On a Dirichlet Series Connected to a Periodic Hurwitz Zeta-Function with Transcendental and Rational Parameter

. In the paper, we construct an absolutely convergent Dirichlet series which in the mean is close to the periodic Hurwitz zeta-function, and has the universality property on the approximation of a wide class of analytic functions.


Introduction
Let a = {a m : m ∈ N 0 = N ∪ {0}} be a periodic sequence of complex numbers with minimal period q ∈ N, 0 < α ⩽ 1 fixed parameter, and s = σ + it a complex variable.The periodic Hurwitz zeta-function ζ(s, α; a) is defined, for σ > 1, by the Dirichlet series If a m ≡ 1, then we have the classical Hurwitz zeta-function ζ(s, α) which has the meromorphic continuation to the whole complex plane with unique simple pole at the point s = 1 with residue 1.The periodicity of the sequence a implies, for σ > 1, the equality ζ(s, α; a) = 1 q s q−1 l=0 a l ζ s, l + α q , (1.1) therefore, the function ζ(s, α; a) also can be meromorphically continued to the whole complex plane with a simple pole at the point s = 1 with residue If a = 0, then ζ(s, α; a) is an entire function.Clearly, ζ(s, 1; {1}) = ζ(s) is the Riemann zeta-function.Thus, the periodic Hurwitz zeta-function is a generalization of the classical Hurwitz and Riemann zeta-functions.Analytical properties of ζ(s, α; a) are governed by the sequence a, and, in particular, by arithmetic of the parameter α.
The function ζ(s, α; a), for some classes of the parameter α, as other zetafunctions, is universal, i. e., its shifts ζ(s + iτ, α; a), τ ∈ R, approximate all analytic functions defined in the strip D = {s ∈ C : 1/2 < σ < 1}.Note that the phenomenon of universality for zeta-functions was discovered by Voronin, in [6] he proved the universality of the Riemann zeta-function.Later, the universality of some other zeta-functions was obtained.By the Linnik-Ibragimov conjecture, all functions in some half-plane given by Dirichlet series, analytically continuable to the left of the absolute convergence half-plane and satisfying some natural growth conditions are universal in the Voronin sense.However, till now the universality of some zeta-functions are not known.The universality of the function ζ(s, α; a) with transcendental α was considered in [3], and with rational α in [4].Denote by K the class of compact subsets of the strip D, and by H(K) with K ∈ K the class of continuous functions on K that are analytic in the interior of K. Let measA stand for the Lebesgue measure of a measurable set A ⊂ R. Then the universality of the function ζ(s, α; a) is described in the following theorem.
It is easily seen that the transcendence of α can be replaced by the linear independence over the field of rational numbers Q of the set The universality of the function ζ(s, α; a) with algebraic irrational α is an open problem.
In the strip D, the function ζ(s, α; a) is defined by using (1.1) and analytic continuation of the Hurwitz zeta-function.Therefore, it is not easy to derive an information on the function f (s) from the inequality It is more convenient to use an absolutely convergent Dirichlet series in place of ζ(s, α; a).This paper is devoted to a realization of the mentioned idea.
Let θ > 1/2 be a fixed number, and, for m ∈ N 0 and u > 0, Since |a m | ⩽ C, m ∈ N 0 , with some C < ∞, the series is absolutely convergent for σ > σ 0 with arbitrary finite σ 0 .Denote by B(X) the Borel σ-field of the space X, by H(D) the space of analytic on D functions endowed with the topology of uniform convergence on compacta, and let γ = {s ∈ C : |s| = 1}.Define the set Ω = m∈N0 γ m , where γ m = γ for all N 0 .With the product topology and pointwise multiplication, the torus Ω is a compact topological Abelian group.Therefore, on (Ω, B(Ω)), the probability Haar measure m H exists, and we obtain a probability space (Ω, B(Ω), m H ). Denote by ω m the mth component of an element ω ∈ Ω, m ∈ N 0 , and, on the probability space (Ω, B(Ω), m H ), define the H(D)-valued random element The latter series, for almost all ω ∈ Ω is uniformly convergent on compact subsets of the strip D. Let P ζ,α,a be the distribution of the random element ζ(s, α, ω; a), i. e., Theorem 2. Suppose that the number α is transcendental, and exists for all but at most countably many ε > 0.
For the case of rational α, define one more infinite-dimensional torus Ω 1 = p∈P γ p , where γ p = γ for all p ∈ P, and P is the set of all prime numbers.Analogically to Ω, we have the probability space (Ω 1 , B(Ω 1 ), m 1H ), where m 1H is the probability Haar measure on (Ω 1 , B(Ω 1 )).Denote by ω 1 (p) the p-th component of an element ω 1 ∈ Ω 1 , p ∈ P, and extend ω 1 (p) to the set N by the formula Denote by χ Dirichlet characters, and by L(s, χ) the corresponding Dirichlet L-functions The functions L(s, χ) have meromorphic continuation to the whole complex plane.Let φ(m) be the totient Euler function.On the probability space where and z denotes the complex conjugate of z ∈ C.
We will derive Theorems 2 and 3 from Theorem 1 by using the approximation of ζ(s, α; a) by ζ u T (s, α; a) in the mean.

Some estimates
In this section, we prove the following equality.

Limit theorems
Let P n , n ∈ N, and P be probability measures on (X, B(X)).Recall that P n converges weakly to P as n → ∞ if, for every real bounded continuous function g on X, There are equivalents of weak convergence in terms of some classes of sets.We will use the following, see, for example, [1].
Lemma 2. P n converges weakly to P as n → ∞ if and only if, for every closed set F ⊂ X, lim sup n→∞ P n (F ) ⩽ P (F ).
For A ∈ B(H(D)), define Lemma 3. Suppose that α is a transcendental number.Then P T,α,a converges weakly to as T → ∞.
The lemma was obtained in [2].The transcendence of α can be replaced by the linear independence over Q for the set L(α).
Let V > 0 be an arbitrary fixed number, and H(D V ) the space of analytic on D V functions.Denote by P 1 T,α,a the analogue of P T,α,a for the space H(D V ).
and (lb + a, bq) = 1 for all l = 0, 1, . . ., q − 1.Then P 1 T,α,a converges weakly to Proof of the lemma is given in [4], Theorem 3. Now we will prove the analogues of Lemmas 3 and 4 for the function Lemma 5. Suppose that α is a transcendental number, and u T → ∞ and u T ≪ T 2 as T → ∞.Then Q T,α,a converges weakly to P ζ,α,a as T → ∞.

Proof.
For where {K l : l ∈ N} ⊂ D is a sequence of compact embedded subsets such that D = ∞ l=1 K l , and if K ⊂ D is a compact set, then K lies in some K l .Then ρ is a metric in the space H(D) inducing its topology of uniform convergence on compacta.
Let ξ T be the random variable defined on a certain probability space ( Ω, A, µ) and uniformly distributed in the interval [0, T ].Define the H(D)-valued random elements Let F be an arbitrary closed set of the space H(D) and ε > 0. Then the set is closed as well.Therefore, by Lemmas 3 and 2, lim sup

Using the inclusion
Since the density of the random variable ξ T is 1/T on [0, T ], and 0 elsewhere, for every measurable function h : Ω → X, we have Therefore, by the definitions of X T,α,a and Y T,α,a , and similarly µ{Y T,α;a ∈ F } = Q T,α,a (F ).
Moreover, in view of Lemma 1 and the definition of the metric ρ, for every ε > 0, These three equalities and (3.1) give lim inf Letting ε → +0 together with Lemma 2 proves the lemma.⊓ ⊔ A) be an analogue of Q T,α,a for the space H(D V ).Using Lemma 4 and repeating a proof of Lemma 5 lead to the following assertion.Lemma 6. Suppose that the hypotheses of Lemma 4 on the parameter α and the sequence a are satisfied, and u T → ∞ and u T ≪ T 2 as T → ∞.Then Q 1 T,α,a converges weakly to P 1 ζ,α,a as T → ∞.
Lemmas 5 and 6 imply the weak convergence for the corresponding measures on (R, B(R)).To see this, recall a preservation property of weak convergence.Let P be a probability measure on (X, B(X)), and h : X → X 1 be a (B(X), B(X 1 ))-measurable mapping.Then the measure P defines the unique probability measure P h −1 on (X 1 , B(X 1 )) by the formula Lemma 7. Suppose that P n , n ∈ N, and P are probability measures on (X, B(X)), h : X → X 1 a continuous mapping, and P n converges weakly to P as n → ∞.Then also P n h −1 converges weakly to P h −1 as n → ∞.
Proof of the lemma can be found in [1].
where K and f (s) are from Theorems 2 and 3.
Lemma 8. Suppose that α is a transcendental number, u T → ∞ and u T ≪ T 2 as T → ∞, and K ⊂ D is a compact set.Then Q T,α,a converges weakly to as T → ∞.
Proof.The mapping h : H(D) → R given by h(g) = sup s∈K |g(s) − f (s)| is continuous.Therefore, the lemma follows from Lemmas 5 and 7. ⊓ ⊔ Similarly, Lemmas 6 and 7 imply the following statement.
Lemma 9. Let K ⊂ D be a compact set, u T → ∞ and u T ≪ T 2 as T → ∞.
Then, under hypotheses of Lemma 4 for the parameter α and sequence a, Q T,α,a converges weakly to as T → ∞.

Proof of universality
Let G n (x), n ∈ N and G(x) be the distribution functions.We recall that G n converges weakly to for every continuity point x of G(x).Moreover, every distribution function has no more than a countable set of discontinuity points.
It is well known that the weak convergence of probability measures on (R, B(R)) is equivalent to that of the corresponding distribution functions.

Proof.
(Proof of Theorem 2).In [3], it is obtained that the support of the measure P ζ,α,a is the space H(D).Therefore, where