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ЧЕБЫШЕВСКИИ СБОРНИК
Том 20. Выпуск 1.
УДК 511.3
DOI 10.22405/2226-8383-2019-20-1-46-65
Совместная дискретная универсальность для L-функций из класса Сельберга и периодических дзета-функций Гурвица
Айдас Бальчюнас — доктор математических наук, ассистент кафедры теории вероятностей и теории чисел, Вильнюсский университет, Литва. e-mail: а. ЬаклипиШдтаИ. сот
Рената Мацайтене — профессор, доктор математических наук, ведущий научный сотрудник, Научный институт Шяуляйского университета, Факультет предпринимательства и технологий Шяуляйской государственной коллегии, Литва. e-mail: [email protected]
Дарюс Шяучюнас — профессор, доктор математических наук, старший научный сотрудник, Научный институт Шяуляйского университета, Литва. e-mail: [email protected]
А. Бальчюнас, Р. Мацайтене, Д. Шяучюнас
Аннотация
Класс Сельберга S составляют ряды Дирихле
коффициенты которых при всяком е > 0 удовлетворяют оценке а(то) тое; существует целое к ^ 0 такое, что (s — 1)fc£(s) является целой функцией конечного порядка; для £ имеет место функциональное уравнение, связывающее s и 1 — s, и эйлерово произведение по простым числам. Штойдинг пополнил класс S условием
где jj означает простые числа. Полученный класс обозначается через S.
Пусть а, 0 < а < 1, - фиксированный параметер, аа = {ат : то G N0} - периодическая последовательность комплексных чисел. Другой объект статьи - периодическая дзета-функция Гурвица С(«, а; а) при а > 1 определяется рядом Дирихле
и мероморфно продолжается на всю комлексную плоскость.
В статье расматривается дискретная универсальность набора
(£(«), С(«, «1; ац),..., С(«, «1; a^),..., с(s, «г; ari),..., С(s, аг; аг1г)),
набором сдвигов
(C(s + ikh), С(s + ikh1,a1; ац),. .., ((s + ikhi, ai, aii1),. .., Q(s + ikhr, ar; ari),. .., С(s + ikhr, ar; arir )),
где h, hi,..., hr положительные числа. При этом требуется линейная независимость над полем рациональных чисел для множества
{(hlogр : р € P), (hj log(m + aj) : т G N0, j = 1,. .. ,r), 2n] ,
где P — множество всех простых чисел.
Ключевые слова: Дзета-функция Гурвица, класс Сельберга, периодическая дзета-функция Гурвица, ряды Дирихле, слабая сходимость, универсальность.
Библиография: 29 названий. Для цитирования:
А. Бальчюнас, Р. Мацайтене, Д. Шяучюнас Совместная дискретная универсальность для L-функций из класса Сельберга и периодических дзета-функций Гурвица // Чебышевский сборник, 2019, т. 20, вып. 1, с. 46-65.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 1.
UDC 511.3 DOI 10.22405/2226-8383-2019-20-1-46-65
Joint discrete universality for L-functions from the Selberg class and periodic Hurwitz zeta-functions
A. Balciunas, R. Macaitiene, D. Siauciunas
Aidas Balciunas - doctor of mathematics, assistant of Department of probability theory and number theory, Vilnius University, Lithuania. e-mail: a. balciunui@gmail. com
Renata Macaitiene - professor, doctor of mathematics, chief researcher of Research Institute, Siauliai University, Siauliai State College, Lithuania. e-mail: [email protected]
Darius Siauciunas - professor, doctor of mathematics, senior researcher of Research Institute, Siauliai University, Lithuania. e-mail: [email protected],
Abstract
The Selberg class S contains Dirichlet series
TO / N
^ a{m)
—T , s = a + %t,
m=l
such that, for every e > 0 a(m) rne; there exists an integer k > 0 such that (s — \)kC(s) is an entire function of finite order; the functions C satisfy a functional equation connecting s with 1 — s, and have a product representation over prime numbers. Steuding introduced a subclass S of S with additional condition
/ \ -i
£ Hp)l2 = n> 0,
p^x
where p runs prime numbers.
Let a, 0 < a < 1, be a fixed parameter, and a = {am : m G N0} be a periodic sequence of complex numbers. The second object of the paper is the periodic Hurwitz zeta-function £(s, a; a) which is defined, for a > 1, by the Dirichlet series
c(s,a; a) = Y^
(to + a)s'
m=0 v '
and is meromorphically continued to the whole complex plane.
The paper is devoted to the discrete universality of the collection
(£(s), C(«, «1; ail),..., c(«, «1; aii!),..., c(«, ar; ari),..., C(«, ar; arir)),
where £(s) G .S^d £(s, ay; a^.) are periodic Hurwitz zeta-functions, i. e., to the simultaneous approximation of a collection
(/(lf),fll(s),...,flh (8),...,frl(8),...,frlr (S))
of analytic functions from a wide class by a collection of shifts
(£(? + (« + ifc^l, «i; an),..., C(s + ifc^l, «I; alij),..., c(s + ikhr, ar; a-l),.. ., C(s + «^r,«r; Orir ^,
where h, hl,..., hr are positive numbers, is considered. For this, the linear independence over the field of rational numbers for the set
{(hlogp : p G P), (hj log(m + aj) : rn G N0, j = 1,. .., r), 2^} ,
where P denotes the set of all prime numbers, is applied.
Keywords: Dirichlet series, Hurwitz zeta-function, periodic Hurwitz zeta-function, Selberg class. universality, weak convergence.
Bibliography: 29 titles. For citation:
A. Balciunas, R. Macaitiene, D. Siauciunas, 2019, "Joint discrete universality for L-functions from the Selberg class and periodic Hurwitz zeta-functions" , Chebyshevskii sbornik, vol. 20, no. 1, pp. 46-65.
In honor of Professor Antanas Laurincikas on the occasion of his 70th birthday
1. Introduction
After a pioneer Voronin's work [27], it is known that some zeta and L-functions are universal in the sense that their shifts approximate a wide class of analytic functions. Also, this universality property was extended to collections of zeta-functions simultaneously approximating a given collections of analytic functions. In other words, some zeta and L-functions are jointly universal in the approximation sense. The first joint universality theorem was obtained also by Voronin. In [28], investigating the joint functional independence of Dirichlet L-functions, he first obtained in a not explicit form their joint universality, see also [10], [11]. A very interesting is the so-called mixed joint universality of zeta and L-functions. In this case, a collection of analytic functions is approximated by the collection of zeta and L-functions consisting of functions having and having no Euler's product over primes. This type of universality was proposed by Mishou in [19] who proved
a mixed joint universality theorem for the Riemann zeta-function ((s), s = a + it, and the Hurwitz zeta-function
ro 1
C(s, a) = ^ --—, a > 1,
n (rn + a)s
m=0 v '
with transcendental parameter a, 0 < a ^ 1. Let D = {s £ C : | < a < 1}. Denote bv K the class of compact subsets of the strip D with connected complements, by H(K) with K £ K the class of continuous functions on K that are analytic in the interior of K, and by H0(K) with K £ K, the subclass of H(K) of non-vanishing functions on K. Then the Mishou theorem is the following statement.
Theorem 1. Suppose that a is transcendental. Let Kl,K2 £ K, fi(s) £ H0(Kl) and f2(s) £ H(K2). Then, for every e > 0;
liminf ^meas < t £ [0,T] : sup (s + ir) — /l(s)| < e, sup (s + it, a) — f2(s)l < el > 0. 1 y seKl seK2 J
In [7], Theorem 1 was extended for zeta-functions with periodic coefficients. Let a = {am: m £ N} and b = {bm : m £ No} No = N U {0}, be a periodic sequences of complex numbers with minimal periods ql £ N and q2 £ N, respectively. Then the periodic zeta-function ((s; a) and periodic Hurwitz zeta-function ((s, a; b), 0 < a ^ 1, are defined, for a > 1, by
c(s; a) = V and ((s,a; b) = V --,
^ ms ^ (m + a)s
m=l m=0
and can be continued meromorphicallv to the whole complex plane with possible simple pole at the point s = 1. The results of [14] were generalized for collections consisting from rl periodic zeta-functions with multiplicative coefficients and r2 periodic Hurwitz zeta-functions with algebraically independent over Q parameters al,... ,ar2. More general results were obtained in the theses of K. Janulis [6] and S. Rackauskiene [22].
The above mentioned universality results for zeta-functions are of continuous type, t in shifts ((s + ir; a) and ((s + ir, a; b) can take arbitrary real values. Reich in [23] proposed an another type of universality when t takes values from a certain discrete set. He used the set {kh : k £ N0} with fixed h > 0. The Reich theorem in the case of Riemann zeta-function is of the following form. In the sequel, #A denotes the cardinality of the set A, and N runs over non-negative integers.
Theorem 2. Suppose that K £ K. and f (s) £ Ho(K). Then, for every e> 0 and h> 0, liminf—1—# <¡0 < k < N : sup fc (s + ikh) — f (s)| < e\ > 0.
N^ro N + 1 [ seK J
Theorem 2 independently by an another method was also proved in [1].
The first discrete version of Theorem 1 was obtained in [3]. Define the set
L(P, a, h,n) = |(logp : p £ P), (log(m + a) : m £ N0), .
Then the main result of [3] is the following theorem.
Theorem 3. Suppose that the set L(P, a, h, ■k) is linearly independent over Q. Let Kl, K2 £ K., and fl(s) £ H0(Kl), f2(s) £ H(K2). Then, for every e> 0,
liminf < k < N : sup fc(s + ikh) — fl(s) <e, sup K(s + ikh, a) — /2(5)! <e \ > 0.
1
n N +1" [ ^ seK^^ y ^^ 7 seK2
In [4], Theorem 3 was generalized for shifts ((s + ¿fcfei^d ((s + a) by using the linear independence over Q of the set
= logp : p G P), (^2 log(m + a) : m G N0), 2^} .
An analogue of Theorem 3 for the functions ((s; a) with multiplicative coefficients and ((s, a; b) was proved in [14]. Finally, in [15], the results of [14] were extended for a wide collections consisting from periodic and periodic Hurwitz zeta-functions.
The aim of this paper is discrete universality theorems for L-functions from the Selberg class and periodic Hurwitz zeta-functions.
The Selberg class S was introduced in [24], and consists of Dirichlet series
oo
£(,) = £ a(m)
ma
m= 1
that satisfy the following axioms:
(i) (Ramanujan conjecture). For every e > 0 the estimate a(m) m£ takes place.
(ii) (analytic continuation). There exists r G N0 such that (s — 1)rC(s) is an entire function of finite order.
(iii) (functional equation). The functional equation
_ f
A£W = wa£(1 — s), A£(s) = C(s)Qs n r(A^s + »3),
3 = 1
with Qj, Xj G R and , w G C, Re^j ^ ^d |w| = 1 is satisfied for all s.
(iv) (Euler product). The product representation over primes
C(s) = n A>(*),
p
where
oo
log £,(.) = £ ^>
1=1 p"
with ^ pie with some d < 2, is valid.
It is well known that the majority of classical zeta and L-functions Me elements of the class S. The first universality results for L-functions from the Selberg class were obtained by J. Steuding in [25] and [26]. The most general universality theorem for the above L-functions is given in [21]. In this theorem, an additional condition that
1
i™ IE1) E№)i2 = 0 w
is required. Moreover, for C G S, let
f
dc = 2 £ Xj,
3 = 1
and
ac = max! -, 1 - — \ , D = Dc = (s £ C : ac < a < 1} . [2 dc )
Denote bv K,c the class of compact subsets of Dc with connected complements, and by H0c with K £ K,c the class of continuous non-vanishing functions on K that are analytic in the interior of K. Thejnain result of the paper [21] is the following theorem. Denote the class S with condition (1) by S.
Theorem 4. Suppose that C £<S. Let K £ Kc and f (s) £ H0c(K). Then, for every e> 0,
liminf 1 meas j t £ [0,T] : sup |C(s + it) - f (s)| < e \ > 0.
T^tt T [ seK J
Joint universality theorems of L-functions from the class S and periodic Hurwitz zeta-functions were proved in [8], [9] and [17].
The discrete version of Theorem 4 was given in [16].
Theorem 5. Under hypotheses of Theorem 4, for every e > 0,
liminf A;. - 1 # |o < k < N : sup |C(s + ikh) - f (s)| <e \ > 0.
1
'W^'tt N + 1 " ^ " ^ I
The aim of this paper is to obtain joint discrete universality for L-functions in the class S and periodic Hurwitz zeta-functions. Such a theorem for a pair (C(s),((s, a; a)) was obtained in [10]. h>0
L(P; a1,... ,ar; h,^) = |(log p : p £ P), (log(m + aj) : m £ No, j = 1,... ,r), .
Theorem 6. Suppose that the set L(P; a1,...,ar; h, ■k) is linearly independen t over Q and C £ S. Le t K £ Kc, Kl,...,Kr £ K, and f (s) £ Hoc(K), fi(s) £ H (K{),..., fr (s) £ H (Kr). Then, for every e > 0;
lim inf —1—# j 0 < k < N : sup |C(s + ikh) - f (s)| < e, N^tt N + 1 [ seK
sup sup |((s + ikh, aj; aj) - fj(s)| < e\ > 0. l^j^r seKj J
Moreover, the limit
Jim lirrT#(0 < k < N : sup |C(s + ikh) - f (s)| < e,
N^tt N + 1 [ seK
sup sup |((s + ikh, aj; aj) - fj(s)| < e> > 0 l^j^rseKj J
exists for all but at most countably many e > 0.
For positive h,h]_,... ,hr, define one more set
L(P; a1,... ,ar; h,h]_,... ,hr; n) = ((h log p : p £ P), (hj log(m + aj) : m £ N0, j = 1,..., r), 2^} .
Then we have the following generalization of Theorem 6
Theorem 7. Suppose that the set L(P; ai,..., ar; h, hi,..., hr; is linearly independent over Q and CeS. Le tK eKc, Ku...,Kr eK, and / (s) e Hoc (K), /i(s) e H (Ki),..., fr (s) e H (Kr). Then, for every e > 0;
liminf < k < N : sup |£(s + ifch) - /(s)| < e,
1
N + 1" ^ ^ " ' seK '
sup sup K(s + ikhj ,aj; aj) — fj(s)| < £ f > 0. i^j^r s£Kj J
Moreover, the limit
Jim A71 < k < N : sup |£(s + ifch) — /(s)| < e,
1
w—rö N + 1 " ^ " " " ' seK '
sup sup K(s + ikhj, aj; a^) — fj (s)| < e> > 0 i^j^rsEKj J
exists for all but at most countably many e > 0.
The latter theorem can be generalized in the following manner. Suppose that aji = {amji : m G No} is a periodic sequence of complex numbers with minimal period qji G N j = 1,... I = 1,..., ¿j. For j = 1,..., r, let be the least common multiple of the periods qj\,..., qji,, and
Aj =
( OOji &0j2 . . . jlj \ tiiji ttij2 . . . jlj
\ aqj-i,ji aqj-i,j2 ... aqj-i,jij J
Theorem 8. Suppose that C G S, the set L(P; a1,..., ar; h, hi,..., hr; ¿s linearly independent over Q and that rank(Aj) = lj, j = 1,..., r. Let K G K.£ and /(s) G H0c(K) and for j = 1,..., r, I = 1,..., lj, let Kji G K, fji(s) G H(Kji). Then, for every e > 0,
lim inf —1—# j 0 < k < N : sup |C(s + ifch) - /(s)| < e, w^^ ^ + 1 [ seK
sup sup sup K(s + ikhj, dj; aji) — /?7(s)| < e > > 0. i^j^r i^l^lj seKji j
Moreover, the limit
lim A71 < k < N : sup |C(s + ifch) — /(s)| < e,
1
sup sup sup K(s + ikhj, aj; aji) — /^7(s)| < e > > 0 <j<~r i^l^lj seKji j
i<j<r i^i^ij seKji exists for all but at most countably many e > 0.
We see that Theorem 6 is a partial case of Theorem 7 with hi = ■ ■ ■ = hr = h, and Theorem 7 is a partial case of Theorem 8 with li = ■ ■ ■ = lr = 1. Therefore, it suffices to prove Theorem 8.
The next section is of probabilistic character. It is devoted to limit theorems on weakly convergent certain probability measures connected to the functions C(s) and ((s, aj; a^).
2. Probabilistic results
Let G be a region on the complex plane, and H(G) be the space of analytic functions on G endowed with the topology of uniform convergence on compacta. We preserve the notation of [17]. Thus, let
r
U = ^ lj, V = u +1, 3=1
and
Hv = Hv(Dc, D) = H(Dc) x HU(D), where Hu(D) = H(D) x ■ ■ ■ x H(D). Denote by B(X) the Borel a-field of the space X, and use the
V
u
notation
Z (j, s,a; a, C) = (C(s), ((s, an; an),..., ((s, an; a^),..., ((s, ar; ari),..., ((s, ar; arir)), where a = (a1,..., ar) and a = (a11,..., a1z1,..., ar1,..., arir). For A £ B(HV) and N £ N0, define
PN (A) = ^ i# (0 < k < N : Z (J + ikh, s + ikh, a; a, C) £ A} ,
where s + ikh = (s + ikh1,... ,s + ikhr). In this section, we will consider the weak convergence of Pn as N ^ m. For the definition of the limit measure, we need a certain Rv-valued random element. Let j = (s £ C : |s| = 1}, and
j=n >, q = n
pep meNo
where = 7 to all p £ ^d 7m = 7 fa all m £ N0. The classical Tikhonov theorem implies that the infinite-dimensional tori j and Q with the product topology and pointwise multiplication are compact topological Abelian groups. Hence,
Q = j x Q1 x ■ ■ ■ x Qr,
where Qj = Q fa aU j = 1,...,r, is again a compact topological Abelian group. Therefore, on (Q, B(Q)), the probabilistic Haar measure mu can be defined, and we obtain the probability space (Q,B(Q),mH)• Denote by 6J(p) the pth component of u £ p £ P, and by (¿j(m) the mth component of (¿j £ Qj, m £ No j = 1,..., r. Moreover, let w = (¿J ,u1,...,ur) be elements of Q. Now, on the probability space (Q, B(Q), mu), define the H^-valued random element Z(s,s,w,a; a, C) by the formula
Z(w) = Z(s,s,u,a; a, C) = (C(s,uj),( (s,ai,wi; an),...,( (s,ai,wi; au1),...,
((s, ar,ur; ari),..., ((s, ar,ur; arir)),
where
C(j, y a(m)uj(m) j £
ms
m=1
with
uj(m) = JJ usl(p), m £ N,
p1- \m
pl+1\m
and, for s G D,
M x ^ amjiWj (m) . 1,1
C(s ,aj ; ajO ^ ^ y , j = l,...,r, Z = l,..., lj.
We observe that, for almost all w G Q, the equality
[peffc=i ^
holds [21] with certain coefficients b(pk).
Denote by Pz the distribution of the random element Z(s, s, w, a; a, C), i.e., Pz is a probability measure on (Hv, B(HV)) defined by
Pz (A) = mH {w G Q : Z (s,s,w,a; a, C) G A} , A G £(HV ).
Now, we are able to state a limit theorem for PN.
Theorem 9. Suppose that C G S, the set, L(P; a1,..., ar ; fa, fa1,..., far ; ¿s linearly independent over Q and that rank( Aj) = lj, j = l,..., r. Then Pn converges weakly to Pz as N ^ oo. Moreover, the support of the measure Pz is the set Se x Hu(D); where
Se = {;?G H (De) : <7(5) = 0or5(S) = 0}.
We divide the proof of Theorem 9 into Lemmas. The first of them deals with weak convergence on the group Q.
Lemma 1. Suppose that the set L(P; a1,..., ar; fa, fa1,..., far; is linearly independent over Q. Then
Qn (A) n^+y {0 ^ N : ((p-kh : p G p) , ((m + a1)-khl : m G Nq) ,...,
((m + ar)-ikhrm G Nq)) G a} , A G £(Q), converges weakly to the Haar measure m h as N ^ o.
proof. We apply the Fourier transform method. Denote by gN(k,i1,..., ir), where k = (kP : kp G G Z, p G P) !1 = (Z 1m : Z1m G Z, m G Nq), ... ,lr = (Zrm : lrm G Z, m G Nq), the Fourier transform of Qn Since the characters of the group Q are of the form [13], [17]
,kv(m\ TT TT , Jj™ i peP j=1meN0
n wkp (P) n n ^T (m),
where the sign ""' shows that only a finite number of integers and ljm are distinct from zero, we have that
9n (k, h,...,lr)= [ (n^ (p) J] H (m)l dQN.
— \ peP i=imeN0 )
Therefore, by the definition of Qn
y n , r ,
9n (k, Î1,...,U = N-Y En P-kkph n n (m + a rkhhm
k=Q peP i=1meN0
l N
^ exp < —k I ^ fa kp logp + ^ ^ fajljm log(m + a,) l> . (2)
N + 1
k=0 I \ peP i=1meN0
Obviously,
* 9n (0,0,..., 0) = 1. (3)
Since the set L(P;a1,...,ar;h,h]_,... ,hr;n) is linearly independent over Q, we have that
exp < -ik I ^ hkp logp + ^ ^ hjljm log(m + aj) I > =1 (4)
[ \peP j=imeNo J )
for (k, 11,... ,lr) = (0,0,..., 0)- Actually, if inequality (4) is not true, then
A <= ^ hkp log p + ^ ^ hjljm log(m + aj) = 2na peP j=i meN0
with a certain a £ Z, and this contradicts the linear independence of the set L(P;a1,... ,ar;h, h1,..., hr; n). Thus, inequality (4) is true, and, in view of (2), we find that, for
(k, h,...,^) = (0,0,..., 0),
1 - exp(-i( N + 1)A}
gN(k,h,..., l_r) =
( N + 1)(1 - exp(-A})' This and (3) show that
( r 1 if (k,l_i,...,l_r) = (00,...,0), m
Jim^gN(k,h,...,lr) = {0 if (k,i_r) = (o,0,...,0), (5)
and the lemma is proved because the right-hand side of (5) is the Fourier transform of the Haar measure rnn- □
The next lemma considers probability measures on the space ( Hv, B(HV)) defined by collections consisting of absolutely convergent Dirichlet series. Let d > 2 be a fixed number, and
vn(m) = exp |- j m,n £ N,
f ( m + aj1
\ \n + aj) j
vn(m, aj) = exp - ( —^—- ) } , m £ N0, n £ N, j = 1,... ,r.
n + aj
Define the functions
oo
C ( s ) = y a(m)Vn(rn)
m*
m=1
and
t ! \ amjlVn(m, aj) .
Cn(S ,o,; M = E (m+ )S , J = 1,...,r, l = 1,..., l3.
m=0 v ■>'
Then it is known that the series for Cn(s) is absolutely convergent for a > ma^2, 1 - 3^) a£
[21], and the series for (n( s, aj; a^) are absolutely convergent for a > 2 [12]. Additionally, we define the series
= ^ a(m)u)(m)v,n(m)
Cn(S, W) = / y
m*
m=1
and
amjWj(m)vn(m,aj) Cn( a, aj; ^ = ^ ' > + )g J ,
. . , i = 1,... ,r, 1 = 1,..., U,
(m + aj )s J
m=0 ■>'
which, obviously, are also absolutely convergent in the above regions. Let, for brevity,
Zn (s, s, a; a, C) = (Cra(s), (n(s, ai; an),..., (n(s, ai ay1),...,
(n(s, ar; ari),..., (n(s, ar; arir)),
and
Zn (s, s, w, a; a, C) = (Cn(s, w), (n(s, wi, ai an),..., (n(s, wi, ai aix),...,
Cn(S, wr, av; ari),..., (n(s, wr, av; a^)),
PN,n(A) = {0 < fc < N : Zn (s + ifch,s + ffch,a; a, C) g!} , A G B(HV).
Lemma 2. Suppose that C G S and the set L(P; ai,..., ar; h, hi,..., hr¿s linearly independent over Q. Then PN,n converges weakly to the measure Pn on (HV, B(HV)) as N ^ to, where Pn = mnu-\ and the function un : Q ^ #V ¿s given 6g/ i/ie formula
un(w) = Zn (s, s, w, a; a, C).
Proof. We have that
un ((p-ikh : p G p) , ((m + ai)-ikhl : m G No) ,..., ((m + ar)-khrm G No))
= Zn (s + ifch, s + ifch, a; a, C)
Therefore,
PN,n = Qwu-i, (6)
where Qn is from Theorem 9, and the equality is understand as Pn,ti(A) = Qn(u-iA), A G ). Moreover, the absolute convergence of the series for Cn(s, w) and (n(s,wj, ay; a-,-1) implies the
un
Theorem 5.1 of [2]. □
Now, we will approximate Z by Zn in the mean. For this, we need the metric in HV. Let G be a region in C. Then it is known [5] that there exists a sequence of compact sets {Ki : I G N}cG such that
G =
i=i
Ki C Ki+i to al 1 I G N ^d if K C D is a compact set, then K C Ki for some I. Taking i,52) = V2 —-'—¡—^---7, <jfi,5-2 Gtf(G),
1+suP seKl 15i(S ) - ^2( s)|
gives a metric in #(D) inducing its topology of uniform convergence on compacta. Define by pc the above metric in H(Dc), and by p the metric in H(D). Let
£= (0, i?ii, . . . , 1,..., 5r1,..., 9rlr ^ / = ( /, /ii, •••, fu 1,..., fn,...,frir) G .
Then
Pv(5-, /) = ma4pc(fl, /), max max p(gjt, /j7) ) is a desired metric in HV inducing its product topology.
Lemma 3. Let C e S. Then the equality 1 N
lim lim sup- V^ pv (Z(s + ikh, s + ikh, a; a, C), Zn(s + ikh, s + ikh, a; a, C)) = 0
n^ NN +1
holds.
PROOF. By the definition of the metric pv, it suffices to prove that
1 N
lim lim sup —- > pc(C(s + ikh), Cn(s + ikh)) = 0,
N^ N + 1
and, for j = 1,..., r, 1 = 1,..., lj,
N
1 "l"hj,aj; ajl), Sn(ö + okhj,aj
1 N
lim lim sup y^p(((s + ikhj,aj; aji), (n(s + ikhj,aj; aji)) = 0.
l—±OG ar , - . /V \ < J
n^tt N^tt N + 1 k=0
However, the first equality was obtained in [21], while the second equality follows from [10]. □
Now, we will consider the limit measure Pn of Lemma 2, and will prove that the sequence (Pn : n £ N} is tight, i.e., for every e > 0, there exists a compact set K = K(e) C Hv such that
Pn(K) > 1 - e
n £ N
Lemma 4. Suppose that C £ S and the set L(P;a1,... ,ar; h,h1 ,...,hr ;n) is linearly independent over Q. Then the sequ,ence (Pn : n £ N} is tight.
Proof. On a certain probability space with the measure define the random variable On by
M 0N = k} = , k = 0,1,...,N.
Define the fl^-valued random element XN,n = XN,n(s, s) = (XN,n(s), XN,n,1,1(s),..., XN,n,1,i 1 (s), ..., XN,n,r,1(s),..., XN,n,r,ir (s)) = Zn(s + iOnh, s + iOnh,a; a, C). Moreover, let
Xn = Xn(s, s) = (Xn(s),Xn,1,l(s), . . .,Xn,1,l 1 (s), . . .,Xn,r,l(s), . . .,Xn,r,l r (s))
be fl^-valued random element with the distribution Pn, where Pn is the limit measure in Lemma 2. Then the assertion of Lemma 2 can be written as
XN,n —-—> Xn,, (7)
N ^tt
V
where —> means the convergence in distribution.
Since the series for Cn(s) is absolutely convergent for a > ac-, we have that, for 2 < & <
1 fT \r <„ , ^ |a(m)|2v"n(m) |a(m)|2
lim 1/ C(a + «№ = £ ^^ (m) < £ ^ ^ < ^ T J0 ^ m2a ^ m2a
J0 m=l m=l
and
1 fT we • m2j ^ la(m)l2v2n(m) log2 m
1 AT ^
lim- / |cn( a + i t )|2 di=V
t^T J0 1 nV n ^ m2a
J0 m=l
^ Cn.\ < oo.
These estimates and an application of the Gallagher lemma [20, Lemma 1.4], which connects discrete and continuous mean squares of some functions, lead, for \ < a < ac and all n G N, to
1 N
limsup V" |Cn(a + ifch)|2 € Cac < to. (8)
NN + 1k=0
Let Km be a compact set from the definition of the metric pc. Then (8) and the Cauchv integral n G N
1 N
limsup —-y sup |Cn(s + ifch)| € CTO < to. (9)
NN + 1k=0 sekm
Let Km be a compact set from the definition of the metric p. Then, in a similar way, we obtain
that, for all j = 1,..., r, Z = 1,..., lj,
1 N
limsup , ., y sup | (n(s + ¿fch,-,«,-; aj7)| €Cj,i,m < to (10)
N^^ iv + 1 k=0 seKm
for all n G N. Let e > 0 be an arbitrary number, and, for m G N,
Mm = Mm(e) = Cm2m+i e-i, = Mjtl>m(e) = CJ)i)m2n+m+i e-i.
n G N
limsup^ < sup |Xw,n(s)| > Mm or I : sup |XW;nj,i(s)| > Mj,i,m N^^ I \ seKm ) V
or (3j,Z : sup |XW)nj,i( s )| > Mjtitm) > J \ seKm J J
€ limsup^ < sup |Xw,n(s)| > Mm > + ^^ limsup^ sup lXN,nj,i(s)| > Mjti,m \ n^^ [seKm ) j=ii=i lseKm )
= limsup —1—# i 0 € k € N : sup |Cn(s + ifch)| > Mm I nN + 1 I sekm J
r lj 1 i 1 + yV" limsup ## 0 : sup | (n(s + ¿fch,-,«,-; a^)| > Mj,i,m\
j=U n7V + 1 I )
1
€
+1
N
n(
limsup , . V sup |Cn(s + ifch)|
k=0 •fs'
r h 1 N
+£ § 'ms"p +1) g |<n(s+' kh"ai; a 0| € 2"
n G N
^ ( sup |Xn(s)| >Mmj or : sup ^¿j| > Mjtt ,m j | € 2" (H)
> |An,i,i | > } € £
vse km ) V seK,
n G N
KV(e) = S (#,5ii,... ,...,5Vi,...,9rir) G HV : sup |#(s)| € Mm, sup n(s)| € M,i,m,..., [ se km
sup |fifHi(s )| € M,ii,m,..., sup 15ri(s)| € Mr,i,m,..., sup 15irir(s)| € Mr,ir,m,m G N ^.
s€Km s£Km s£Km )
Then the set Kv(e) is compact in Hv, and, in virtue of (11),
/i{xn e Kv(e)} ^ 1 -e
for all n e N, or equivalently,
Pn(Kv(e)) > 1 -e
for all n e N This shows that the sequence {Pn : n e N} is tight. □
proof. [Proof of Theorem 9] Since, bv Lemma 4, the sequence {Pn : n e N} is tight, in virtue of the Prokhorov theorem [2, Theorem 6.1], it is relatively compact. Therefore, every sequence of {Pn} contains a subsequence {Pnk} such that Pnk converges to a certain probability measure P on (Hv, B(Hv)) as k ^ < Hence,
Xnk P. (12)
Now, define the Hv-valued random element Xn by the formula
XN = XN (§, s) = Z(s + iQnh, s + idNh,a; a, C).
Then an application of Lemma 3 shows that, for every e > 0,
lim lim sup pv (XN,XNn) ^ e} n—~ N—to
= lim lim sup—1—#{0 ^ k ^ N :
N—to N + 1
pv(Z(s + ikh, s + ikh,or, a, C), Zn(s + ikh, s + ikh, a; a, C)) ^ e} 1 N
^ lim limsup—-^tt^Pv((s + ikh,s + ikh,a; a,C),Zn(s + ikh,s + ikh,a; a, C)) = 0.
n—~ n—to (N + 1)e
The latter equality, relations (7), (11) and Theorem 4.2 of [2] imply that
XN P, (13)
n —to
or, in other words, Pn converges weakly to P as N ^ <x>. Moreover, (13) shows that the measure P is independent of the choice of the sequence {Xnk }. Therefore,
Xn —P.
n—y^o
This means that P^s N ^ «, converges weakly to the limit measure P of Pn as n ^ <. Denote by
X = (Xo,Xi ,...,Xr), Xj = (Xjh...,Xji.), j = 1,...,r,
the Hv-valued random element with distribution P. Moreover, let Pn,0, Pn,1,..., Pn,r be the
Pn Pn, o
of the H(D)-valued random element
a(m)Co(m)
C(s ,w)=>---, seDc
Z—/ rn-
m-
m=1
as n ^ < The linear independence over Q of the set L(P; o1,... ,or;h,h1,... ,hr;n) implies that for the sets
L(aj) = {log(m + a.j) : m e N0}, j = 1,... ,r.
Therefore, repeating the arguments of [12], we obtain that Pnj- converges weakly to the distribution of the Hli -valued random element
0 = 0(u) = (C(s, aj, U; ajh),..., (((s, a, U; aji.) as n ^ to, j = 1,..., r- This and the definition of the random element X show that
X0 = £(s , U) and Xj = Q, j = 1,...,r. Therefore, P is the distribution of the HV-valued random element
(£(h,U), Ci,..., Cr),
in other words, P^ converges weakly to the distribution Pz of the random element Z. It remains to find the support of Pz.
It is known [21] that the support of the random element £(s,U) is the set Sc. Denote by rhH the Haar measure on ( Q, B(Q)), and by mHj- the Haar measure on (Qj, B(Qj)), j = 1,..., r. Then we have that mH is the product of the measures m^d mHii,..., mH,r- This means that, for
A = A0 xAi x ■ ■ ■ x Ar, A0 G B(H(Dc)), Aj G B(H^(D)), 3 = 1,...,
mH(A) = rhh(A0) ■ mH,i(Ai) ■ ■ ■ mH,r(Ar). (14)
H( . c) H( . )
B(HV) = B(H(Dc)) x B(H1 (D)) x ■ ■ ■ x B(Hr(D)).
Hence, it suffices to consider the measure mH on sets of the type (14). Since the sets L(aj) are linearly independent over Q and rank( Aj) = lj, j = 1,..., r, we have that the support of (j is the set Hli(D), j = 1,..., r [13]. Therefore, using the equality (14), we obtain that
mH{u G Q : Z(u) G A} = rhH{U G h : £(s, U) G A0} ■ mH,i{wi G Q : Ci(ui) G Au} ■ ■ ■
mH,r {ur G Qr : (r (ur) G Ar }.
This, the minimality of the support and the supports of the random elements £(s, U), Ci(ui),..., (r(ur) imply that the support of the measure Pz is the set Sc x HM(D). The theorem is proved. □
3. Proof of universality
First we recall the Mergelvan theorem on the approximation of analytic functions by polynomials [18].
Lemma 5. Let K c C be a compact set, with connected com,piements, and /(s) be a continuous
function on K and analytic in the interior of K. Then, for every e > 0, there exists a polynomial ( )
sup |/(s ) - p(s)| < e.
s£K
PROOF. [Proof of Theorem 8] In view of Lemma 5, there exist polynomials p(s) and Pji(s) such that
/(*) - ep( < 6- (15)
sup
se kc
2
and
suP Uji (s ) - Pfl 00| < ^ i^...^ i^..^ h. (16)
sEKji 2
Define the set
Ge = jfo, gii,..., g h 1,..., gri,..., grir ) e H" : sup f(s) - ep{s)
tv
seKjz
< 2
sup sup sup | fjl ( s ) - Pjl (s)| < -
KJ^r ij seKji 2
.
Then, by the second part of Theorem 9, the set G£ is an open neighborhood of the element (ep(s\ pn,... pu 1,..., pri,..., prir) of the support of the measure Pz. Hence,
Pz(G£) > 0. (17)
Moreover, by Theorem 9 and the equivalent of weak convergence of probability measures in terms of open sets ([2, Theorem 2.1]), we have that
liminf PN (G£) >Pz G).
N ^TO
PN G£
To prove the second assertion of the theorem, define the set
( ,
GGe = { (g, gii,..., gih,..., gri,..., 9rir) e Hv : sup | g(s) - f(s)l < e,
seK£
sup sup sup |gjt (s) - fjl (s)| < £ L
KJ^r i^i^^ seKji J
Then the boundaries dG£1 and dG£2 do not intersect for different positive ei and e2- Hence, the set G£ is a continuity set of the measure Pz (Pz(dG£) = fa all but at most countablv many e > 0. Using of Theorem 9 and the equivalent of weak convergence of probability measures in terms of continuity sets ([2, Theorem 2.1]) yields the equality
lim Pn(G£) = PzG) (18)
N
for all but at most countablv many e > Inequalities (15) and (16) imply that G£ C G£. Therefore, in virtue of (17), we have that Pz(G£) > the definitions of PN and G£, and (18) prove the
second assertion of the theorem. □
4. Conclusions
In the paper, the joint discrete universality of the L-functions from the modified Selberg class and periodic Hurwitz zeta-functions is obtained. This means that wide collections of analytic functions ( f, f11,..., fu 1,..., fr1,..., frir ) can be approximated by discrete shifts
(C(s + ikh), ((s + ikh1 ,a1 ; an),..., ((s + ikh1,a1; a^ 1 ),..., ((s + ikhr, ar; ar 1),...,
((s + ikhr,ar; arir)).
For this, the linear independence over Q for the set
{(h logp : p e P), (hj log( m + aj) : m e No, j = 1,..., r), 2n} ,
a1 , . . . , ar h; h1 , . . . , h
numbers, is applied.
L
from the Selberg class. For this, the linear independence over Q for the set
j(hfc logp : p G P, k = 1,..., m), (hj log(m + a^) : m G N0, j = 1,..., r), would be used.
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8. Kacinskaite R., Matsumoto K. The mixed joint universality for a class of zeta-functions // Math. Nachr. 2015. Vol. 288, №16. P. 1900-1909.
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Получено 09.01.2019
Принято к печати 10.04.2019
