A Weighted Version of the Mishou Theorem

In 2007, H. Mishou obtained a joint universality theorem for the Riemann and Hurwitz zeta-functions ζ(s) and ζ(s, α) with transcendental parameter α on the approximation of a pair of analytic functions by shifts (ζ(s+ iτ), ζ(s+ iτ, α)), τ ∈ R. In the paper, the Mishou theorem is generalized for the set of above shifts having a weighted positive lower density. Also, the case of a positive density is considered.


Introduction
The Riemann zeta-function ζ(s), s = σ + it, and the Hurwitz zeta-function ζ(s, α) with parameter 0 < α 1 are defined, for σ > 1, by the Dirichlet series ζ(s) = ∞ m=1 1 m s and ζ(s, α) = general. The definitions of ζ(s) and ζ(s, α) are similar, however, their analytic properties are quite different. For example, since the function ζ(s), for σ > 1, has the Euler product over primes , ζ(s) = 0 in the half-plane σ > 1, while the function ζ(s, α) has zeros in that half plane if α = 1 or 1/2. On the other hand, the functions ζ(s) and ζ(s, α) have a common feature, they are universal in the sense that their shifts ζ(s + iτ ) and ζ(s + iτ, α) approximate wide classes of analytic functions. We recall that universality of the function ζ(s) was discovered by S.M. Voronin in [31]. For a statement of the Voronin theorem, it is convenient to use the following notation. For D = {s ∈ C : 1/2 < σ < 1}, denote by 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 H 0 (K) the subclass of H(K) of non-vanishing functions on K. Then the modern version of the Voronin theorem, see, for example, [1,6,13,30] asserts that, for every K ∈ K, f (s) ∈ H 0 (K), and ε > 0, The latter inequality shows that there are infinitely many shifts ζ(s + iτ ) approximating a given function f (s) ∈ H 0 (K). Here measA denotes the Lebesgue measure of a measurable set A ⊂ R. Universality of the Hurwitz zeta-function is a more complicated problem. At the moment, the following result is known. Suppose that α is a transcendental or rational = 1, 1/2. Then, for every K ∈ K, f (s) ∈ H(K), and ε > 0, The case of rational α was obtained by Voronin [32] and B. Bagchi [1], while the case of transcendental α was treated by S.M. Gonek [6], and, by a different method, in [23]. In [14], the transcendence of α was replaced by a weaker condition on the linear independence of the set L(α) = {log(m + α) : m ∈ N 0 = N ∪ {0}} over the field of rational numbers Q.
The aim of this paper, is a joint weighted universality theorem for the functions ζ(s) and ζ(s, α). The weighted universality of zeta-functions was began to study in [12]. In weighted universality theorems, the positivity of a lower density of the shifts approximating a given analytic function is replaced by the positivity of that weighted analogue. Let w(τ ) be positive function for τ T 0 > 0 such that and, for every interval [a, b] ⊂ [T 0 , ∞), the variation V b a w satisfies the inequality V b a w cw(a) with certain c > 0. Moreover, let I(A) denote the indicator function of the set A. Under the above hypotheses on the weight function w, it was obtained in [12] that, for every K ∈ K, f (s) ∈ H 0 (K), and ε > 0, A weighted discrete universality for ζ(s) was proved in [25]. Weighted universality theorems for periodic zeta-functions were obtained in [26,27].
A weighted universality theorem for the Hurwitz zeta-function was proved in [3]. Denote by W the above class of weight functions. Theorem 1. Suppose that α is transcendental and w ∈ W . Let K ∈ K and f (s) ∈ H(K). Then, for every ε > 0, The main result of this paper is the following weighted theorem.
For the proof of Theorem 2, we will use the probabilistic approach based on weak convergence of probability measures in the space of analytic functions.

A weighted limit theorem on the product of two tori
In what follows, we denote by B(X) the Borel σ-field of the space X, by P the set of all prime numbers, and N 0 = N ∪ {0}.
In this section, we will consider the weak convergence for Theorem 3. Suppose that α is transcendental and w ∈ W . Then Q T,w converges weakly to the Haar measure m H as T → ∞.
Proof. The characters of the group Ω are of the form where the sign " " means that only a finite number of integers k p and l m are distinct from zero. Therefore, the Fourier transform g T,w (k, l), k = (k p : k p ∈ Z, p ∈ P), l = (k p : l m ∈ Z, m ∈ N 0 ), of Q T,w is defined by Therefore, by the definition of Q T,w , Suppose that (k, l) = (0, 0). Then Actually, if the latter inequality is not true, then From this, it follows that is a rational number. However, this contradicts the transcendence of α. If all l m = 0, then p∈P k p log p = 0 because the set {log p : p ∈ P} is linearly independent over the field of rational numbers. Thus, (2.3) is true. Now, by (2.1), we find Then the latter series are absolutely convergent for σ > 1/2, see [13,23], respectively. For brevity, let ζ n (s, α) = (ζ n (s), ζ n (s, α)) .
Consider the function u n : Ω → H 2 (D) given by u n (ω) = ζ n (s, ω, α). Since the above seeries are absolutely convergent for σ > 1/2, the function u n (ω) is continuous. For A ∈ B(H 2 (D)), define Then we have P T,n,w (A) = Q T,w (u −1 A). Thus, the equality P T,n,w = Q T,w u −1 is true. This, the continuity of u n , Theorem 3 together with Theorem 5.1 of [4] lead to the following theorem.
Theorem 4. Suppose that α is transcendental and w ∈ W . Then P T,n,w converges weakly to the measure V n def = m H u −1 n as T → ∞. The measure V n plays an important role in the proof of the limit theorem for where ζ(s, α) = (ζ(s), ζ(s, α)) . From the proof of the Mishou theorem [29], the following properties of V n follows. On the probability space (Ω, B(Ω), m H ), define the H 2 (D)-valued random element and let P ζ be the distribution of ζ(s, ω, α), i. e., P ζ (A) = m H ω ∈ Ω : ζ(s, ω, α) ∈ A , A ∈ B(H 2 (D)).
Moreover, let S = {g ∈ H(D) : g(s) = 0 or g(s) ≡ 0}. Under the above notation, we have Lemma 1. Suppose that α is transcendental. Then V n converges weakly to P ζ as n → ∞. Moreover, the support of P ζ is the set S × H(D).
To prove that P T,w , as T → ∞, also converges weakly to the measure P ζ , some approximation of ζ(s, α) by ζ n (s, α) is needed.

A limit theorem for ζ(s, α)
Now we are ready to prove the weak convergence for P T,w as T → ∞.
Theorem 6. Suppose that α is transcendental and w ∈ W . Then P T,w converges weakly to the measure P ζ as T → ∞.
Proof. On a certain probability space with measure µ, define a random variable θ T,w by Consider the H 2 (D)-valued random element X T,n,w = X T,n,w (s) = ζ n (s + iθ T,w , α).
Then, in view of Theorem 4, where Y n is the H 2 (D)-valued random element with the distribution V n . Lemma 1 implies the relation Moreover, an application of Theorem 5 shows that, for every ε > 0, w(τ )ρ ζ(s+iτ, α), ζ n (s+iτ, α) dτ =0, (5.2) where the H 2 (D)-valued random element X T,w = X T,w (s) is defined by X T,w (s) = ζ(s + iθ T,w , α). Now, relations (5.1)- (5.2) show that all hypotheses of Theorem 4.2 from [4] are satisfied. Therefore, we obtain that and this is equivalent to the assertion of the theorem.

Proof of universality
Theorem 2 follows easily from Theorem 6 and the Mergelyan theorem on the approximation of analytic functions by polynomials [28].
Proof. (Proof of Theorem 2). By the Mergelyan theorem, there exist polynomials p 1 (s) and p 2 (s) such that Define the set We observe that, in virtue of Lemma 1, (e p1(s) , p 2 (s)) is an element of the support of the measure P ζ . Since G ε is an open neighbourhood of an element of the support of P ζ , the inequality is true. Therefore, using the equivalent of the weak convergence of probability measures in terms of open sets and taking into account Theorem 6, we have Hence, by the definitions of P T,w and G ε , Then the boundaries ∂Ĝ ε1 and ∂Ĝ ε2 do not intersect for different positive ε 1 and ε 2 . This shows that the setĜ ε is a continuity set of the measure P ζ for all but at most countably many ε > 0. Therefore, using the equivalent of weak convergence of probability measures in terms of continuity sets, we obtain by Theorem 6 that lim T →∞ P T,w (Ĝ ε ) = P ζ (Ĝ ε ) (6.4) for all but at most countably many ε > 0. Moreover, inequalities (6.1) imply the inclusion G ε ⊂Ĝ ε . Thus, by (6.2), the inequality P ζ (Ĝ ε ) > 0 holds. This, the definitions of P T,w andĜ ε , and (6.4) prove the second assertion of the theorem.