ЧЕБЫШЕВСКИЙ СБОРНИК Том 17 Выпуск 3 МОДИФИКАЦИЯ ТЕОРЕМЫ МИШУ

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1 A MODIFICATION OF THE MISHOU THEOREM 35 ЧЕБЫШЕВСКИЙ СБОРНИК Том 7 Выпуск 3 УДК 59.4 МОДИФИКАЦИЯ ТЕОРЕМЫ МИШУ А. Лауринчикас, (г. Вильнюс, Литва), Л. Мешка (г. Вильнюс, Литва) Аннотация В 2007 г. Г. Мишу доказал совместную теорему унивурсальности для дзета-функции Римана ζ(s) и дзета-функции Гурвица ζ(s, α) с трансцендентным параметром α об одновременном приближении пары функций из широкого класса аналитических функций сдвигами (ζ(s + iτ), ζ(s + iτ, α)), τ R. 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 ε > 0, за исключением счетного множества значений ε, где ε точность приближения. Результаты аналогичного типа также получены для сложных функций F ( ζ(s), ζ(s, α)) для некоторых классов операторов F в пространстве аналитических функций. Ключевые слова: дзета-функция Гурвица, дзета-функция Римана, пространство аналитических функций, универсальность. Библиография: 2 названий. MODIFICATION OF THE MISHOU THEOREM A. Laurinčikas, (Vilnius, Lithuania), L. Meška (Vilnius, Lithuania) Abstract 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 (ζ(s + iτ), ζ(s + iτ, α)) with transcendental α, τ R, and that the set of such τ 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 ε > 0, where ε is the accuracy of approximation. We also obtain similar results for composite functions F (ζ(s), ζ(s, α)) for some classes of operator F. Keywords: universality. Bibliography: 2 titles.. Introduction Hurwitz zeta-function, Riemann zeta-function, space of analytic functions, Let ζ(s), s = σ + it, be the Riemann zeta-function. In 975, S. M. Voronin discovered [2] the universality property of ζ(s) which means that a wide class of non-vanishing analytic functions can be approximated by shifts ζ(s + iτ), τ R. The non-vanishing of approximated functions is connected to the existence of Euler s product over primes for ζ(s). Now let 0 < α be a fixed parameter, and ζ(s, α) denotes the Hurwitz zeta-function which is defined, for α >, by the series ζ(s, α) = m=0 (m + α) s,

2 36 A. LAURINČIKAS, L. MEŠKA and can be meromorphically continued to the whole complex plane. Clearly, ζ(s, ) = ζ(s), and ζ ( s, ) 2 = (2 s )ζ(s). For other values of the parameter α, the function ζ(s, α) has no Euler product. It is well known that the Hurwitz zeta-function with transcendental or rational, 2 parameter α is also universal in the above sense, however, its shifts ζ(s + iτ, α) approximate not necessarily non-vanishing analytic functions. The universality of ζ(s, α) with algebraic irrational α is an open problem. Some other zeta-functions are also universal in the Voronin sense. The universality for zetafunctions of certain cusp forms was obtained in [2], for periodic zeta-functions was studied in [20] and [5], while the works [2], [4] and [5] are devoted to periodic Hurwitz zeta-functions. Universality theorems for Lerch zeta-functions can be found in []. A very good survey on universality of zetafunctions is given in [6]. In [9], 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 (ζ(s), ζ(s, α)) with transcendental α. For the statement of the Mishou theorem, we need some notation. Let D = { s C : 2 < σ < }. Denote by K the class of compact subsets of the strip D with connected complements. Moreover, let H(K), K K, be the class of continuous functions on K which are analytic in the interior of K, and let H 0 (K), K 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 R. Then H. Mishou proved [9] the following theorem. Theorem. Suppose that α is transcendental number. Let K, K 2 K, and f (s) H 0 (K ), f 2 (s) H(K 2 ). Then, for every ε > 0 inf T T meas τ [0; T ] : sup ζ(s + iτ) f (s) < ε, > 0. sup 2 ζ(s + iτ, α) f 2 (s) < ε Mixed joint universality theorems are also proved in [3], [7] and [0]. Our aim is to replace " inf"in Theorem by "". In the case of the function ζ(s), this was done in [3] and [8], and, in the case of ζ(s, α), a similar theorem was obtained in [4]. Let P be the set of all prime numbers, N 0 = N {0}, and L(α, P) = {(log(m + α) : m N 0 ), (log p : p P)}. Theorem 2. Suppose that the set L(α, P) is linearly independent over the field of rational numbers Q. Let K, K 2 K, and f (s) H 0 (K ), f 2 (s) H(K 2 ). Then the it T T meas τ [0; T ] : sup ζ(s + iτ) f (s) < ε, sup ζ(s + iτ, α) f 2 (s) < ε > 0 2 exists for all but at most countably many ε > 0. For example, if α is transcendental, then the set L(α, 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 (ζ(s), ζ(s, α)) with some operators F : H 2 (D) H(D). Let S = {g H(D) : g(s) 0 or g(s) 0}. Then, for example in [9], the following assertion was obtained.

3 A MODIFICATION OF THE MISHOU THEOREM 37 Theorem 3. Suppose that α is transcendental, and that F : H 2 (D) H(D) is a continuous operator such that, for every open set G H(D), the set (F G) (S H(D)) is non-empty. Let K K and f(s) H(D). Then, for every ε > 0, inf T T meas τ [0; T ] : sup F (ζ(s + iτ), ζ(s + iτ, α)) f(s) < ε > 0. More general results are obtained in [0]. Clearly, the transcendence of α in Theorem 3 can be replaced by a linear independence over Q of the set L(α, P). Therefore, we will prove the following theorem. Theorem 4. Suppose that the set L(α, P) is linearly independent over Q, and that F, K and f(s) are the same as in Theorem 3. Then the it T T meas τ [0; T ] : sup exists for all but at most countably many ε > 0. F (ζ(s + iτ), ζ(s + iτ, α)) f(s) < ε > 0 () Now, let V be an arbitrary positive number, D V = { s C : 2 < σ <, t < V} and S V = {g H(D V ) : g(s) 0 or g(s) 0}. For brevity, we use the notation H 2 (D V, D) = H(D V ) H(D). Theorem 5. Suppose that the set L(α, P) is linearly independent over Q, and that K and f(s) are the same as in Theorem 3, and V > 0 is such that K D V. Let F : H 2 (D V, D) H(D V ) be a continuous operator such that, for each polynomial p = p(s), the set (F {p}) (S V H(D V )) is non-empty. Then the it () exists for all but at most countably many ε > 0. For example, Theorem 5 implies the modified universality of the functions c ζ(s) + c 2 ζ(s, α) and c ζ (s) + c 2 ζ (s, α) with c, c 2 C {0}. Let a,..., a r be arbitrary distinct complex numbers, and H a,...,a r (D) = { g H(D) : (g(s) a j ) H(D), j =,..., r }. Theorem 6. Suppose that the set L(α, P) is linearly independent over Q, and F : H 2 (D) H(D) is a continuous operator such that F (S H(D)) H a,...,a r (D). When r =, let K K, and f(s) H(K) and f(s) a on K. Then the it () exists for all but at most countably many ε > 0. If r 2, K D is an arbitrary compact subset, and f(s) H a,...,a r (D), then the it () exists for all but at most countably many ε > 0. The case r = with a = 0 shows that, for F (g (s), g 2 (s)) = e g (s)+g 2 (s), the it () exists for all but at most countably many ε > 0. If r = 2 and a =, a 2 =, then, for example, for F (g (s), g 2 (s)) = cos(g (s) + g 2 (s)) and f(s) H, (D), the it () exists for all but at most countably many ε > 0. Theorem 7. Suppose that the set L(α, P) is linearly independent over Q, F : H 2 (D) H(D) is a continuous operator, K D is a compact subset, and f(s) F (S H(D)). Then the it () exists for all but at most countably many ε > 0.

4 38 A. LAURINČIKAS, L. MEŠKA 2. Lemmas In this section, we present probabilistic theorems on the weak convergence of probability measures in the space of analytic functions. Let γ = {s C : s = }, and Ω = p γ p and Ω 2 = γ m, where γ p = γ for all p P, and γ m = γ for all m N 0. By the Tikhonov theorem, the tori Ω and Ω 2 with the product topology and operation of pointwise multiplication are compact topological Abelian groups. Similarly, Ω = Ω Ω 2 is also a compact topological Abelian group. Therefore, denoting by B(X) the Borel σ-field of the space X, we have that, on (Ω, B(Ω)), the probability Haar measure m H can be defined, and we obtain the probability space (Ω, B(Ω), m H ). Denote by ω (p) and ω 2 (m) the projections of ω Ω and ω 2 Ω 2 to the coordinate spaces γ p, p P, and γ m, m N 0, respectively, and, on the probability space (Ω, B(Ω), m H ), define the H 2 (D)-valued random element ζ(s, ω), ω = (ω, ω 2 ) Ω, by the formula where and Moreover, let m=0 ζ(s, α, ω) = (ζ(s, ω ), ζ(s, α, ω 2 )), ζ(s, ω ) = p ζ(s, α, ω 2 ) = ( ω ) (p) p s m=0 ω 2 (m) (m + α) s. P ζ (A) = m H ( ω Ω : ζ(s, α, ω) A ), A B(H 2 (D)), i.e., P ζ is the distribution of the random element ζ(s, ω). We set ζ(s, α) = ( ζ(s), ζ(s, α) ), and P T (A) def = T meas { τ [0, T ] : ζ(s + iτ, α) A }, A B(H 2 (D)). Lemma. Suppose that the set L(α, P) is linearly independent over Q. Then P converges weakly to P ζ as T. Proof. The lemma for transcendental α is proved in [9], Theorem, however, the transcendence of α is used only for the linear independence of the set L(α, P). Let X and X 2 be two metric spaces, and let the function u : X X 2 be (B(X ), B(X 2 ))- measurable. Then every probability measure P on (X, B(X )) induces on (X 2, B(X 2 )) the unique probability measure P u (A) given by the formula P u = P (u A), A B(X 2 ). It is well known that if u is a continuous function, then it is (B(X ), B(X 2 ))-measurable. In the sequel, the following property of weakly convergent probability measures will be very useful. Lemma 2. Suppose that P n, n N, and P are probability measures on (X, B(X )), the function u : X X 2 is continuous, and P n converges weakly to P as n. Then P n u also converges weakly to P u as n.

5 A MODIFICATION OF THE MISHOU THEOREM 39 The lemma is Theorem 5. from []. Lemma 3. Suppose that the set L(α, P) is linearly independent over Q, and F : H 2 (D) H(D) is a continuous operator. Then P T,F (A) def = T meas { τ [0, T ] : F ( ζ(s + iτ, α) ) A }, A B(H(D)), converges weakly to P ζ F as T. Proof. The definitions of P T and P T,F imply that P T,F = P T F. Therefore, the continuity of F and Lemmas and 2 prove the lemma. Let V > 0, and, for A B ( H 2 (D V, D) ), P T,V (A) = T meas { τ [0, T ] : ζ(s + iτ, α) A }, P ζ,v (A) = m H ( ω Ω : ζ(s, α, ω) A ). Lemma 4. Suppose that the set L(α, P) is linearly independent over Q, and F : H 2 (D V, D) H(D V ) is a continuous operator. Then P T,F,V (A) def = T meas { τ [0, T ] : F ( ζ(s + iτ, α) ) A }, A B(H(D V )), converges weakly to P ζ,v F as T. Proof. Clearly, the function u V : H 2 (D) H 2 (D V, D) given by the formula ( ) u V (g (s), g 2 (s)) = g (s), g 2 (s), g, g 2 H(D), s DV is continuous, and, P T,V = P T u V. Therefore, Lemmas and 2 imply that P T,V converges weakly to P ζ,v as T. Since P T,F,V = P T,V F, we have that P T,F,V converges weakly to P ζ,v F as T. Now we consider the supports of the it measures P ζ, P ζ F, P ζ,v and P ζ,v F. Lemma 5. Suppose that the set L(α, P) is linearly independent over Q. Then the support of the measure P ζ is the set S H(D). Proof. Denote by m H and m 2H the probability Haar measures on (Ω, B(Ω )) and (Ω 2, B(Ω 2 )), respectively. Then we have that m H is the product of m H and m 2H, i.e., if A = A A 2, where A B(Ω ) and A 2 B(Ω 2 ), then m H (A) = m H (A )m 2H (A 2 ). (2) The space H 2 (D) is separable, therefore, B(H 2 (D)) = B(H(D)) B(H(D)). Thus, it suffices to consider the measure P ζ on the sets A = A A 2, A, A 2 H(D). It is known [20] that the support of the measure m H (ω Ω : ζ(s, ω ) A), A B(H(D)) (3)

6 40 A. LAURINČIKAS, L. MEŠKA is the set S. The linear independence of L(α, P) implies that of the set L(α) = {log(m+α) : m N 0 }. Therefore, the case r = of Theorem from [6] gives that the support of the measure is the set H(D). Since m 2H (ω 2 Ω 2 : ζ(s, α, ω 2 ) A), A B(H(D)), (4) P ζ (A) = m H ( ω Ω : ζ(s, α, ω) A ), A B(H 2 (D)), in view of (2), we have that, for A = A A 2, P ζ (A) = m H (ω Ω : ζ(s, ω ) A ) m 2H (ω 2 Ω 2 : ζ(s, α, ω 2 ) A 2 ). Therefore, the lemma follows from remarks on supports of the measures (3) and (4), and minimality property of a support. Lemma 6. Suppose that the set L(α, P) is linearly independent over Q, and F : H 2 (D) H(D) is a continuous operator such that, for every open set G H(D), the set (F G) (S H(D)) is non-empty. Then the support of the measure P ζ F is the whole of H(D). Proof. We apply standard arguments. Let g H(D) be an arbitrary element, and G be its any open neighborhood. Since the operator F is continuous, the set F G is open, too. Therefore, by the hypothesis of the lemma, F G is an open neighborhood of a certain element of the set S H(D). Hence, by Lemma 5, P ζ (F G) > 0. Therefore, P ζ F (G) = P ζ (F G) > 0. Since g and G are arbitrary, this proves the lemma. In what follows, the Mergelyan theorem on the approximation of analytic functions by polynomials will be exceptionally useful [7]. Lemma 7. Suppose that K 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 ε > 0, there exists a polynomial p(s) such that sup f(s) p(s) < ε. Lemma 8. Suppose that the set L(α, P) is linearly independent over Q, and V > 0. Then the support of P ζ,v is the set S V H(D). Proof. Let g be an arbitrary element of S V H(D), and G be its open neighborhood. The function u V defined in the proof of Lemma 4 is continuous. Therefore, by the definition of u V, the set u V G is open and non-empty. Really, it is well known, see, for example, [8], that the approximation in the space H(D) coincides with the uniform approximation on compact sets with connected complements. Therefore, by Lemma 7, there exists a polynomial p(s) such that p(s) G. Since the polynomial p(s) is an entire function, p(s) also belongs to u V G. Thus, the set u V G is non-empty, and is an open neighborhood of an element from S H(D). Therefore, by Lemma 5, P ζ (u V G) > 0. Hence, P ζ,v (G) = P ζ u V (G) = P ζ(u V G) > 0. Clearly, if (g, g 2 ) S H(D), then also (g, g 2 ) S V H(D). Therefore, Hence, m H ( ω Ω : ζ(s, α, ω) SV H(D) ) m H ( ω Ω : ζ(s, α, ω) S H(D) ) =. P ζ,v (S V H(D)) =.

7 A MODIFICATION OF THE MISHOU THEOREM 4 9. Suppose that the set L(α, P) is linearly independent over Q. Let F : H 2 (D V, D) Lemma H(D V ) be a continuous operator such that, for each polynomial p = p(s), the set (F {p}) (S V H(D)) is non-empty. Then the support of the measure P ζ,v F is the whole of H(D V ). Proof. Let g be an arbitrary element of H(D V ), and G be its arbitrary open neighbourhood. Then, by Lemma 7, there exists a polynomial p(s) G. Therefore, the hypotheses of the lemma imply that the set F G is open and contains an element of the set S V H(D). Thus, in virtue of Lemma 8, P ζ,v (F G) > 0. From this, it follows that P ζ,v F (G) = P ζ,v (F G) > 0, and the lemma is proved because g and G are arbitrary. Lemma 0. Suppose that the set L(α, P) is linearly independent over Q, and the operator F : H 2 (D) H(D) satisfies the hypotheses of Theorem 6. Then the support of the measure P ζ F contains the closure of the set H a,...,a r (D). Proof. Since F (S H(D)) H a,...,a r (D), for each element g H a,...,a r (D), there exists an element (g, g 2 ) S H(D)) such that F (g, g 2 ) = g. If G is an arbitrary open neighborhood of g, then we have that the open set F G is an open neighborhood of a certain element of S H(D). Therefore, in view of Lemma 5, P ζ (F G) > 0. Hence, P ζ F (G) = P ζ (F G) > 0. This shows that the element g lies in the support of the measure P ζ F. Since g is an arbitrary element of H a,...,a r (D), we have that the support of P ζ F contains the set H a,...,a r (D). However, the support is a closed set, therefore, it contains the closure of H a,...,a r (D).. Suppose that the set L(α, P) is linearly independent over Q, and F : H 2 (D) Lemma H(D) is a continuous operator. Then the support of P ζ F is the closure of F (S H(D)). Proof. Let g be an arbitrary element of F (S H(D)), and G is its any neighborhood. Then, by Lemma 5, P ζ (F G) > 0. Hence, P ζ F (G) > 0. Moreover, by Lemma 5 again, P ζ F (F (S H(D))) = P ζ (S H(D)) =. Therefore, the support of P ζ F is the closure of F (S 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 B(X) is a continuity set of the probability measure P on (X, B(X)) if P ( A) = 0, where A is the boundary of A. Lemma 2. Let P n, n N, and P be probability measures on (X, B(X)). Then P n, as n, converges weakly to P if and only if, for every continuity set A of P, P n(a) = P (A). n

8 42 A. LAURINČIKAS, L. MEŠKA A proof of the lemma can be found in[], Theorem 2.. Proof of Theorem 2. Put G ε = (g, g 2 ) H 2 (D) : sup g (s) f (s) < ε, sup g 2 (s) f 2 (s) < ε. 2 Then G ε is an open set in H 2 (D). Moreover, G ε = (g, g 2 ) H 2 (D) : sup g (s) f (s) < ε, sup g 2 (s) f 2 (s) = ε 2 (g, g 2 ) H 2 (D) : sup g (s) f (s) = ε, sup g 2 (s) f 2 (s) < ε 2 (g, g 2 ) H 2 (D) : sup g (s) f (s) = ε, sup g 2 (s) f 2 (s) = ε. 2 Therefore, if ε > 0, ε 2 > 0 and ε ε 2, then G ε G ε2 =. Hence, we have that P ζ ( G ε ) > 0 for at most a countable set of values of ε > 0. This means that the set G ε is a continuity set of P ζ for all but at most countably many ε > 0. Therefore, by Lemmas and 2, or, by the definition of G ε, T T meas τ [0; T ] : ζ(s + iτ) G ε = P ζ (G ε ), T T meas τ [0; T ] : sup ζ(s + iτ) f (s) < ε, sup 2 ζ(s + iτ, α) f 2 (s) < ε = P ζ (G ε ) (5) for all but at most countably many ε > 0. By Lemma 7, there exist polynomials p (s) and p 2 (s) such that (6) and sup f (s) e p (s) < ε 2 sup f 2 (s) p 2 (s) < ε 2 2. (7) In view of Lemma 5, { e p(s), p 2 (s) } is and element of the support of the measure P ζ. Therefore, putting ^G ε = (g, g 2 ) H 2 (D) : sup g (s) e p(s) < ε 2, sup g 2 (s) p 2 (s) < ε, 2 2 we obtain that P ζ ( ^G ε ) > 0. Inequalities (6) and (7) show, that for (g, g 2 ) ^G ε, and sup g (s) f (s) < ε sup g 2 (s) f 2 (s) < ε. 2 Thus, we have that ^G ε G ε. Hence, P ζ (G ε ) P ζ ( ^G ε ) > 0. This together with (5) proves the theorem.

9 A MODIFICATION OF THE MISHOU THEOREM 43 Proof of Theorem 4. Define the set G,ε = g H(D) : sup g(s) f(s) < ε. Then we have that G,ε is a continuity set of the measure P ζ F for all but at most countably many ε > 0.Hence, in view of Lemmas 3 and 2, T = T T meas { τ [0; T ] : F ( ζ(s + iτ, α) ) } G,ε T meas τ [0; T ] : sup =P ζ F (G,ε ) F (ζ(s + iτ), ζ(s + iτ, α)) f(s) < ε (8) for all but at most countably many ε > 0. By Lemma 7, there exists a polynomial p(s) such that Define ^G,ε = sup f(s) p(s) < ε 2. (9) g H(D) : sup g(s) f(s) < ε 2 The polynomial p(s), by Lemma 6, is an element of the support of the measure P ζ F. Hence, P ζ ( ^G,ε ) > 0. Obviously, for g ^G,ε, by (9), sup g(s) f(s) < ε. Therefore, ^G,ε G,ε, P ζ F (G,ε ) P ζ F ( ^G,ε ) > 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. Proof of Theorem 6. The case r =. By Lemma 7, there exists a polynomial p(s) such that. sup f(s) p(s) < ε 4. (0) By hypotheses of the theorem, f(s) a on K. Therefore, in view of (0), p(s) a on K as well if ε is small enough. Thus, we can define a continuous branch of log(p(s) a ) which will be an analytic function in the interior of K. Using Lemma 7 once more, we find a polynomial p (s) such that sup p(s) a e p(s) ) < ε 4. () Now we put f (s) = e p (s) + a. Then f (s) H(D) and f (s) a. Therefore, by Lemma 0, f (s) is an element of the support of the measure P ζ F. Define G,ε = g H(D) : sup g(s) f (s) < ε 2.

10 44 A. LAURINČIKAS, L. MEŠKA Then G,ε is an open neighborhood of f (s), thus, P ζ F (G,ε ) > 0. Now consider the set ^G,ε = g H(D) : sup g(s) f(s) < ε. Similarly as in the proof of the above theorems, we observe that G,ε is an continuity set of the measure P ζ F for all but at most countably many ε > 0. Therefore, taking into account Lemmas 3 and 2, we have that T = T Clearly, by (0) and (), T meas τ [0; T ] : F ( ζ(s + iτ, α) ) ^G,ε T meas τ [0; T ] : sup F (ζ(s + iτ, α)) f(s) < ε = P ζ F ( ^G,ε ). (2) sup f(s) f (s) < ε 2. Therefore, if g G,ε, then g ^G,ε, i.e., G,ε ^G,ε. Since P ζ F (G,ε ) > 0, we have that P ζ F ( ^G,ε ) > 0. This inequality together with (2) proves the theorem in the case r =. Now let r 2. Define G 2,ε = g H(D) : sup g(s) f(s) < ε Since f(s) H a,...,a r (D), we have by Lemma 0, that f(s) is an element of the support of P ζ F. Moreover, G 2,ε is an open neighborhood of f(s). Therefore,. P ζ F (G 2,ε ) > 0. (3) On the other hand, G 2,ε is a continuity set of the measure P ζ F for all but at most countably many ε > 0. Therefore, in view of Lemmas 3 and 2, and (2) T T meas τ [0; T ] : sup F (ζ(s + iτ, α)) f(s) < ε = T T meas { τ [0; T ] : F ( ζ(s + iτ, α) ) } G 2,ε = Pζ F (G 2,ε ) > 0. Proof of Theorem 7. We repeat the proof of the case r 2 of Theorem 6, and, in place of Lemma 0, we apply Lemma. 4. Conclusions It was well known that the Riemann zeta-function ζ(s) and Hurwitz zeta-function ζ(s, α) with transcendental or rational parameter α are universal in the Voronin sense, i.e., their shifts ζ(s + iτ) and ζ(s + iτ, α), τ R, approximate functions from wide classes. H. Mishou obtained a joint universality theorem for ζ(s) and ζ(s, α). He proved that the set of shifts (ζ(s + iτ), ζ(s + iτ, α)) with transcendental α approximating a pair of given analytic functions has a positive lower density.

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