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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 19. Выпуск 1
УДК 511.3 DOI 10.22405/2226-8383-2018-19-1-124-137
Совместная дискретная универсальность дзета-функций Jlepxa
Антанас Лауринчикас — доктор физико-математических наук, профессор, Действительный член АН Литвы, заведующий кафедрой теории вероятностей и теории чисел Вильнюсского университета.
e-mail: [email protected]
Аста Минцевич — докторант кафедры теории вероятностей и теории чисел, Вильнюсский университет.
Аннотация
После 1975 г. работы Воронина известно, что некоторые дзета и L-функции универсальны в том смысле, что их сдвигами приближается широкий класс аналитических функций. Рассматриваются два типа сдвигов: непрерывный и дискретный.
В работе изучается универсальность дзета-функций Лерха L(X,a,s), s = а + it, которые в полуплоскости а > 1 определяются рядами Дирихле с членами е2пгХт(т + a)-s с фиксированными параметрами A € Ми а, 0 < а < 1,и мероморфно продолжаются на всю комплексную плоскость. Получены совместные дискретные теоремы универсальности для дзета-функций Лерха. Именно, набор аналитических функций fi(s),...,fr(s) одновременно приближаются сдвигами L(X\, а\, s + ikh),..., L(Xr, ar, s + ikh), к = 0,1, 2,..., где h > 0 - фиксированное число. При этом требуется линейная независимость над полем рациональных чисел множества {(log(m + ay) : т € No, j = 1,... ,r), Доказательство теорем универсальности использует вероятностные предельные теоремы о слабой сходимости вероятностных мер в пространстве аналитических функций.
Ключевые слова: дзета-функция Лерха, пространство аналитических функций, слабая сходимость, теорема Мергеляна, универсальность.
Библиография: 18 названий. Для цитирования:
А. Лауринчикас, А. Минцевич. Совместная дискретная универсальность дзета-функций Лерха // Чебышевский сборник. 2018. Т. 19, вып. 1, С. 138-151.
CHEBYSHEVSKII SBORNIK Vol. 19. No. 1
UDC 511.3 DOI 10.22405/2226-8383-2018-19-1-124-137
Joint discrete universality for Lerch zeta-functions1
Antanas Laurincikas — doctor of physics-mathematical sciences, professor, Member of the Academy of Sciences of Lithuania, Head of the chair of probability theory and number theory, Vilnius university.
e-mail: [email protected]
Asta Mincevic — doctoral student in the department of probability theory and number theory, Vilnius u niversitv.
Abstract
After Voronin's work of 1975, it is known that some of zeta and L-functions are universal in the sense that their shifts approximate a wide class of analytic functions. Two cases of shifts, continuous and discrete, are considered.
The present paper is devoted to the universality of Lerch zeta-functions L(X, a, s), s = a + it, which are defined, for a > 1, by the Dirichlet series with terms e2"Ara(to+a)-s with parameters A € R Mid a, 0 < a < 1, and by analytic continuation elsewhere. We obtain joint discrete universality theorems for Lerch zeta-functions. More precisely, a collection of analytic functions f1(s),... ,fr (s) simultaneously is approximated by shifts L(X1,a1, s+ikh),..., L(Xr ,ar, s+ikh), k = 0,1, 2,..., where h > 0 is a fixed number. For this, the linear independence over the field of rational numbers for the set {(log(TO + aj) : to € No, j = 1,... ,r), is required. For the proof, probabilistic limit theorems on the weak convergence of probability measures in the space of analytic function are applied.
Keywords: Lerch zeta-function, Mergelyan theorem, space of analytic functions, universality, weak convergence.
Bibliography: 18 titles. For citation:
A. Laurincikas, A. Mincevic, 2018, "Joint discrete universality for Lerch zeta-functions" , Chebyshevskii sbornik, vol. 19, no. 1, pp. 138-151.
1The research of the first 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.
Dedicated to the 100th birthday of Nikolai Mikhailovich Korobov
1. Introduction
In [18], see also [4], S.M. Voronin discovered the universality of the Riemann zeta-function ((s), s = a+it, that a wide class of analytic functions can be approximated bv shifts ((s+ir),r e R. After Voronin's work, various authors extended his universality theorem for some other zeta- and L-functions, and classes of Dirichlet series. One of universal zeta-functions is the Lerch zeta-function L(X, a, s) with parameters A e R and a, 0 < a ^ 1, which is defined, for a > 1, by the Dirichlet series
The function L(X, a, s) was introduced and studied independently bv R. Lipschitz [14] and M. Lerch [13]. The analytic properties of L(X, a, s) depend on the parameters A and a, and in particular case, this is true for the analytic continuation to the whole complex plane. If A e Z, then L(X, a, s) is an entire function, while, for A e Z, L(X, a, s) reduces to the Hurwitz zeta-function
which is analytically continued to the whole complex plane, except for a simple pole at the point s = 1 with residue 1. In virtue of the periodicity of e2mXm, it suffices to suppose that 0 < A ^ 1. The theory of the Lerch zeta-function is given in [7].
The first universality result for the function L(X, a, s) was obtained in [5]. Let
K be the class of compact subsets of the strip D with connected complements, and let H(K) with K e £ denote the class of continuous functions on K that are analytic in the interior of K. Let meas A denote the Lebesgue measure of a measurable set A c R. Then it was obtained in [5] that if a is transcendental, then for K eK, f (s) e H(K), 0 < A ^ 1 and every e > 0,
The case of rational a is more complicated. Some conditional result in this direction has been obtained in [7]. If both a and A are rational, then the function L(a,X,s) becomes the periodic Hurwitz zeta-function, and, for it, an universality theorem of type of [9] is true. In this case, a certain condition connecting a and A is involved.
The universality of L(a, X, s) with algebraic irrational a is an open problem. The case of a with linearly independent set L(a) = {log(m + a) : m e No = N U {0}} over the field of rational numbers Q can be viewed as a certain approximation to that problem, see [17] and [6].
For the function L(a, X, s), also a discrete universality when t in L(a, X, s + ir) takes values from a certain discrete set is considered. One of the simplest discrete sets is the arithmetic progression {kh : k e N0} with h > 0. Denote by # A the cardinality of the set A If a is transcendental and the number exp{^f} is rational, then it is known [3], [8] that, for K e K, f (s) e H(K), 0 < A ^ 1 and every e > 0,
liminf—1—# j0 < k < N : sup lL(X,a,s + ikh) - f(s)| <A> 0. NN + 1 [ seK )
Let, for h > 0
L(a,h,n) = |(log(m + a) : m e N0), ^ j .
Then, in [12], the transcendence of a and rationalitv of exp{} were replaced bv the linear independence over Q of the set L(a, h, n).
The aim of this paper is joint discrete universality theorems for Lerch zeta-functions. We note that the joint universality for Lerch zeta-functions is an interesting problem connecting algebraic properties of the parameters a\,...,ar and Ai,...,Ar with analytic properties of a collection L(X1,a1,s),..., L(Xr, ar ,s), therefore, there are many results of such a kind. The first joint universality theorem for Lerch zeta-functions was proved in [10], [11].
Theorem 1. Suppose that the parameters a1,..., ar are algebraically independent over Q; X1 = I1 ,...,Xr = (a1,q1) = I,..., (ar ,qr) = 1, are rational num bers, k is the least common multiple of q1,..., qr, and that the rank of the matrix
02kwi\i g2kwi\2 g2k'Ki\.
is equal to r. For j = 1,... ,r, let Kj g Km d fj g H (Kj ). Then, for e very e > 0 ^meas <J t g [0,'i '] :
l^j^r seK.
Let
liminf —meas < t g [0,T] : sup sup lL(Xj,(Xj,s + it) — fj(s)| < ^ > 0.
T—^oo J
L(a1,..., ar) = {(log(m + a1) : m e N0),..., (log(m + ar) : m e N0)} .
Then in [16], under the hypothesis that the set L(a1,..., ar) is linearly independent over Q, it was obtained that the inequality of Theorem 1 is true for all 0 < A ^ 1, j = 1,... ,r.
We will focus on joint discrete analogues of the above results. For h > 0, define the set
L(a1,... ,ar;h,^) = |(log(m + a1) : m e N0),..., (log(m + ar) : m e N0), .
Then we have
Theorem 2. Suppose that the set L(a1,...,ar; h, ■k) is linearly independen t over Q. For j = 1,... ,r, let Kj e K, fj e H(Kj) and 0 < Xj ^ 1. Then, for every e> 0,
lim iQf A7 , ^ k ^ N : sup sup |L(Aj, aj, s + ikh) - fj(s)| <e\ > 0.
1
nN + 1 " 1 " l^j^rseKj Theorem 2 has the following modification.
Theorem 3. Suppose that the set L(a\,...,ar ; h, ■k) is linearly independen t over Q. For 'j g K, fj g H (Kj ) and 0 < Xj
xr
j = 1,... ,r, let Kj G K, fj G H (Kj ) and 0 < Xj ^ 1. Then the limit
1
NN + 11 l^j^r seK.
lim A7 | 1 # { 0 ^ k ^ N : sup sup lL(Xj, aj ,s + ikh) - fj (s)| <e \ > 0
exists for all but at most countably many e > 0.
The proofs of Theorems 2 and 3 are based on statistical properties of Lerch zeta-functions, more precisely, on limit theorems of weakly convergent probability measures in the space of analytic functions.
2. Discrete limit theorems
Denote by B(X) the Borel a-field of the space X. We recall that D = {s e C : 2 < a < 1}. Denote by H (D) the space of analytic functions on D endowed with the topology of uniform convergence on compacta. In this section, we consider the weak convergence of probability measures defined on (H(D), B(H(D))).
We use the notation 7 = {s e C : |s| = 1}, and define
Q = H 7m,
m=0
where 7m = 7 for all m e N0. Then, by the famous Tikhonov theorem, the torus Q with the product topology and pointwise multiplication is a compact topological Abelian group. Putting
Qr = Qi x ■ ■ ■ x Qr,
where Qj = Q for j = 1,..., r, by the Tikhonov theorem again, we have that Qr is a compact topological Abelian group. Therefore, on (Qr, B(Qr)), the probability Haar measure mu can be defined. This gives the probability space (Qr, B(Qr),mn)• Denote by mjH the probability Haar measure on (QJ, B(QJ)), j = 1,... ,r. Then we have that
mu = miH x ■ ■ ■ x mrH.
Let Wj be the elements of Qj, j = 1,..., r, and w = (wi,..., denote the elements of Qr. Moreover, denote by (¿j (m) the projection of an element (¿j e Qj to the circle 7m, m e N0, j = 1,... ,r. Now, on the probability space (Qr,B(Qr),mn) define the Hr(^)-valued random element L(X,a,s,w), where X = (Xi,..., Xr^d a = (ai,..., ar), by
L(X,a, s, w) = (Li(Xi, ai, s, wi),..., Lr (Xr, ar, s, )),
where
r .. . ^ e2mX?(m) .
Lj (Xj, (Xj ) = , 2 = 1,...,r.
m=0 v J/
We note that the latter series are uniformly convergent on compact subsets of the strip D [7], thus, they define the H(^)-valued random elements.
Having the above definitions, we state a joint discrete limit theorem for Lerch zeta-functions.
Theorem 4. Suppose that the set L(ai,... ,ar; h, ■k) is linearly independe nt over Q. Then
PN(A) {0 < k < N : L(X,a,s + ikh) e A} , A e B(Hr(D)),
converges weakly to the distribution Pl of the random element L(X,a,s,u) as N ^ to.
We remind that, for A e B(Hr(D)),
Pl(A) = mH {w e Qr : L(X, a,s,u) e A} .
We divide the proof of Theorem 4 into lemmas. The first of them deals with the weak convergence of probability measures on (Qr, B(Qr)), and for that the linear independence of the set L(ai,... ,ar; h,n) is essentially applied. Let, for A eB(Qr),
Qn(A) = ^#{0 < k < N :((m + ai)-lkh : m e No),..., ((m + ar)-lkh : m e No)) e .
Lemma 1. Suppose that the set L(a\,... ,ar; h, k) is linearly independe nt over Q. Then Qn converges weakly to the Haar measure m^ as N ^
Proof.
We consider the Fourier transform of Qn- Since characters of the group Qr are of the form
n n ^ (m)
j=1 m=0
where only a finite number of integers kjm are distinct from zero, we have that the Fourier transform 9n(ki,.. .,kr) kj = (kjm : kjm G Z, m G N0), j = 1,... ,r, of QN is
„ r x N r x
9n (Ml,--, kr) = ja r n n (m)d QN = — Enn (m + <*i )-khk]m
j=1m=0 k=0 j=1m=0
N ( r ^
= NT1 E exp { -ikh E E' kim log(m + ai) \ , ^
+ k=0 y j=i m=0 J
where means that only a finite number of integers kjm are distinct from zero. Clearly,
9n (0,-..,0) = 1- (2)
Since the set L(ai,... ,ar; h, n) is linearly independent over Q,
r X
exp < - ih E E log(m + a.j) > = 1
[ j=i m=0 j
for (ki,... ,kr) = (0,...,0). Actually, if this inequality is not true, the
I
h E E kim log(m + ai) - = 0
j=i m=0
with I G Z, and this contradicts the linear independence of the set L(ai,... ,ar; h,n). Thus, in this case, we find by (1) that
1 - exp{-(N + 1)ih T!j=i E'X=0kjm log(m + aj)|
gN (k^ ...,kr) =-t-?-—-rv.
(N + 1) ^ - exp j-t hE j=iT, X=0 kjm log(m + a) j J
This and (2) show that
r 1 if (ki,...,kr ) = (0,..., 0), ^inx gN ^ ...,kr)^0 if ..., ^) = (0,..., 0).
Since the right-hand side of the latter equality is the Fourier transform of the Haar measure mn, the lemma is proved. □
Now, we will apply Lemma 1 to obtain a joint limit theorem in the space of analytic functions for functions given by absolutely convergent Dirichlet series connected to Lerch zeta-functions. Let a > 2 be a fixed number, and, for m G N0 and n G N,
i_ /m + ajV 1 1 + a ) j
mm + a '
v^ma) = ex^ _ ( n J ) }, j = i,...,r.
Define
and
where
and
L,n(X,a,s) = (Ln(Xi,ai,s),... ,Ln(Xr,ar,s)) Ln(X,a,s,u) = (Ln(Xi,ai,s,ui),.. .,L,n(Xr,ar,s,wr)),
Ln(X3,a3, 8) = ^ , J = 1,...,r,
m=0 ■>'
^ e27TiXimUj(m)Vn(m,aj) . Ln(X3, a.j ,S,U)=Y:-^-, ; = 1,...,r,
m=0 v ■>'
It is known [7] that the series for Ln(Xj ,a3, s) and Ln(Xj ,a3 ,s,Uj) are absolutely convergent for
® > 2-
The next lemma deals with weak convergence for
PN,n(A) = {0 < k ^N : Ln(X,a, s + ikh) e A} , A e B(Hr (D)).
Define the function un : Qr ^ Hr (D) by the formula
un(cv) = Ln(X, a, s,u), w e Q.
Since the series for Ln(Xj ,a3 ,s,Uj), j = 1,..., r, are absolutely convergent for a > 2, the function un is continuous, hence it is (B(Qr), B(Hr(_D)))-measurable. Therefore, the measure mu induces [1] on (Hr(D), B(Hr(D))) the unique probability measure Pn = mnu-1, where, for A e B(Hr(D)),
Pn(A) = mHu-i(A) = mH (u-1A).
Lemma 2. Suppose that the set L(ai,..., ar; h, ■k) is linearly independent over Q. Then PN,n converges weakly to Pn as N ^ to.
Proof.
Let Qn be defined in Lemma 1. Then the definitions of PN,n, Qn and un show that,for every A e B(Hr(D)),
1
+ 1
(( m + ar)-ikh : m e N0)) e u-1A} = QN(u-1A),
PN,n(A) = # {0 < k ^N : (((m + ai)-lkh : m e N0),...
i.e., Pn,ti = Qnu-1. This, Lemma 1, the continuity of un and Theorem 5.1 from [1] show that Pn,, converges weakly to the measure mnu-1 as N ^ to.
Now, we will approximate L(X,a, s) by Ln(X,a, s). For g1, g2 e H(D), let
( ) = ^ 1 suP 8eK 19i(s) - 92(s)| P(9192) = = 1 + SUpse^ igi(s) - g2(s),
where {K : I e N} is a sequence of compact subsets of the strip D such that
D = Qki ,
Kl C Ki+i for al lk N ^d if K C D is a compact subset, then K Cf^ome I. The proof of the existence of the sequence {Kl : I £ N} can be found, for example, in [2]. The metric p induces the topology of the space H(D) of uniform convergence on compacta. The metric p in Hr (D) inducing the product topology is defined by
p(lv l2) = ^P^l,'
where g1 = (gn,...gir), g2 = (^l,... 92r) £ Hr(D). □ Lemma 3. For all a and h> 0,
1 N
lim limsup —- > p (L(X,a, s + ikh), Ln(X, a, s + ikh)) = 0.
n^x NN + 1
Proof.
The definition of the metric p shows that the equality of the lemma follows from the equalities
1 N
lim lim sup ——-^p(Lj(Xj,aj,s + ikh),Ln(Xj,s + ikh)) =0, k=0
j = 1,... ,r, that were obtained in Lemma 3 of [12]. □ We recall that the measure Pn was defined in Lemma 2.
Lemma 4. Suppose that the set L(a1,..., ar; h,ft) is linearly independent over Q. Then the sequence {Pn : n £ N} is tight, i.e., for every e > 0, there exists a compact subset K = K( ) C H ( D)
Pn(K) > 1 - e
for all n £ N. Proof.
Consider the marginal measures of Pn, i.e., the measures
/ \
Pnj(A) - Pn
H (D) x ■■■ x H (D) xA x H (D) x ■ ■ ■ x H (D)
A £ B(H(D)),
\ i-i J
where j = 1,... ,r. The linear independence of the set L(a1,... ,ar; h, implies that for L(aj ,h, ft), j = 1,... ,r. Therefore, in view of the proof of Lemma 5 from [12], we have that Pnj converges weakly to the distribution P^. of the random element Lj(Xj,aj) as n ^ «, j = 1,...,r. Hence, the sequence {Pn,j : n £ N} is relatively compact, j = 1,...,r. Since the set H(D) is complete and separable, by the inverse Prokhorov Theorem [1, Theorem 6.2], the sequence {Pn,j : n £ N} is tight, j = 1,... ,r. Thus, for every e > 0 there exists a compact subset Kj C H (D)
Pn(Kj) > 1 - p j = 1'...'r' for all n £ N. The set K = K1 x ■ ■ ■ x Kr is compact in Hr(D). Moreover,
Pn(H (D) \K) = Pn (Jj i(H (D) \K3 ) ) < > J Pn,,(H (D) \ K3) < e
for all n £ N i.e.,the sequence {Pn : n £ N} is tight. □
For convenience, we recall one result from [1]. Suppose that ( S, ^)-valued random elements Yn, X1n, X2n,... are defined on the same probability space with measure P, and that the space S is separable.
v>) < É
7 i=i
Xkn-> Xk
n^ro
and
Xk X.
k^-ro
Moreover, for every e > 0; let
Then Yn X.
lim lim sup P{p(Xkn,Yn) ^ e} = 0.
k^-x n^tx
The lemma is Theorem 4.2 from [1].
Proof of Theorem 3. By Lemma 4 and the Prokhorov theorem fl, Theorem 6.1], the sequence {Pn : n e N} is relatively compact. Hence, every subsequence of Pn contains a subsequence {Pnk} such that Pnk converges weakly to a certain probability measure P on (Hr(D), B(Hr(D))) as k ^ to. Therefore, denoting by Xn = Xn(s) the Hr(^D)-valued random element having the distribution Pn, we have that
Moreover, by Lemma 2,
Xnk P. (3)
fc^-ro
XN,n —-—> xn, (4)
N ^ro
where the Hr(_D)-valued random element XN,n = XN,n(s) is defined by
XN,n(s) = Ln(X, a,s + i0N), and 9n is a random variable defined on a certain probabilitv space (Q, T, P) by the formula
P(0W = kh) = ^py, k = 0,1,...,N.
Define one more Hr (^D)-valued random element
Yn = Yn (s) = L(X,a,s + i 0N). Then, in view of Lemma 3, for every e > 0, lim limsupP( q(Xnti,Yn) ^ e)
n^ro N^ro - '
= lim limsup —1—# {0 ^ k ^ N : p (L(X,a, s + ikh),Ln(X,a, s + ikh)) ^ e}
n^ro n^ro N + 1 -
1 N
^ lim limsup—-—7 g(L(X,a,s + ikh), Ln(X, a, s + ikh)) = 0.
n^ro N^ro (N + 1)£fe=0~
This equality together with relations (3) and (4) shows that all hypotheses of Lemma 5 are satisfied. Therefore, we obtain the relation
Yn -——> P. (5)
N ^ro
Thus, we have that Pn converges weakly to P as N ^ to. Moreover, the relation (5) shows that the measure P is independent of the choice of the subsequence Pnk. Since the sequence Pn is relatively compact, hence we obtain that
Xn
n^-ro
This means that Xn converges weakly to P as n ^ to. The latter remark allows easily to identify the measure P. Actually, in [16], it was obtained that, under hypothesis that the set L(a\,..., ar) is linearly independent over Q,
1meas {re [0,T] : L(\,a,s + it) e A} , A e B(Hr (D)), (6)
also converges weakly to the limit measure P of Pn as n ^ to, and that P coincides with Pl. Obviously,the linear independence of the set L(a\,... ,ar; h, n) implies that of the set L(a\,..., ar).
Therefore, Pn also converges weakly to Pl which is the limit measure of Pn. The theorem is proved. □
3. Proofs of universality
We remind the Mergelvan theorem on approximation of analytic functions by polynomials [15].
Lemma 6. Let, K be a compact subset on the complex plane with connected complement, and let f(s) be a function continuous on K and analytic in the interior of K. Then, for every e > 0; there exists a polynomial p(s) such that
sup | f(s) - p(s)| < £.
s£K
We also need the explicit form of the support of the measure Pl. We recall that the support of Pl is a closed minimal set Sl such that Pl(Sl) = 1. The set Sl consists of all g e Hr(D) such that, for every open neighbourhood G of g, the inequality Pl(G) > 0 is true.
Lemma 7. The support of the measure Pl is the whole of Hr(D).
Proof.
It was observed above that Pl is the limit measure of (6). Thus, the lemma follows from [16],
□
We also recall two equivalents of the weak convergence of probability measures. Let Pn, n e N, and P be probability measures on (X, B(X)). The set A e B(X) is called a continuity set of P if P(dA) = 0, where dA is the boundary of A.
Lemma 8. The following statements are equivalent: 1° Pn converges weakly to P; 2° for every open set G C X,
limmfPn(G) ^ P(G),
n^-ro
3° for every continuity set A of the measure P,
lim Pn(A) = P(A).
n^ro
The lemma is a part of Theorem 2.1 from [1]. Proof of Theorem 2.
In view of Lemma 6, there exist polynomials p\(s),..., pr (s ) such that
Consider the set
g
sup sup 1 fj(s) - Pj(s^ (7)
1 ^j^r sEKj 2
Ge = i (gi,..., gr) G Hr(D) : sup sup |gj(s) - pj(s)| < J
1 ^j^r se Kj 2
Then the set Ge is open, and,by Lemma 7, is a neighborhood of the collection (p 1(s),...,pr(s)) which is an element of the support of the measure PL. Therefore, the inequality
PL(Ge) > 0 (8)
is satisfied. Hence, by Theorem 4 and 2° of Lemma 8,
lim inf Pn (Ge) >Pl(Ge) > 0. (9)
N
PN Ge
0 < k ^N : sup su^'.........' ^
lim inf # < 0 ^ k ^ N : sup sup |L(Xj,dj,s + ikh) - pj(s)| < ^ > 0. (10)
k £ N
sup sup |L( Xj,a.j,s + ikh) — pj(s)| < -.
l^j^rseKj 2
k
sup sup |L( Xj, a.j ,s + ikh) — fj (s)| < e.
l^j^rsEKj
□
Proof of Theorem 3.
Consider the set
Ge = < (91,..., gr) £ Hr (D) : sup sup | gj (s) - fj (s)| < e I Kj^r se Kj
Then the set Ge is open. Moreover, the boundary dGe lies in the set
jfo1,...,
1,...,gr) £Hr(D) : sup sup |gj(s) - fj(s)| =e >.
seKj J
Therefore, dGei HdGe2 = 0 for positive e1 = From this, it follows that PL(Ge) > 0 for at most countablv many e > 0, i.e., the set Ge is a continuity set of PL for all but at most countablv many e > 0. Hence, by Theorem 4, and 1° and 3° of Lemma 8, the limit
lim Pn (Ge) = PL(Ge) (11)
N
exists for all but at most countablv many e > 0. Moreover, it is not difficult to see that if (g 1,..., gr) £ Ge, where Ge is defined in the proof of Theorem 2, then, taking into account (7), we find that
sup sup | gj ( s) - fj (s)| < sup sup |gj (s) - pj (s)| + sup sup | fj (s) - pj (s)| < e.
se Kj Kj^r seKj 1<j<:rseKj
This shows that Ge C Ge. Since, by (9), PL(Ge) > 0, the monotonicitv of the measure gives the inequality PL(Ge) > 0. This inequality and (11) prove the theorem. □
4. Conclusions
The Lerch zeta-function L(A, a, s), s = a + it, with parameters A G R and 0 < a ^ 1 is defined, for a > 1, bv the series
L(X,a, s) =
œ g 2тггХт
r(m + a)s'
m=0
and by analytic continuation elsewhere. In the paper, it is obtained that a collection of Lerch zeta-functions ( L( X\,a\, s),...,L(Xr ,ar, s)) has a discrete universality property, i.e., a wide class of analytic functions can be approximated by shifts L(X\, ai,s + ikh),..., L(Xr, ar,s + ikh), h > 0, k = 0,1,2,____For this, the linear independence over Q of the set
|(log(m + (x3) : m e N0,j = 1,..., r),
is required. More precisely, if K\,..., Kr are compact subsets of the strip {se C : 2 < & < 1} with connected complements, and f\( s),..., fr (s) are functions continuous on K\,... ,Kr and analytic in the interior of K\,..., Kr, respectively, then, for every e > 0 ,
liminf —1—0 ^k ^ N : sup sup lL(Xj,aj,s + ikh) - fj(s)| < e ¡> > 0.
NN + 1 ^
It is possible to consider a more general situation, i.e., to consider the approximation of fi(s),..., fr (s) by different shifts L(X\,a\, s + i kh\),..., L(Xr, ar, s+i khr) with h\ > 0,... ,hr > 0. For this case, a new more general method than that of the paper is required, and it will be developed in a subsequent paper.
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
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2. Conway J.B. Functions of one complex variable. Berlin: Heidelberg; N. Y.: Springer, 1978. 167 p.
3. Ignataviciute J. Discrete universality of the Lerch zeta-function // Abstracts 8th Vilnius Conference on Prob. Theory. Vilnius, Lithuania, 2002. P. 116-117.
4. Воронин C.M., Карацуба А. А. Дзета-функция Римана. M.: Физматлит, 1994. 376 с.
5. Laurincikas A. The universality of the Lerch zeta-function // Liet. Matem. Rink. 1997. Vol. 37. P. 275—280, 367-375
6. Laurincikas A. On the joint universality of Hurwitz zeta-functions // Siauliai Math. Semin. 2008. Vol. 3(11). P. 169-187.
7. Laurincikas A., Garunkstis R. The Lerch Zeta-Function. Dordrecht; Boston; London: Kluwer Academic Publishers, 2002. 189 p.
8. Laurincikas A., Macaitiene R. The discrete universality of the periodic Hurwitz zeta-function // Integral Transforms. Spec. Funct. 2009. Vol. 20. P. 673-686.
9. Laurincikas A., Macaitiene R., Mochov D., Siauciunas D. Universality of the periodic Hurwitz zeta-function with rational parameter. 2017 (submitted).
10. Laurincikas A., Matsumoto K. The joint universality and functional independence for Lerch zeta-functions // Nagova Math. Journal. 2000. Vol. 157. P. 211-227.
11. Laurincikas A., Matsumoto K. Joint value-distribution theorems on Lerch zeta-functions. II // Lith. Math. Journal. 2006. Vol. 46. P.332-350.
12. Laurincikas A., Mincevic A. Discrete universality theorems for the Lerch zeta-function // Anal. Probab. Methods Number Theory. A. Dubickas et al. (Eds). P. 87-95.
13. Lerch M. Note sur la fonction К(w,x, s) = Era>o exp{2mnx}(n + w)-s // Acta Math. 1887. Vol. 11. P. 19-24. >
14. Lipschitz R. Untersuchung einer aus vier Elementen gebildeten Reihe // J. Reine Angew. Math. 1889. Vol. 105. P. 127-156.
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16. Mincevic A., Siauciunas D. Joint universality theorems for Lerch zeta-functions // Siauliai Math. Semin. 2017. Vol. 12(20). P. 31-47.
17. Mincevic A., Vaiginvte A. Remarks on the Lerch zeta-function // Siauliai Math. Semin. 2016. Vol. 11(19). P. 65-73.
18. Воронин C.M. Теорема об "универсальности" дзета-функции Римана // Изв. АН СССР. Сер.: Математика. 1975. Т. 39. С. 475-486 = Math. USSR Izv. 1975. Vol. 9. P. 443-453.
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