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14
ЧЕБЫШЕВСКИИ СБОРНИК
Том 17 Выпуск 3
УДК 519.14
МОДИФИКАЦИЯ ТЕОРЕМЫ МИШУ
А. Лауринчикае, (г, Вильнюс, Литва), Л, Мешка (г, Вильнюс, Литва)
Аннотация
В 2007 г. Г. Мишу доказал совместную теорему унивурсальности для дзета-функции Римана С (в) и дзета-функции Гу рвица С (в, а) с трансцендентным параметром а об одновременном приближении пары функций из широкого класса аналитических функций сдвигами (С (в + гт), £ (в + гт, а)), т € М. Он получил, что множество таких сдвигов, приближающих данную пару аналитических функций, имеет положительную нижнюю плотность. В статье получено, что множество таких сдвигов имеет положительную плотность для всех е > 0 за исключением счетного множеств а значений е, где е — точность приближения.
Результаты аналогичного типа также получены для сложных функций Г ( £ (в), С (в, а)) для некоторых классов операторов Г в пространстве аналитических функций.
Ключевые слова: дзета-функция Гурвица, дзета-функция Римана, пространство аналитических функций, универсальность.
Библиография: 21 названий.
The Mishou theorem asserts that a pair of analytic functions from a wide class can be approximated by shifts of the Riemann zeta and Hurwitz zeta-functions (Z(s + ¿t), Z(s + ¿t, a)) with transcendental a, t g R, and that the set of such t has a positive lower density. In the paper, we prove that the above set has a positive density for all but at most countably many e > 0, where £ is the accuracy of approximation. We also obtain similar results for composite functions F(Z(s),Z(s, a)) for some classes of operator F.
Keywords: Hurwitz zeta-function, Riemann zeta-function, space of analytic functions, universality.
Bibliography: 21 titles.
1. Introduction
Let Z(s), s = a + it, be the Riemann zeta-function. In 1975, S. M. Voronin discovered [21] the universality property of Z(s) which means that a wide class of non-vanishing analytic functions can be approximated by shifts Z(s + ¿t), t G R. The non-vanishing of approximated functions is
Z(s)
Now let 0 < a ^ 1 be a fixed parameter, and Z(s, a) denotes the Hurwitz zeta-function which is defined, for a > 1, by the series
MODIFICATION OF THE MISHOU THEOREM
A. Laurincikas, (Vilnius, Lithuania), L. Meska (Vilnius, Lithuania)
Abstract
and can be meromorphically continued to the whole complex plane. Clearly, ((s, 1) = ((s), and
((s, 1) =(2s - 1)Z(s).
For other values of the parameter a, the function ((s, a) has no Euler product. It is well known that the Hurwitz zeta-function with transcendental or rational = 1, \ parameter a is also universal in the above sense, however, its shifts ((s + ir, a) approximate not necessarily non-vanishing analytic functions. The universality of ((s, a) with algebraic irrational a is an open problem.
Some other zeta-functions are also universal in the Voronin sense. The universality for zeta-functions of certain cusp forms was obtained in [12], for periodic zeta-functions was studied in [20] and [15], while the works [2], [4] and [5] are devoted to periodic Hurwitz zeta-functions. Universality theorems for Lerch zeta-functions can be found in [11]. A very good survey on universality of zeta-functions is given in [16].
In [19], H. Mishou began to study the so-called mixed joint universality. In this case, a collection of analytic functions are simultaneously approximated by shifts of a collection of zeta-functions consisting from functions having the Euler product and having no such a product. H. Mishou considered the pair (((s), ((s, a)) with transcendental a. For the statement of the Mishou theorem, we need some notation. Let D = {s e C : i <ct< l}. Denote by K the class of compact subsets of the strip D with connected complements. Moreover, let H(K), K e K, be the class of continuous functions on K which are analytic in the interior of K, and let H0(K), K e K, be the subclass of H(K) consisting from non-vanishing functions on K. Denote by measA the Lebesgue measure of a measurable set A C R. Then H. Mishou proved [19] the following theorem.
Theorem 1. Suppose that a is transcendental number. Let Ki,K2 e K, and /i(s) e H0(Ki); /2(s) e H(K2). Then, for every e > 0
liminf ^measjr e [0; T] : sup |((s + ir) — /i(s)| < e,
T I seKl
sup |Z(s + ir, a) — /2(s)| < e ^ > 0.
sK
seK 2
Mixed joint universality theorems are also proved in [3], [7] and [10].
lim inf lim Z(s)
done in [13] and [18], and, in the case of ((s, a) a similar theorem was obtained in [14]. Let P be the set of all prime numbers, No = N U {0}, and
L(a, P) = {(log(m + a) : m e N0), (logp : p e P)} .
Theorem 2. Suppose that the set L(a, P) is linearly independent over the field of rational numbers Q. Let Ki,K2 e K , and /i(s) e H0(Ki) /2(s) e H(K2). Then the limit
lim ^measj r e [0; T] : sup |( (s + ir) — /i(s)| < e, sup |( (s + ir, a) — /2(s)| < ^ > 0
T ^ s€K1 s€K2 J
e > 0
For example, if a is transcendental, then the set L(a, P) is linearly independent over Q.
Let H(G) be the space of analytic functions on G equipped with the topology of uniform convergence on compacta. In [9], universality theorems were proved for the functions F(((s), ((s, a)) with some operators F : H2(D) ^ H(D). Let
S = {g e H(D) : g(s) = 0 or g(s) = 0} . Then, for example in [9], the following assertion was obtained.
Theorem 3. Suppose that a is transcendental, and that F : H2(D) ^ H(D) is a continuous operator such that, for every open set G C H(D) the set (F-1G) n (S x H(D)) is non-empty. Let K G K and f (s) G H(D). Then, for every e > 0,
liminf —meas< t G [0; T] : sup |F(Z(s + iT), Z(s + iT, a)) - f (s)| < 4 > 0. t^^ T I seK J
More general results are obtained in [10].
Clearly, the transcendence of a in Theorem 3 can be replaced by a linear independence over Q of the set L(a, P). Therefore, we will prove the following theorem.
Theorem 4. Suppose that the set L(a, P) is linearly independent over Q, and that F, K and f(s)
lim ¿measi t G [0; T] : sup |F (Z (s + iT ),Z (s + iT,a)) - f (s)| < e) > 0 (1)
T I s£K J
e > 0
Now, let V be an arbitrary positive number, Dy = {s e C : 2 < ^ < 1, |t| < V} and
Sv = {g G H(Dv) : g(s) = 0 or g(s) = 0} . For brevity, we use the notation H2(DV, D) = H(DV) x H(D).
Theorem 5. Suppose that the set L(a, P) is linearly independe nt over Q, and th at K and f (s) are the same as in Theorem 3, and V > 0 is such that K C Dy- Let F : H2(Dy, D) ^ H(Dy) 6e a continuous operator such that , for each polynomial p = p(s), iAe sei (F-1{p}) n (Sy x H(Dy))
e > 0
For example, Theorem 5 implies the modified universality of the functions
c1Z(s) + c2Z(s, a^d c1Z'(s) + c2Z'(s, a) with c1, c2 e C \ {0}. Let a1,..., ar be arbitrary distinct complex numbers, and
Hai,...,ar(D) = {g G H(D) : (g(s) - aj)-1 G H(D), j = 1,..., r} .
Theorem 6. Suppose that the set L(a, P) is linearly independent over Q, and F : H2(D) ^ H(D) is a continuous operator such that F(S x H(D)) D Hai,...,ar(D^. Wften r = 1, lei K G K,
f(s) G H(K) f(s) = a1 K e > 0. // r ^ 2 K C D ¿s an arbitrary compact subset, and f (s) G Hai,...,ar(D); iften ifte Hmfi (1)
e > 0
The case r = 1 with a1 = 0 shows that, for F(g1(s),g2(s)) = egl(s)+g2the limit (1) exists for all but at most countablv many e > 0. If r = 2 and a1 = 1 a2 = —1, then, for example, for F(g1(s),g2(s)) = cos(g1(s) + g2(s)^d f(s) G H1)-1(D), the limit (1) exists for all but at most e > 0
Theorem 7. Suppose that the set L(a, P) is linearly independe nt over Q, F : H 2(D) ^ H (D) ¿s a continuous operator, K C D is a compact subset, and f (s) G F(S x H(D)). Tften the limit (1)
e > 0
2. Lemmas
In this section, we present probabilistic theorems on the weak convergence of probability measures in the space of analytic functions. Let y = {s e C : |s| = 1}, and
<x
Qi = n Yp and Q2 = n Ym,
p m=0
where yp = Y f°r aU P e ^^d Ym = Y f°r aU m e N0. By the Tikhonov theorem, the tori Qi and Q2 with the product topology and operation of pointwise multiplication are compact topological Abelian groups. Similarly, Q = Qi x Q2 is also a compact topological Abelian group. Therefore, denoting by B(X) the Borel a-field of the space X, we have that, on (Q,B(Q)), the probability Haar measure mH can be defined, and we obtain the probability space (Q, B(Q),mH). Denote by wi(p^d w2(m) the projections of wi e Qi and w2 e Q2 to the coordinate spaces yp P e P, and Ym, m e N0, respectively, and, on the probability space (Q, B(Q),mH), define the H2(D)-valued random element ((s, w), w = (wi,w2) e Q, by the formula
((s, a, w) = (((s, wi), ((s, a, W2)),
where
z c.wi) = n (1 - "
p
and
N ^ W2(m) Z (s,a,W2) = > 7-■—-.
^ (m + a)s
m=0
Moreover, let
Pz(A) = mH (w e Q : ((s,a,w) e A) , A e B(H2(D)), i.e., Pz is the distribution of the random element ((s,w). We set ((s, a) = (((s), ((s, a)), and
Pt(A) d=f ^meas {r e [0,T] : ((s + ir, a) e A} , A e B(H2(D)).
Lemma 1. Suppose that the set L(a, P) is linearly independent over Q. Then P converges weakly to Pz as T ^ to.
Proof. The lemma for transcendental a is proved in [19], Theorem 1, however, the transcendence of a is used only for the linear independence of the set L(a, P).
□
Let Xi and X2 be two metric spaces, and let the function u : Xi ^ X2 be (B(Xi), B(X2))-measurable. Then every probability measure P on (Xi, B(Xi)) induces on (X2, B(X2)) the unique Pu-i(A)
Pu-i = P(u-iA), A e B(X2).
It is well known that if u is a continuous function, then it is (B(Xi), B(X2))-measurable.
In the sequel, the following property of weakly convergent probability measures will be very useful.
Lemma 2. Suppose that Pn, n e N, and P are probability measures on (Xi, B(Xi)); the function u : Xi ^ X2 is continuous, and Pn converges weakly to P as n ^ to. Then Pnu-i also converges weakly to Pu-i as n ^ to.
The lemma is Theorem 5.1 from [1].
Lemma 3. Suppose that the set L(a, P) is linearly independent over Q, and F : H2(D) ^ H(D) is a continuous operator. Then
Pt,f(A) d=f 1meas {t g [0,T] : F (((s + iT, a)) G A} , A G B(H(D)), converges weakly to PzF-1 as T ^ to.
PROOF. The definitions of PT and PT,F imply that PT,F = PTF-1. Therefore, the continuity F
□
Let V > 0, md, for A e B (H2(Dy ,D)),
Pt,v(A) = ^meas {t g [0, T] : Z(s + iT, a) e A} ,
Pc,y(A) = mg (w e Q : Z(s, a, w) e A .
Lemma 4. Suppose that the set L(a, P) is linearly independe nt over Q, an d F : H 2(Dy, D) ^ H(Dy) ¿s a continuous operator. Then
PT,F,y (A) d=f ^meas {t g [0, T] : F (Z(s + iT, a)) e A} , A e B(H(Dy)), converges weakly to Pz,yF-1 as T ^ to.
Proof. Clearly, the function uy : H2(D) ^ H2(Dy, D) given bv the formula
uV(g1(s),g2(s)) = g1(s) n , g2(s) I , g1,g2 G H(D),
\ seDy /
is continuous, and, PT,y = PTu-1. Therefore, Lemmas 1 and 2 implv that PT,y converges weakly to Pz,V as T ^ to. Since PT)F)V = PT,VF-1, to have that PT)F)V converges weakly to Pz,yF-1 as T ^to. ' ' ' ' '
□
Now we consider the supports of the limit measures Pz, PzF-1, Pz,y and Pz,VF-1.
Lemma 5. Suppose that the set L(a, P) is linearly independe nt over Q. Then the support of the measure Pz is the set S x H(D).
Proof. Denote bv m1H and m2H the probability Haar measures on (Q1, B(Q1)) and (Q2, B(Q2)), respectively. Then we have that mH is the product of m1H and m2H, i.e., if A = A1 x A2, where A1 e B(Q1) md A2 e B(Q2), then
mg (A) = m1H (A1 )m2g (A2). (2)
The space H2(D) is separable, therefore, B(H2(D)) = B(H(D)) x B(H(D)). Thus, it suffices to consider the measure Pz on the sets A = A1 x A2, A1, A2 e H (D). It is known [20] that the support of the measure
m1H (W1 G Q1 : Z(s, W1) G A), A e B(H(D)) (3)
is the set S. The linear independence of L(a, P) implies that of the set L(a) = {log(m+a) : m e N0}. Therefore, the case r = 1 of Theorem 11 from [6] gives that the support of the measure
m2H (w2 e Q2 : Z(s,a,w2) e A), A e B(H(D)), (4)
H(D)
Pz(A) = mH (w e Q : ((s,a,w) e A) , A e B(H2(D)),
in view of (2), we have that, for A = Ai x A2,
Pz (A) = miH (wi e Qi : ((s, wi) e Ai) m2H (w2 e Q2 : Z (s, a, w2) e A2).
Therefore, the lemma follows from remarks on supports of the measures (3) and (4), and minimality property of a support.
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Lemma 6. Suppose that the set L(a, P) is linearly independent over Q, and F : H2(D) ^ H(D) is a continuous operator such that, for every open set G C H(D) the set (F-iG) n (S x H(D)) is non-empty. Then the support of the measure PzF-i is the whole of H(D).
Proof. We apply standard arguments. Let g e H(D) be an arbitrary element, and G be its
F F- i G
F-iG
S x H(D). Hence, by Lemma 5, Pz(F-iG) > 0. Therefore,
P<zF-i(G) = Pz(F-iG) > 0. Since g and G are arbitrary, this proves the lemma.
□
In what follows, the Mergelvan theorem on the approximation of analytic functions by polynomials will be exceptionally useful [17].
Lemma 7. Suppose that K C C is a compact subset with connected complement, and f (s) is a continuous function on K which is analytic in the interior of K. Then, for every e > 0; there exists p(s)
sup |f(s) — p(s)| < e.
Lemma 8. Suppose that the set L(a, P) is linearly independe nt over Q, an d V > 0. Then the support of Pz,y is the set Sy x H(D).
Proof. Let g be an arbitrary element of Sy x H(D), and G be its open neighborhood. The function uy defined in the proof of Lemma 4 is continuous. Therefore, by the definition of uy, the set u-iG is open and non-empty. Really, it is well known, see, for example, [8], that the
H(D)
p(s) p(s) e G
Since the polynomial p(s) is an entire function, p(s) also belongs to u-iG. Thus, the set u-iG is
S x H(D)
Pz(u-iG) > 0 Hence, Pz,y(G) = Pzu-i(G) = Pz(u-iG) > 0 Clearly, if (gi,g2) e S x H(D), then also (gi,g2) e Sy x H(D). Therefore,
mH (w e Q : ((s, a, w) e Sy x H(D)) ^ mH (w e Q : ((s, a, w) e S x H(D)) = 1.
Hence,
Pz,y (Sy x H(D)) = 1.
□
Lemma 9. Suppose that the set L(a, P) is linearly independent over Q. Let F : H2(Dy, D) ^ H(Dy) he a continuous operator such that, for each polynomial p = p(s), iAe set (F-1{p}) n (Sy x H(D)) is non-empty. Then the support of the measure Pz,yF-1 is the whole of H(Dy).
Proof. Let g be an arbitrary element of H(Dy), and G be its arbitrary open neighbourhood. Then, by Lemma 7, there exists a polynomial p(s) G G. Therefore, the hypotheses of the lemma imply that the set F-1G is open and contains an element of the set Sy x H(D). Thus, in virtue of Lemma 8, Pz,y(F-1G) > 0. From this, it follows that
Pc,VF-1(G) = Pzy (F-1G) > 0, and the lemma is proved because g and G are arbitrary.
□
Lemma 10. Suppose that the set L(a, P) is linearly independent over Q; and the operator F : H2(D) ^ H(D) satisfies the hypotheses of Theorem 6. Then the support of the measure PzF contains the closure of the set Hai)...,ar(D).
PROOF. Since F(S x H(D)) D Hai;...;ar(D), for each element g G Hai)...,ar(D), there exists an element (g1 ,g2) G S x H(D)) such that F(g1,g2) = g. If G is an arbitrary open neighborhood of g, then we have that the open set F-1G is an open neighborhood of a certain element of S x H(D). Therefore, in view of Lemma 5, Pz(F-1G) > 0. Hence,
PzF-1(G) = Pz(F-1G) > 0.
This shows that the element g lies in the support of the measure PzF-1. Since g is an arbitrary element of Hai)...,ar(D), we have that the support of PzF-1 contains the set Hai)...,ar(D). However, the support is a closed set, therefore, it contains the closure of Hai)...,ar(D).
□
Lemma 11. Suppose that the set L(a, P) is linearly independent over Q, and F : H2(D) ^ H(D) is a continuous operator. Then the support of PzF-1 is the closure of F(S x H(D)).
g F(S x H(D)) G
by Lemma 5, Pz(F-1G) > 0. Hence, PzF-1(G) > 0. Moreover, by Lemma 5 again,
PzF-1 (F(S x H(D))) = Pz (S x H(D)) = 1.
Therefore, the support of PzF-1 is the closure of F(S x H(D)).
" □
3. Proof of universality theorems
We will apply the equivalent of the weak convergence of probability measures in terms of continuity sets. We remind that A e B(X) is a continuity set of the probabilitv measure P on (X, B(X)) if P(dA) = 0, where dA is the boundary of A.
Lemma 12. Let Pn; n e N, and P be probability measures on (X, B(X)). Then Pn, as n ^ to, converges weakly to P if and only if, for every continuity set A of P,
lim P„(A) = P(A).
n^-rx
A proof of the lemma can be found infl], Theorem 2.1. Proof of Theorem 2. Put
Ge = ^ (gi,g2) e H (D) : sup |gi(s) — fi(s)| < e, sup |g2(s) — f2(s)| < e
I sg—i sg—
Then Ge is an open set in H2(D). Moreover,
dG£ = <j (gi,g2) e H (D) : sup |gi(s) — fi(s)| < e, sup |g2(s) — f2(s)| = e
sg—i sg—
U <J (gi,g2) e H2(D) : sup |gi(s) — fi(s)| = e, sup |g2(s) — f2(s)| < e seKi se—
^ (gi,g2) e H (D) : sup |gi(s) — fi(s)| = e, sup |g2(s) — f2(s)| = e i seKi se—
Therefore, if ei > 0 e2 > 0 and ei = e2, then dG£1 n dG£2 = 0. Hence, we have that Pz(dGe) > 0 for at most a countable set of values of e > 0. This means that the set Ge is a continuity set of Pz for all but at most countablv many e > 0. Therefore, by Lemmas 1 and 12,
lim ¿measjr e [0; T] : ((s + ir) e Ge) = Pz(Ge), t^^ 11 ~ J —
1
T ^^
or, by the definition of Ge,
lim ^measjr e [0; T] : sup |((s + ir) — fi(s)| < e,
1 L seKi
sup |((s + ir,a) — f2(s)| <4 = Pz(Ge) (5)
se— J ^
for all but at most countablv many e > 0. By Lemma 7, there exist polvnomials pi(s) and p2(s) such that
epi(s)| e
sup |fi(s) — epi(s)| <- (6)
and
sG—l ' 2
e
sup |f2(s) — p2(s)| < -. (7)
sg—2 2
In view of Lemma 5, {epi(s),p2(s)} is and element of the support of the measure Pz- Therefore, putting
G£ = j(gi,g2) e H2(D) : sup |gi(s) — epi(s)| < 2, sup |g2(s) — p2(s)| < 2) ,
i sg—i 2 sg-2 2 j
we obtain that Pz(Ge) > 0. Inequalities (6) and (7) show, that for (gi,g2) e Ge,
sup |gi(s) — fi(s)| < e
sg—i
and
sup |g2(s) — f2(s)| < e.
sg—2
Thus, we have that Ge C Ge. Hence, Pz(Ge) ^ Pz(Ge) > 0. This together with (5) proves the theorem.
□
Proof of Theorem 4. Define the set
GM = g e H(D) : sup |g(s) - f (s)| < e . I seK J
Then we have that G1;£ is a continuity set of the measure P^F-1 for all but at most countablv many e > 0.Hence, in view of Lemmas 3 and 12,
lim —meas {r G [0; T] : F (Z(s + ît, a)) G GM}
= lim — measit g [0; T] : sup |F(Z(s + ît), Z(s + ît, a)) - f (s)| < e) T L seK J
=PÇF-1(GM) (8)
for all but at most countablv many e > 0. By Lemma 7, there exists a polvnomial p(s) such that
Define
£
sup |f (s) - p(s)| < -. (9)
seK 2
GM = <j g g H(D) : sup |g(s) - f (s)|
seK -
The polynomial p(s), by Lemma 6, is an element of the support of the measure P^F 1- Hence, P<Z(G 1,e) > 0. Obviously, for g e G 1)£, by (9),
sup |g(s) - f (s)| < e.
seK
Therefore, G1)£ C G1)£, P^F-1(G1;£) ^ P^F-1(G1;£) > 0, and the theorem follows from (8).
' ' " ' " ' □
proof of Theorem 5. We follow the proof of Theorem 4, and use Lemma 4 in place of Lemma 3, and Lemma 9 in place of Lemma 6.
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Proof of Theorem 6. The case r = 1. By Lemma 7, there exists a polvnomial p(s) such that
£
sup |f (s) - p(s)| <-. (10)
seK 4
By hypotheses of the theorem, f (s) = a1 on K. Therefore, in view of (10), p(s) = a1 on K as well if e is small enough. Thus, we can define a continuous branch of log(p(s) — a1) which will be an analytic function in the interior of K. Using Lemma 7 once more, we find a polynomial p1 (s) such that
sup |p(s) - fl! - epi(s))| <e. (11)
seK 4
Now we put f (s) = Then f1(s) e H(D) and f1(s) = a^. Therefore, by Lemma 10, f1(s)
is an element of the support of the measure P^F-1. Define
Gm = { g e H (D) : sup |g(s) - f1(s)| <e
L seK 2
Then £i,£ is an open neighborhood of /^s), thus, PzF 1(Gi,e) > 0. Now consider the set
Gi,e = g e H (D) : sup |g(s) - / (s)| <e . I seK J
Similarly as in the proof of the above theorems, we observe that G1,e is an continuity set of the measure Pc F-1 for all but at most countablv many e > 0. Therefore, taking into account Lemmas 3 and 12, we have that
lim — meas {r e [0; T] : F (Z(s + ir, a)) e Gi,e )
T—^ T L )
= lim ¿meaJr e [0; T] : sup |F(Z(s + ir, a)) - /(s)| <4 = PzF-1 (Gi,e). (12)
T—<x T I _ J _
Clearly, by (10) and (11),
e
sup |/(s) - /i(s)| < -.
seK 2
Therefore, if g e G^e, then g e G^e, i-e., Gi,e C G^e- Since PzF-i(Gi,e) > 0, we have that PzF-i(G/i,e) > 0. This inequality together with (12) proves the theorem in the case r = 1. Now let r ^ 2. Define
G2,e = g e H(D) : sup |g(s) - /(s)| < e . I J
Since /(s) e Hai,...,ar(D), we have by Lemma 10, that /(s) is an element of the support of PzF-i. Moreover, G2,e is an open neighborhood of /(s). Therefore,
P<zF-i (G2,e) > 0. (13)
On the other hand, G2,e is a continuity set of the measure PzF-i for all but at most countablv many e > 0. Therefore, in view of Lemmas 3 and 12, and (12)
lim ^measj r e [0; T] : sup |F (Z (s + ir, a)) - / (s)| < el T—<^ T I seK " J
= lim ^meas {r e [0; T] : F (Z(s + ir, a)) e G2,e} = PzF-i(G2,e) > 0.
T—t —
□
Proof of Theorem 7. We repeat the proof of the case r ^ 2 of Theorem 6, and, in place of Lemma 10, we apply Lemma 11.
□
4. Conclusions
Z(s) Z(s, a)
transcendental or rational parameter a are universal in the Voronin sense, i.e., their shifts Z(s + ir) and Z(s + ir, a) r e R, approximate functions from wide classes. H. Mishou obtained a joint universality theorem for Z(s) and Z(s, a). He proved that the set of shifts (Z(s + ir), Z(s + ir, a)) a
In the paper, it is observed that the set of the above shifts has a positive density for all but at most countablv many values of e > 0, where e is accuracy of approximation.
Also, it is obtained that composite functions F(Z(s),Z(s, a)) for some classes of operators F
H(D)
shifts F(Z(s + ¿t), Z(s + ¿t, a)) approximating a given analytic function with accuracy e > 0 has a positive density for all but at most countablv many values of e.
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Vilnius University. Получено 27.06.2016 г. Принято в печать 12.09.2016 г.
