A Weighted Universality Theorem for Periodic Zeta-Functions

The periodic zeta-function ζ(s; a), s = σ + it is defined for σ > 1 by the Dirichlet series with periodic coefficients and is meromorphically continued to the whole complex plane. It is known that the function ζ(s; a), for some sequences a of coefficients, is universal in the sense that its shifts ζ(s + iτ ; a), τ ∈ R, approximate a wide class of analytic functions. In the paper, a weighted universality theorem for the function ζ(s; a) is obtained.


Introduction
Let s = σ + it be a complex variable, and a = {a m : m ∈ N} be a periodic sequence of complex numbers with minimal period k ∈ N. The periodic zetafunction ζ(s; a) is defined, for σ > 1, by the Dirichlet series ζ(s; a) = ∞ m=1 a m m s .
The Hurwitz zeta-function ζ(s, α) with parameter α, 0 < α 1, is given, for If the later quantity is equal to zero, then the function ζ(s; a) is entire one.
In 1975, S.M. Voronin discovered [16] the universality of the Riemann zetafunction ζ(s) = ζ(s, 1) on the approximation of a wide class of analytic functions by shifts ζ(s + iτ ), τ ∈ R. After Voronin's work, various authors observed that some other zeta-functions also are universal in the Voronin sense. The attention was also devoted to the periodic zeta-function. The first universality results for periodic zeta-function was obtained by B. Bagchi in [1] and [2], and by different methods in [13] and [14]. We will recall Theorem 11.8 from [14]. Denote by K the class of compact subsets of the strip D = s ∈ C : 1 2 < σ < 1 with connected complements, and by H(K), K ∈ K, the class of continuous functions on K which are analytic in the interior of K. Let meas A stand for the Lebesgue measure of a measurable set A ⊂ R. Theorem 1. [14]. Suppose that a m is not a multiple of a character mod k satisfying a m = 0 for (m, k) > 1. Let K ∈ K and f (s) ∈ H(K). Then, for every ε > 0, In [14], also the upper bounds for the density of universality of ζ(s; a) were obtained.
We note that the assumptions of Theorem 1 imply that the sequence a is not multiplicative. We recall that the sequence a is multiplicative if a mn = a m a n for all coprime m, n ∈ N. The universality of ζ(s; a) with multiplicative sequence a was obtained in [11]. Denote by H 0 (K), K ∈ K, the class of continuous non-vanishing functions on K, which are analytic in the interior of K. Theorem 2. [11]. Suppose that the sequence is multiplicative and ∞ α=1 |a p α | p α 2 c < 1 for all primes p. Let K ∈ K and f (s) ∈ H 0 (K). Then the assertion of Theorem 1 is true.
The universality of periodic zeta-functions is not a simple problem. It turns out, as it was observed in [5], that not all periodic zeta-functions are universal in the Voronin sense. Moreover, in [5], a new restricted universality property for ζ(s; a) was introduced. For K ∈ K, the height h(K) of K is defined by Then in [5] the following theorem has been obtained.
Theorem 3. There exists a positive constant c = c(a) such that, for every K ∈ K of height h(K) c, every f (s) ∈ H(K) and every ε > 0, the inequality of Theorem 1 is true.
Also, in [5], the necessary and sufficient conditions of the universality for ζ(s; a) with prime k were obtained. In [15], the universality of the function ζ(s; a) with prime k satisfying the condition where ϕ(k) is the Euler function, was considered. A joint universality theorem for periodic zeta-functions was proved in [9]. The joint universality of periodic and periodic Hurwitz zeta-functions was studied in [4] and [7].
The aim of this paper is to discuss the weighted universality of the function ζ(s; a). The universality of this type for the Riemann zeta-function was considered in [6].
Let w(t) be a positive function of bounded variation on Let I A stand for the indicator function of the set A. Then the following statement holds.
Theorem 4. Suppose that the function w(t) satisfies all above conditions, and that the sequence a is as in Theorem 2. Let K ∈ K and f (s) ∈ H 0 (K). Then, for every ε > 0, We note that in [6] a certain additional condition generalizing the classical Birkhoff-Khintchine theorem was used. We do not need that condition.

Limit theorems
Denote by B(X) the Borel σ-field of the space X, and by H(D) the space of analytic functions on D equipped with the topology of uniform convergence on compacta. This section is devoted to a limit theorem on weakly convergent probability measures in the space (H(D), B(H(D))).
Let γ def = {s ∈ C : |s| = 1} be the unit circle on the complex plane. Define Ω = p γ p , where γ p = γ for all primes p. With the product topology and pointwise multiplication, the torus Ω is a compact topological Abelian group. Therefore, the probability Haar measure m H on (Ω, B(Ω)) can be defined. This gives the probability space (Ω, B(Ω), m H ). Let ω(p) stand for the projection of ω ∈ Ω to the coordinate space γ p . Moreover, let We note that the latter product converges uniformly on compact subsets of D for almost all ω ∈ Ω. Moreover, for almost all ω ∈ Ω, We start with a weighted limit theorem on the torus. Let, for A ∈ B(Ω), where P is the set of all prime numbers.
Lemma 1. Q T,w converges weakly to the Haar measure m H as T → ∞.
Proof. Denote by g T,w (k), k = (k p : k p ∈ Z, p ∈ P), the Fourier transform of the measure Q T,w . Since characters χ of Ω are of the form where only a finite number of integers k p are distinct from zero, we have that Hence, by the definition of Q T,w , where only a finite number of integers k p are distinct from zero. It is well known that the set {log p : p ∈ P} is linearly independent over the field of rational numbers Q. Therefore, in view of (2.1), and, for k = 0, using properties of w(t), we find that Since the series for ζ n (s, ω; a) is absolutely convergent for σ > 1 2 [11], the function u n is continuous one. We setP n = m H u −1 n , where, for A ∈ B(H(D)), Lemma 2. P T,n,w converges weakly toP n as T → ∞.

Proof.
Clearly, u n p −iτ : p ∈ P = ζ n (s + iτ ; a). Therefore, This, the continuity of u n , Lemma 1 and Theorem 5.1 of [3] prove the lemma. Now we will approximate ζ(s; a) by ζ n (s; a). Let, for g 1 , g 2 ∈ H(D), where {K l : l ∈ N} is a sequence of compact subsets of the strip D such that D = ∪ ∞ l=1 K l , K l ⊂ K l+1 for all l ∈ N, and if K ⊂ D is a compact, then K ⊂ K l for some l. Then ρ is a metric on H(D) which induces its topology of uniform convergence on compacta.

Proof.
Consider the series where 1 2 < σ < 1 and σ > θ. Suppose that σ 1 2 and 2π |t| πx. Then, see, for example, [8], Let K be a compact subset of the strip D. Then, using (2.5) and the contour integration, we obtain that withσ < 0  Theorem 5. The measure P T,w converges weakly to P ζ as T → ∞. Moreover, the support of P ζ is the set {g ∈ H(D) : g(s) = 0 or g(s) ≡ 0}.

Proof.
On a certain probability space (Ω, F, P), define a random variable η T by By Lemma 2, we have that P T,n,w converges weakly toP n as T → ∞. Define X T,n = X T,n (s) = ζ n (s + iη T ; a).
Then the assertion of Lemma 2 can be written as where D − → denotes the convergence in distribution, andX n is the H(D)-valued random element having the distributionP n . We will prove that the family of probability measures {P n : n ∈ N} is tight. Since the series for ζ n (s; a) is absolutely convergent for σ > 1 2 , it is not difficult to see that, for σ > 1 2 , Then the set H ε is uniformly bounded on every compact set of D, thus it is compact subset of H(D). Moreover, by (2.9) P(X n (s) ∈ H ε ) 1 − ε for all n ∈ N. Hence,P n (H ε ) 1 − ε for all n ∈ N, i.e., the family {P n } is tight. Therefore, by the Prokhorov theorem [3], it is relatively compact. Hence, every sequence of {P n } contains a subsequence {P nr } such thatP nr converges weakly to a certain probability measure P on (H(D), B(H(D))), i.e.,  [3] show that Hence, P T,w converges weakly to P as T → ∞. The latter relation also implies, that the measure P in (2.10) is independent of the choice of subsequenceP nr . ThusX orP n converges weakly to P . This means that P T,w , as T → ∞, converges weakly to the limit measure ofP n , as n → ∞. It remains to identify the measure P .
In [11], the measure was considered, and it was obtained that P T converges weakly to P ζ as T → ∞. Moreover, in the proving process, it was observed that P T , as P T,w , also converges weakly to the limit measure ofP n as n → ∞, i.e, to the measure P . From these remarks, we have that P coincides with P ζ . In [11] it is also noted that the support of the measure P ζ is the set {g ∈ H(D) : g(s) = 0 or g(s) ≡ 0}. The theorem is proved.

Universality
The proof of Theorem 4 is quite standard and is based on Theorem 5 and the Mergelyan theorem on the approximation of analytic functions by polynomials [12].