (n(s + ip(r), F)) dr = 0.

п^ж т^ж T - To JT0

Доказательство. [Proof] Let в be from the definition of vn(m), and

L(s) = |Г (I) ns, n £ N,

where T(s ) denotes the Euler gamma-function. Then the function (n(s, F) has the representation [10]

1 rd+KX d7 K

Us ,F ) = — ((s + z,F)ln(z)-, a>-.

2iti Je-ioo z 2

K D.

we get

1 ''т

f sup (((s + ip(r),F), (n(s + ip(T),F)) dr

Jtо seK

< I IL(& + iu)^—1— I |C(<7 + it + iu + ip(T),F)Idr]du,

J ж \T - TO JTо J

as T ^ to, where a < 0 § < & < ^d t is bounded by a constant depending on K. Lemma 3 implies that with t € R, for § < a < ^r,

I \ ((a + it + iu + ip(r),F)\dT ^(t I \ ((a + it + iu + ip(r), F)\2dr) ^a,KT(1 + \u\). J to \ Jto /

Therefore,

1 rT гж

-- sup (C(s + i<(t),F), Cn(s + i<p(r),F))dr <a,K \In(a + iu)\(1 + \u\)du,

1 — To J то seK Joo

о

as Т ^ ж. Hence,

1 T

limlimsup —- sup (((s + i<p(r),F), (n(s + i<p(r),F))dr = 0.

п^ж T^ж Т — To JT0 seK

So, the lemma follows from the definition of the metric p. □

[0,1] and defined от a certain probability space with measure ц. Define the H(D)-valued random

XT, n, F

xt,U,f = xt,U,f (s) = (n(s + i<(m,F). Then the assertion of Lemma 2 can be written as

XT,n,F —Xn,F, (7)

D

where —> means the convergence in distribution, and Xn,F is the H(D)-valued random element with the distribution Pn,F .Here Pn,F is the same limit probability measure as in Lemma 2.

Now, we will prove that the family [Pn,F ■ n e N} is tight, i.e., for every e > 0, there exists a compact set К = К(e) С H(D) such that Pn,F(K) > 1 — e for all n e N. Let К С D be a compact set. Then, by the integral Cauchv formula,

sup \((s + i<p(t),F)\<< f \((z + i<p(T),F)\\dz\, seK о к Jlk

where Lk is a simple closed contour lying in D and enclosing the set K, and 5k is the distance of Lk from the set K. Hence,

f sup K(s + i<p(r),F)\dT < ^f \dz\ f K(Re(z) +Im(z) + i<p(r),F)\dT <к Т.

Jt0 seK OK Jlk J to

I To seK OK JLK jto

This with Lemma 4 shows that

1 rT

[ sup \(^n(s + i<p(r),F)\dT ^Ci < ж, (8)

Jto sEK,

sup lim sup . __^

neN t^^ J — to JT0 seKi

where [Ki : I £ N} is the sequence of compact subsets of D from the definition of metric p.

Now, let the e be an arbitrary positive number, and Mi = Mi(e) = Cl2le-1. Then, from (8), we have

1 <-t

sup lim sup ц< sup \XT,n,F (s)\ > e > ^ suplimsup —-— i sup \ (n(s + i<p(r), F)\dr ^ ^,

neN t^ж [seK; J neN t^ж Т — To JТо seKt 2

and, by (7),

1

ц<| sup \Xn,F(s)\ >e j> < 2 (9)

for all n e N. Define the set К = К(e) = {g e H(D) : supseKl \g(s)| ^ M, I e N} . Then К is a compact set in H(D), and, bv (9),

v{xn,F e К > 1 for all n e N, or, by definition of Xn,F,

Pn,F(К) > 1 — £

for all n e N, thus the family <^Pn,F : n e N| is tight. Therefore, by the Prokhorov theorem (see

Theorem 6.1 in [2]), it is relatively compact, i.e., every sequence of {Pn,F} contains a weakly convergent subsequence. Thus, there exists {Pnr,F} С {Pn,F} such that {Pnr,F} converges weakly to a certain probability measure PF on (H(D), B(H(D))) as r ^ <, or, in terms of convergence in distribution, we say

Xnr,F Pf (10)

H( D)

XT,F = XT,F (s) = (n(s + <(£ T ),F). Then, in view of Lemma 4, for every e > 0, lim limsup^ {p(XTF,XTnF\ ^ e}

= lim lim sup —-—meas {re [to,T] : p (((s + i<p(r),F), (n (s + i<p(r),F)) ^ e} t^X T - To

1 fT

< lim limsup—-- p(((s + i<р(т), F), (n(s + i<p(r),F))dr = 0.

T^x (T - To)£ JT0

This together with (7) and (10) shows that all hypotheses of Theorem 4.2 of [2] are fulfilled, therefore,

Xt f-^ Pf ,

T^<x>

or Pt,f converges weakly to the limit measure PF of Pn,F as T ^ <.

The final step is to identify the measure PF. For this, we will use a simple observation. It is known [3], [7] that

Imeas {т e [0, T] : ((s + it,F) e A} , A e B(H(D))),

as T ^ <, converges weakly to the limit measure PF of Pn,F, and that PF = P^F. Moreover, the support of P^,F is the set SF .Therefore, PT,F also converges weakly to P^,F as T ^ <. □

3. Proofs of universality theorems

Доказательство. [Proof of Theorem 2] Define the set

G£ = { geH (D):sup \g (s) - e^s)\ < £-\ , I seK 2)

where p(s) is a polynomial satisfying

sup

seK

f(s) - s) < -. (11)

( )

by polynomials (see [13]).

By the second part of Theorem 4, the function belongs to the support of the measure P^,f . Therefore,

PC,F(Ge) > 0. (12)

Since Ge is an open set, by the first part of Theorem 4 and the equivalent of weak convergence of probability measures in terms of open sets, we have that

lim^nfPT,F(G£) > P(,F(Ge).

This, the definition of PT,F and inequality (12) give

liminf-

T ^ж Т — T0

meas \t e [tq, Т] ■ sup \((s + i<p(j),F) — ep(s)

e K

< 2 > 0

□

Доказательство. [Proof of Theorem 3] Define the set

Ge = geH (D):sup \g (s) — f(s)\ <£

se K

Then the boundarv dGE oi Ge lies in the set

[geH (D):sup \ g(s) — f(s)\ = Л e K

Therefore, dGei n dGe2 = 0 for = £2, £1, £2 > 0. Hence, for at most countablv many e > 0, the sets dGe have a positive P^,F measure. Using Theorem 4 and equivalent of weak convergence of probability measures in terms of continuity sets, we obtain that

lim Pt,F (Ge) = PC,F (Ge)

T ^ж

(13)

> 0. Ge

of (11), we obtain that Ge C Ge, and thus, by (12), P^,F(Ge) > 0. This, the definition of PT,F and (13) prove the theorem. □

4. Conclusions

In the paper, a generalized version of the Laurincikas-Matsumoto universality theorem for zeta functions of certain cusp forms ((s,F) is proved in two different forms. Namely, it is shown that the shifts ((s + i<p(r),F), where <р(т) belongs to a certain class of differentiable functions U(r0) can approximate with a given accuracy all non-vanishing analytic functions defined in the strip {se C ■ I < a < , where к is the weight of the form F, and the lower density of the set of such shifts is positive.

СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ

1. Bagchi В. The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, PhD Thesis, //Calcutta, Indian statistical Institute. 1981.

2. Billingslev P. Convergence of Probability measures // Wiley, New York. 1968.

3. Dubickas A., Laurincikas A. Distribution modulo 1 and the discrete universality of the Riemann zeta-function // Abh. Math. Semin. Hambg. 2016. Vol. 86, P. 79-87.

4. Gonek S. M. Analytic properties of zeta and L-functions // Math. Z. 1982. Vol. 181, P. 319-334.

5. Hecke E. Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I // Math. Ann. 1937. Vol. 114, 1-28; II. ibid. P. 316 351.

6. Good A. On the distribution of values of Riemann's zeta-function // Act. Arith. 1981. Vol. 38, P. 347-388.

7. Laurincikas A. Limit theorems for the Riemann zeta-function // Kluwer, Dordrecht, Boston, London.1996.

8. Laurincikas A. The universality theorem // Lith. Math. J. 1983. Vol. 23, P. 283-289.

9. Laurincikas A. The universality theorem II // Lith. Math. J. 1984. Vol. 24, P. 143-149.

10. Laurincikas A., Matsumoto K. The universality of zeta-functions attached to certain cusp forms // Acta Arith. 2001. Vol. 98, P. 345-359.

11. Лауринчикас А., Мешка Л. Уточнение неравенства универсальности // Матем. заметки. 2014. Т. 96, С. 905—910,

12. Laurincikas A., Meska L. On the modification of the universality of the Hurwitz zeta-function // Nonlinear Anal. Model. Control. 2016. Vol. 21, P. 564-576.

13. Мергелян C.H. Равномерные приближения функций комплексного переменного // УМН. 1952. Т. 7, №2. С. 31-122.

14. Reich А. Zur Universalität und Hvpertranszendenz der Dedekindschen Zetafunktion // Abh. Braunschweig. Wiss. Ges. 1982. Vol. 33, P. 197-203.

15. Воронин C.M. Теорема об "универсальности" дзета-функции Римана // Изв. АН СССР. Сер. матем. 1975. Т. 39. С. 475-486.

REFERENCES

1. Bagchi, В. 1981, "The statistical behavior and universality properties of the Riemann zeta-function and other allied Dirichlet series", PhD Thesis, Calcutta, Indian statistical Institute.

2. Billingslev, P. 1968, "Convergence of Probability measures", Wiley, New York.

3. Dubickas, A., Laurincikas, A. 2016, "Distribution modulo 1 and the discrete universality of the Riemann zeta-function", Abh. Math. Semin. Hambg., vol. 86, pp. 79-87.

4. Gonek, S.M. 1982, "Analytic properties of zeta and L-functions", Math. Z., vol. 181, pp. 319— 334.

5. Hecke, E. 1937, "Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung I", Math. Ann., vol. 114, 1-28; II. ibid. pp. 316—351.

6. Good, A. 1981, "On the distribution of values of Riemann's zeta-function", Act. Arith., vol. 38, pp. 347-388.

7. Laurincikas, A. 1996, "Limit theorems for the Riemann zeta-function", Kluwer, Dordrecht, Boston, London.

8. Laurincikas, A. 1983, "The universality theorem", Lith. Math. ,J., vol. 23, pp. 283-289.

9. Laurincikas, A. 1984, "The universality theorem II", Lith. Math. J., vol. 24, pp. 143-149.

10. Laurincikas, A., Matsumoto, K. 2001, "The universality of zeta-functions attached to certain cusp forms", Acta Arith., vol. 98, pp. 345-359.

11. Laurincikas, A., Meska, L. 2014, "Sharpening of the universality inequality", Math. Notes, vol. 96, pp. 971—976,

12. Laurincikas, A., Meska, L. 2016, "On the modification of the universality of the Hurwitz zeta-function", Nonlinear Anal. Model. Control, vol. 21, pp. 56 I 576.

13. Mergelvan, S.N. 1952, "Uniform approximations of functions of a complex variable", Uspehi Matern. Nauk (N.S.), vol. 7, pp. 31-122.

14. Reich, A. 1982, "Zur Universalität und Hvpertranszendenz der Dedekindschen Zetafunktion", Abh. Braunschweig. Wiss. Ges., vol. 33, pp. 197-203.

15. Voronin, S. M. 1975, "Theorem on the "universality" of the Riemann zeta-function", Math. USSR Izv., vol. 9, pp. 443-453.

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