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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 19. Выпуск 3.
УДК 511.3 DOI 10.22405/2226-8383-2018-19-3-219-230
О совместном распределении значений дзета-функций Гурвица1
Францкевич Виолета — докторант, Институт математики, факультет математики и информатики, Вильнюсский университет. e-mail: [email protected]
Лауринчикас Антанас — доктор физико-математических наук, профессор, Действительный член Академии наук Литвы, ведущий научный сотрудник, Институт математики, факультет математики и информатики, Вильнюсский университет. e-mail: [email protected]
Шяучюнас Дариус — доктор математических наук, профессор кафедры математики Шяуляйского университета, старший научный сотрудник, Исследовательский институт, Шяуляйский университет. e-mail: [email protected]
Аннотация
Хорошо известно, что некоторые дзета и L-функции универсальны в смысле Воронина, т.е., ими приближается широкий класс аналитических функций. Некоторые из этих функций также совместно универсальны. В этом случае, набор аналитических функций одновременно приближается набором дзета-функций. В статье рассматривается проблема, связанная со совместной универсальностью дзета-функций Гурвица. Известно, что дзета-функции Гурвица С(s,«i),...,C(s,ar) совместно универсальны, если параметры а\,..., аг алгебраически независимы над полем рациональных чисел Q, или в более общем случае, если множество {log(rn + aj) : т g No, j = 1,... ,г} линейно независимо над Q. Мы рассматриваем случай произвольных параметров а1,... ,аг и получаем, что существует непустое замкнутое множество функций Fai..,ar пространства Hr(D) аналитических в полосе D = {s g C : 1 < а < 1} такое, что для любых компактных множеств Ki,...,Kr с D, функций (fi,...,fr) G Fai,...,ar и всякого £ > 0 множество |r g R : supseKi |£(s + ir, aj) — fj(s)| < e| имеет положительную нижнюю плот-
ность. Также рассматривается случай положительной плотности этого множества.
Ключевые слова: вероятностная мера, дзета-функция Гурвица, пространство аналитических функций, слабая сходимость, универсальность.
Библиография: 14 названий. Для цитирования:
В. Францкевич, А. Лауринчикас, Д. Шяучюнас. О совместном распределении значений дзета-функций Гурвица // Чебышевский сборник, 2018, т. 19, вып. 3, с. 219-230.
1 Исследование второго автора финансируется Европейским Социальным фондом по направлению "Повышение квалификации исследователей путем внедрения научно-исследовательских проектов мирового уровня" n0. 09.3.3-ЬМТ-К-712-01-0037.
CHEBYSHEVSKII SBORNIK Vol. 19. No. 3.
UDC 511.3
DOI 10.22405/2226-8383-2018-19-3-219-230
On joint value distribution of Hurwitz zeta-functions2
Franckevic Violeta — doctoral student, Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University. e-mail: [email protected]
Laurincikas Antanas — Full member of the AS in Lithuania, doctor of physical and mathematical sciences, professor, chief researcher, Institute of Mathematics, Faculty of Mathematics and Informatics, Vilnius University. e-mail: [email protected]
Siauciunas Darius — doctor of mathematical sciences, professor of the department of mathematics of the Siualiai University, Research Institute, Siauliai University. e-mail: [email protected]
It is well known that some zeta and L-functions are universal in the Voronin sense, i.e., they approximate a wide class of analytic functions. Also, some of them are jointly universal. In this case, a collection of analytic functions is simultaneously approximated by a collection of zeta-functions. In the paper, a problem related to joint universality of Hurwitz zeta-functions is discussed. It is known that the Hurwitz zeta-functions ((s,«i),...,C(s,ar) are jointly universal if the parameters a1 ,...,ar are algebraically independent over the field of rational numbers Q, or, more generally, if the set {log(rn + aj) : m g No, j = 1,...,r} is linearly independent over Q. We consider the case of arbitrary parameters a1,...,ar and obtain that there exists a non-empty closed set Fai,...,ar of the space Hr(D) of analytic functions on the strip D = {s g C : 1 < a < 1} such that, for every compact sets K1,... ,Kr c D,
fi,...,fr G Fai,...,ar and £ > 0, the set |r g R : sup^.^ supseKj |C (s + ir, a.j) - fj (s)| < e| has a positive lower density. Also, the case of positive density of the latter set is discussed.
Keywords: Hurwitz zeta-function, probability measure, space of analytic functions, universality, weak convergence.
Bibliography: 14 titles. For citation:
V. Franckevic, A. Laurincikas, D. Siauciunas, 2018, "On joint value distribution of Hurwitz zeta-functions" , Chebyshevskii sbornik, vol. 19, no. 3, pp. 219-230.
1. Introduction
The Hurwitz zeta-function ((s,a), s = a + it, with parameter a, 0 < a ^ 1, is defined, for a > 1, by the Dirichlet series
2 The research of the second author is funded by the European Social Fund according to the activity "Improvement of researchers qualification by implementing world-class R&D projects" of Measure No. 09.3.3-LMT-K-712-01-0037.
Abstract
Памяти Юрия Владимировича Линника посвящается
and has the analytic continuation to the whole complex plane, except for a simple pole at the point s = 1 with residue 1. For a = 1, the Hurwitz zeta-function reduces to the Riemann zeta-function
c(s) = y—, i,
m=l
and
c(s, 2) =(2S - 1) c(s).
Thus, ((s, a) is a generalization of the Riemann zeta-function. The function ((s) has the Euler product over primes
c (•) = I! (1 - £•
while the function ((s, a), except for the values a = 1 and a = 2, has no such a product. This fact reflects in value distribution differences of the functions ((s) and ((s,a). For example, it is well known that ((s) = 0, while the function ((s, a) has infinitely many zeros for all a = 1, 2 in the half plane a > 1. On the other hand, the functions ((s) and ((s, a;) for some closes of the parameter a have a common property of the approximation of a wide class of analytic functions. This interesting property is called universality, and for the function ((s) was obtained by S.M. Voronin [12]. For modern statements of universality theorems it is convenient to use the following notation. Let D = {s £ C : 1 < a < 1}. Denote by K the class of compact subsets of the strip D with connected complements, and by H0(K) with K £ K the class of continuous non-vanishing functions on K that are analytic in the interior of K. Then the modern Voronin universality theorem, see, for example, [7], says that for every K £K, f £ H0(K) and e > 0,
lim inf ^meas < t £ [0, T] : sup (s + ir) - f (s) | < ^ > 0.
T^X 1 [ seK J
The later inequality shows that there are infinitely many shifts ((s + ir, a) approximating with accuracy e a function f (s) £ H0(K). Yuri Vladimirovich Linnik knew the Voronin theorem
and highly valued it. Moreover, Il'dar Abdulovich Ibragimov imformed the second author that Yu. V. Linnik had a conjecture that all Dirichlet series satisfying some natural growth conditions are universal in the Voronin sense. Now this conjecture is called the Linnik-Ibragimov conjecture (or problem), see, for example, [11].
The universality of the Hurwitz zeta-function differs slightly from that of the function ((s). Denote by H (K) with K £ K the class of continuous functions on K that are analytic in the interior of K. Thus, H0(K) C H(K) fo all K £ K. Then the following universality theorem for the function ((s,a) is known.
Theorem 1. Suppose that the parameter a is transcendental or rational = 1, Let K £ K and f (s) £ H(K). Then, for every e > 0;
lim inf ^mea^ t £ [0,T] : sup |((s + it, a) - f (s)| < ^ > 0. (1)
T^X 1 I seK J
The theorem in the case of rational a was already known to Voronin [14]. In a slightly different form, the theorem was obtained independentlv bv S.M. Gonek and B. Bagchi in their theses [5],
Unfortunately, the universality of ((s, a) with algebraic irrational parameter a is an open problem. This problem is closely connected to linear independence over the field of rational numbers Q of the set L(a) = |log(m + a) : m £ No = N U {0}}. Denote by H(D) the space of analytic functions on D endowed with the topology of uniform convergence on compacta. Then, in [2], the following result towards to universality problem of ((s, a) with algebraic irrational a was obtained.
Theorem 2. Suppose that the parameter a is algebraic irrational. Then there exists a closed non-empty set, Fa c H(D) such that, for every compact set, K c D, f (s) g Fa and e > 0; the inequality (1) is true.
Some of zeta-functions are also jointly universal. In this case, a collection of analytic functions are simultaneously approximated by a collection of zeta-functions. The first joint universality results belong to S.M. Voronin. In [13], he considered the joint functional independence of Dirichlet F-functions, and, for this, he applied their joint universality. It is clear, that in the case of joint universality, the approximating zeta-functions must be in some sense independent. For Hurwitz zeta-functions this independence in [10] was described by the algebraic independence over Q of the parameters a1,... ,ar. In [8], the algebraic independence was replaced by the linear independence over Q for the set
F(a1,..., ar) = {log(m + aj) : m g no, j = 1,... ,r}.
Thus, the following theorem is known [8].
Theorem 3. Suppose that the set F(a1,..., ar) is linearly independe nt over q. Fo r j = 1,... ,r, let Kj g k an d fj (s) g H (Kj). Then, for e very e > 0;
liminf ^meas < t g [0, T] : sup sup (s + it, aj) — fj(s)| < o > 0. T ^ j
The aim of this paper is to prove a joint generalization of Theorem 2, i.e., to prove a certain theorem on joint approximation by the functions ((s,a{),...,((s,ar) without using any independence condition.
Theorem 4. Suppose that the numbers a.j, 0 < ttj < 1, ttj = \, j = 1,...,r; are arbitrary. Then there exists a closed non-empty set, Fai..,ar c Hr(D) such that, for every compact sets K1,...,Kr c D, (fi,...,fr) g Fai...,ar and e> 0,
liminf ^meas ^ t G [0, T] : sup sup (s + it, a.j) — fj(s)| < e) > 0.
i | i^j^rseKj
Theorem 4 has the following modification.
Theorem 5. Suppose that the numbers aj, 0 < a.j < 1 a.j = j = l,...,r, are arbitrary. Then there exists a closed non-empty set, Fait.,ar c Hr(D) such that, for every compact sets Ki,..., Kr c D and (fi,..., fr) g Fai..,ar, the limit
1
T 1 L ' i^j^rseK.
lim — meas ^ t g [0, T] : sup sup (s + it, aj) — fj(s)| < ^ > 0
e > 0
For the proof of above theorems we will apply the probabilistic approach. This is influenced in a certain sense by Yu. V. Linnik who was an expert not only in number theory but also in probability theory and mathematical statistics.
2. Auxiliary results
In this section, we will prove a joint limit theorem for the functions ((s, a1),..., ((s, ar) in the space of analytic functions. Denote by B(X) the Borel a-field of the space X, and, for A c B(Hr (D)), define
Pr,a(A) = ^meas {t e [0, T] : ((s + it, a) e A] , where a = (a1,..., ar) and
C(s,a) = (C(s,ai),...,((s,ar)).
Theorem 6. Suppose that the numbers a.j, 0 < ttj < 1, ttj = \, j = l,...,r, are arbitrary. Then, on (Hr(D), B(Hr(D))), there exists a probability measure P„ such that PT,a converges weakly to Pa as T ^ to.
We divide the proof of Theorem 6 into lemmas.
Denote by 7 the unit circle on the complex plane, and define the set
q = n ,
meNo
where 7m = 7 fa all m e No- By the classical Tikhonov theorem, the infinite-dimensional torus Q with the product topology and pointwise multiplication is a compact topological Abelian group. Define one more set
r
Q = n Q,
3=1
where Q = Q fa all j = 1,..., r. Then again by the Tikhonov theorem, Qr is a compact topological Abelian group. Denote by w = (w1,...,), w1 e Q1,... ,wr e Qr, the elements of Qr Moreover, let (¿j(m) be the m-th component of the element (¿j e Q, j = 1,... ,r, m e N0. For A e B(Qr), define
QtAa) = ^meas {t e [0, T] : (((m + 0:1)^ : m e No) ,..., ((m + ar)-lT : m e No)) e A} .
Lemma 1. On (Qr, B(Qr)), there exists a probability measure Qa such th at Qr,a converges weakly to Q^ as T ^ to.
Proof. We apply the Fourier transform method. The dual group of Qr is isomorphic to
r
Q = ^^ ^¡^ Zmj,
3=1 mei0
where Zmj = Z fa all j = 1,... ,r,m e N0. The element k = (kmj : kmj e Z, j = 1,... ,r, m e N0) e Q, where only a finite number of integers kmj are distinct from zero, acts on Qr by
w ^ = n n (m).
3=1 mei0
Therefore, the Fourier transform §t©of Qr,a is of the form
9T (k)= (nil"?™ (m)| dQT^,
/Qr \j=1 meNo
where the sign " '" shows that only a finite number of integers kmj are distinct from zero. Thus, by the definition of Qr,a,
1 t'T r ' 1 t'T
от(Ю = J П П (ffl + а1 )-гТкт]dr = 1 exP i-TE E ^ lQg(m + ) } dr• (2)
Define two collections of integers
{к'} = \kmj : ^ ^ log( m + a^) = 0
j=1 m&N0
and
{k''} = <kmj : ^ ^ kmj log(m + «j) = 0 > .
[ i=1 m€No J
Obviously, in view of (2),
* 9t (k) = 1 (3)
for k G {k'}- If k G {k''}) then integrating in (2), we find that
7j=iL meN0 log(m + ai
1 - exp { — tyj $=I £ ' meNo kmj log(m + a) ) 9T (k) = - 1 ' J
iT Ei=l E rneNo log(m + ai)
This and (3) show that
& - «Mi* k G{k»}.
The right-hand side of the later equality is continuous in the discrete topology. Therefore, by a continuity theorem for probability measures on compact groups, we obtain that Qt,u, as T ^ converges weakly to a probability measure Q« on (Qr, B(Qr)) defined by the Fourier transform
9(h) =
{
1 if к g M, 0 if к g {k"}.
The lemma is proved. □
Unfortunately, the limit measure Qa in Lemma 1 is given by its Fourier transform, we do not know the explicit form of Qa, and this reflects in Theorems 4 and 5 with non-effective set Fax,..,ar-For example, if the set L(a1,..., ar) is linearly independent over Q, then
9(M) = {
1 if k = 0,
0 if k = 0,
and we have that the limit measure Qa coincides with the Haar measure on (Qr, B(Qr)).
The next lemma is a joint limit theorem in the space Hr (D) for absolutely convergent Dirichlet series.
Let Go be a fixed number. For m G N0 and n G N, set
ao -
Vn(m>a)=exp{ } , j = 1>-,r,
n + a
and define the functions
vn(m, aj)
С (s a-Vn(m m a) i — 1 r
m=0
Proof. For Uj E Qj, define the functions
It is known [9] that the series for (n(s, dj) are absolutely convergent for a > For brevity, let
(n(s ,a) = ((n(s ,ai),..., (n(s, ar)),
and
PT,n,a(A) = Imeas {r E [0,T] : (Js + iT,a) e a} , A e B(Hr(D)).
Lemma 2. On ( Hr(D), B(Hr( D))); there exists a probability measure Png such that Pt<, converges weakly to Pn,g as T ^ to.
runctic
^ Uj(m)Vn(m,a.j) .
<n(s'Uj'dj) = £ m»+<m>• ■ , = 1--r-
Since luj( m)| = 1, the series for (n(s,Uj,aj) is also absolutely convergent for a > Let
(n(s ,u,a) = ((n(s ,ui,ai),..., (n(s ,dr)). Consider the function un,g : ^ Hr (D) given by the formula
un,a(u) = (n(s ,U,a).
In virtue of the absolute convergence of the series for (n(s, Uj ,aj), j = 1,... ,r, the function un is continuous. Moreover,
Un,a{((m + ai)-lT :m e N0) ,..., ((m + a)-%T : m E No)) = (n(s + it, a).
Therefore, for every A E B(Hr(D)),
PT,n,a(A) =
= ^meas {t e [0,T] : {((m + ai)-ir : m E No) ,..., ((m + dr)-iT : m E No)} E u^A} =
T
= QT,a(un,gA).
Hence, PT,n,a = QT,gUnia - Therefore, Theorem 5.1 of [3], Lemma 1 and the continuity of the function un,g imply that PT,n,g converges weakly to the measure Pn,g = Qgu-g as T ^ to, where Qg is the limit measure in Lemma 1. □
The next step of the proof of Theorem 6 consists of the approximation of ((s,a) by (n(s,a). For this, we recall the metric in the space Hr(D). It is known, see, for example, [4], that there exists a sequence of compact sets [Ki : I E N}cD such that
D = Qk ,
z=i
K C Ki+i{oy al He N, and, for every compact set K C D, there exists Ki such that K CK\. Let, for gi,g2 E H(D),
( ) = ^ i suPseKt 19i(s) -g2(s)| P(9i,92) = ¿^ 1 + sup^, Ms)-g2(s)l.
Then p is a metric in the space H(D) inducing the topology of uniform convergence on compacta. Now, setting, for gi = (gn,.. .,g^), g2 = (g2i,...,§2r) E Hr(D),
P(lv l2) = 1maxr P(9!j, 92j)
gives a metric in the space Hr (D) inducing its product topology.
Lemma 3. The equality
lim limsup1 I p(((s + ir,a),( (s + it, a}) dr = 0 n^ T ^ 1 J0 -n J
holds.
proof. The proof of the lemma does not depend on the arithmetic of the numbers a\,...,ar, and can be found in [8], Lemma 7. □
Now, we consider the sequence [Pn,a ■ n £ N}, where Pn,c is the limit measure in Lemma 2.
Lemma 4. The sequence Pn,c is tight, i.e., for every e > 0; there exists a compact set, K = K£ C Hr (D) such that '
Pn,a(K) > 1 - £
for all n £ N.
Proof. For an arbitrary a, 0 < a < 1, define
Pt,uAA) = fme&s {t £ [0, T] ■ (n(s + ir,a) £ A} , A £ B(H(D)),
and denote by Pn,a the limit measure of PTin,a as T ^ <x>. Then, in [2], it was obtained that the sequences {Pn,a ■ n £ N} is tight. Hence, the sequences
{Pn,aj ■ n £ N}, j = 1,...,r,
are tight. Clearly, Pn,aj ^re the marginal measures of the measure Pn,a, i.e.,
I \
P'n,aj — Pn
H(D) x • • • x H(D) xA x H(D) x • • • x H(D)
, A £ B(H(D)), (4)
V
j-1
/
j = 1,..., r. the sequence {Pn,aj} is tight, for every e > 0, there exists a compact set
Kj = Kj (e) C H (D) such to
Pn,aj (Kj) > 1 - -, j = 1,
(5)
for all n £ N. We p ut K = K\ x^ • •x Kr. Then the set K is compact in the space Hr (D). Moreover, in view of (4) and (5),
/
Pn,a(Hr (D) \ K )= P,
•nc^
(
r
u
3 = 1
\
H (D)x-^xH (D)
\
<
^ ^ Pn,a 3 = 1
x (H(D) \ Kj) x H(D) x • • • x H(D)
V i-1 J J
I \
H(D) x • • • x H(D) x(H(D) \ Kj) x H(D) x-^x H(D)
V
i-1
/
^ Pn,aj (H(D) \ K3) < ^
3 = 1
3 = 1
£
- = £ r
for all n £ N. Therefore,
Pn,a(K) > 1 - £
r
for all n E N The lemma is proved. □
PROOF.[Proof of Theorem 6] We will use the language of convergence in distribution (—>). Let the random variable d be defined on a certain probability space with measure and be uniformly distributed on [0,1]. Define the Hr(D)-valued random element by the formula
XT,n,a = XT,n,a(s) = (n(s + i 0T,a).
Moreover, let Xn,— = Xn,—(s) be the Hr(-Devalued random element having the distribution Pn,—. Then the assertion of Lemma 2 can be written in the form
XT,n,a ———> Xn—. (6)
Since the sequence {Pn,— : n E N} is tight, by the Prokhorov theorem ([3, Theorem 6.1]), it is relatively compact. Therefore, there is a subsequence [Pnk,—} C {Pn,—} such that Pnk,— converges weakly to a certain probability measure P— on (Hr(D), B(Hr(D))) as k ^ <x>. In other words, we have the relation
Xnk,— --> P—. (7)
k^x
Define one more Hr (D)-valued random element XT,— by the formula
Xt,— = Xt,—(S) = C(s + i 0T,a).
Then, the application of Lemma 3 shows that, for every e > 0,
lim lim sup ^ {p (Xt,— , XT,n,—) > 1
= lim lim sup — meas {tE [0,T]:pf ((s + ir,a),( (s + iT,a)} ^ e\
n^x T ^x T I -n ) J
^ lim lim sup— I p( ((s + i r,a),( (s + i r,a^dr = 0.
n^x T^x eT Jo ~n J
The latter equality together with relations (6) and (7) shows that all hypotheses of Theorem 4.2 of [3] are satisfied. Therefore, we obtain the relation
xt— —P—,
T^x -
which is equivalent to the weak convergence of PT,— to P— as T ^ The theorem is pr oved. □
3. Proof of Theorems 4 and 5
Theorems 4 and 5 follow easily from Theorem 6. For this, the notion of the support of a probability measure is applied. Denote by F—1,...,—r the support of the limit measure P— in Theorem 6. We remind that F—1 t...,—r C Hr(D) is a minimal closed set such that P-(F-1,...,-r) = 1. The set F—1,...,—r consists of all elements g E Hr(D) such that, for every open neighborhood G of g, the inequality P—(G) > 0 is satisfied.
Also, we will use two equivalents of the weak convergence of probability measures. We recall that a set A is a continuity set of the probability measure P if P(dA) = 0, where dA is the boundary of the set A.
Lemma 5. Let Pn, n E N, and P be the probability measures on (X, B(X)). Then the following statements are equivalent:
1° Pn converges weakly to P as n ^ to; 2° For every open set G C X;
liminf Pn(G) ^ P(G);
n—
3° For every continuity set A of the measure P,
lim Pn(A) = P(A).
n—TO
The lemma is Theorem 2.1 of [3].
PROOF.[Proof of Theorem 4] Suppose that Fai,...,ar is the support of the measure Pa- Then Fai,..,ar is non-empty closed set of the space Hr(D).
Let (fi,..., fr) G Fai,...,ar, Ki,..., Kr are compact sets of the strip D and e > 0. Define
G£ = ^ (gi, ...,gr) G Hr (D) : sup sup Igj (s) - fj (s)| < ^ . [ i^j^rse Kj J
Then the set G£ is an open neighborhood of the element (fi,..., fr) which belongs to the support of the measure Pa- Therefore,
Pa(G£) > 0. (8)
1° 2°
liminf PT,a(G£) > Pa(Ge). T
This, the definitions of PT,a and G£, and (7) show that
1
_________—:______ _ L„, _ j .
i \ Kj^r seKj
liminf-meas<| t g [0, T] : sup sup (s + it, aj) — fj(s)| < O > 0.
□
PROOF. [Proof of Theorem 5] We use the same notation as in the proof of Theorem 4. We observe that the boundaries dG£l and dG£2 do not intersect for different positive ei and e2- Therefore, Pa(G£) > 0 for at most count ably many e > 0. This shows that that the set G£ is a continuity set
of the measure Pa for all but at most countablv many e > 0. Therefore, using Theorem 6, 1° and 3°
lim PT,a(G£) = Pa(G£) > 0
T—TO _
exists for all but at most countablv many e > 0. Thus, the definitions of PT,a and G£ prove the □
4. Conclusions
The Hurwitz zeta-function ((s, a) depends on the parameter a whose arithmetic properties influence the analytic behavior of ((s,a), including the universality. The universality problem is related to the linear independence over Q of the set
F(a) = [log(m + a) : m G No}.
If the parameter a is algebraic irrational, them we have not much information on the set F(a), it is only known by the Cassels theorem that at least 51 percent of elements F(a) in the sense of density
are linearly independent over Q. However, there is not any idea how to use the Cassels theorem for the proof of universality.
A similar situation arises in the investigation of the joint universality for Hurwitz zeta-functions. The linear independence of the set
L(a\,..., ar) = {log(m + aj) : m £ No, j = 1,..., r}
leads to joint universality for the functions ((s ,a),..., ((s ,ar). In the paper, we search a way how to avoid involving of the set L(a\,... ,ar). Without using any information about the set L(ai,... ,ar), we prove that there exists a closed non-empty set of analytic functions such that the collections of those functions can be approximated bv shifts ( ((s + iт, a),..., ((s + iT, ar))• It remains a very difficult problem to describe the mentioned set of analytic functions.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
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2. Бальчюнас А., Дубицкас А., Лауринчикас А. О дзета-функции Гурвица с алгебраическим иррациональным параметром // Матем. заметки. 2019. Т. 105, №2. С. 179-186.
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7. Laurincikas A. Limit Theorems for the Riemann Zeta-Function. Dordrecht, Boston, London: Kluwer Academic Publishers, 1996.
8. Laurincikas A. On the joint universality of Hurwitz zeta-functions // Siauliai Math. Semin. 2008. V. 3(11). P. 169-187.
9. Laurincikas A., Garunkstis R. The Lerch Zeta-Function. Dordrecht, Boston, London: Kluwer Academic Publishers, 2002.
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REFERENCES
1. Bagchi, B. 1981, The statistical behavior and universality properties of the Riem,ann zeta-function and other allied Dirichlet series, Ph. D. Thesis, Indian Statistical Institute, Calcutta.
2. Balciunas, A., Dubickas, A., Laurincikas, A. 2019, "On the Hurwitz zeta-function with algebraic irrational parameter", Mat. Zametki, vol. 105, No 2, pp. 179-186. (in Russian). = Math. Notes, vol. 105, No 2, pp. 173-179.
3. Billingslev, P. 1968, Convergence of Probability Measures, Wiley, New York.
4. Conway, J. B. 1978, Functions of one complex variable., Springer, Berlin, Heidelberg, New York.
5. Gonek, S. M. 1979, Analytic properties of zeta and F-functions, Thesis, University of Michigan, Ann Arbor.
6. Karatsuba, A. A., Voronin, S.M. 1992, The Riemann zeta-function, Walter de Gruvter, Berlin.
7. Laurincikas, A. 1996, Fim.it Theorems for the Riem,ann Zeta-Function, Kluwer Academic Publishers, Dordrecht, Boston, London.
8. Laurincikas, A. 2008, "On the joint universality of Hurwitz zeta-functions", Siauliai Math. Semin., vol. 3(11), pp. 169-187.
9. Laurincikas, A., Garunkstis R. 2002, The Ferch Zeta-Function, Kluwer Academic Publishers, Dordrecht, Boston, London.
10. Nakamura, T. 2007, "The existence and the non-existence of joint ¿-universality for Lerch zeta-functions" // J. Number Theory, vol. 125, pp. 424-441.
11. Steuding, J. 2007, Value-Distribution of F-Functions, Lecture Notes Math. vol. 1877, Springer, Berlin, Heidelberg, New York.
12. Voronin, S.M. 1975, "Theorem on the "universality" of the Riemann zeta-function", Izv. Akad. Nauk SSSR., vol. 39, pp. 475-486 (in Russian) = Math. USSR Izv., vol. 9, pp.443-453.
13. Voronin, S.M. 1975, "On the functional independence of Dirichlet L-functions", Acta Arith., vol. 27, pp. 493-503 (in Russian).
14. Voronin, S.M. 1977, Analytic properties of Dirichlet generating functions of arithmetic objects, doct. diss., MIAS, Moscow (in Russian).
Получено 21.08.2018 Принято к печати 10.10.2018
