A weighted limit theorem for periodic Hurwitz zeta-function

. In the paper, a weighted limit theorem for weakly convergent probability measures on the complex plane for the periodic Hurwitz zeta function is obtained


Introduction
Let a = {a m : m ∈ N 0 }, N 0 = N ∪ {0}, be a periodic with the least period k ∈ N sequence of complex numbers, and α ∈ (0, 1] be a fixed parameter. The periodic Hurwitz zeta-function ζ(s, α; a), s = σ + it, is defined, for σ > 1, by Dirichlet series ζ(s, α, a) = where ζ(s, α) = ∞ n=0 1 (n + α) s , σ > 1, is the classical Hurwitz zeta-function. Since the function ζ(s, α) has a simple pole at s = 1 with residue 1, equality (1) gives analytic continuation for ζ(s, α; a) to the whole complex plane, except maybe, for a simple pole at s = 1. If then the function ζ(s, α; a) is entire, while, in the case a = 0, the point s = 1 is a simple pole with residue a. Denote by B(S) the class of Borel sets of the space S, and by meas{A} the Lebesgue measure of a measurable set A ⊂ R. Suppose that α is transcendental. Then in [2], by the way, it was obtained that, for σ > 1 2 , probability measure 1 T meas t ∈ [0, T ]: ζ(σ + it, α; a) ∈ A , A ∈ B(C), converges weakly to the explicitly given probability measure on (C, B(C)) as T → ∞.
The aim of this note is to prove a weighted limit theorem on the complex plane for the function ζ(s, α; a). Let w(t) be a positive function of bounded variation on Also, we require that, for σ > 1 2 , σ = 1, and all v ∈ R, the estimate should be satisfied. Denote by I A (t) the indicator function of a set A, and define the probability measure

Theorem 1.
Suppose that α is transcendental, σ > 1 2 , and that the weight function satisfies the condition (2). Then, on (C, B(C)), there exists a probability measure P σ such that the measure P T,σ,w converges weakly to P σ as T → ∞.

Auxiliary rezults
We start with a weighted limit theorem on the infinite-dimensional torus. Let where γ m = {s ∈ C: |s| = 1} for all m ∈ 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 can be defined. Define the probability measure

Lemma 1.
Suppose that α is transcendental. Then the probability measure Q T,w converges weakly to m H as T → ∞.
Proof. Denote by Z the set of all integers. Then the dual group (the character group) where only a finite number of integers k m , m ∈ N 0 , are distinct from zero, acts on Ω by Therefore, the Fourier transform g T,w (k) of the measure Q T,w is Since α is transcendental, the set {log(m + α): m ∈ N 0 } is linearly independent over the field of rational numbers. Therefore, ∞ m=0 k m log(m+α) = 0 if and only if k = 0.
where, as above, only a finite number of integers k m are distinct from zero. This, together with (3), shows that and the lemma follows from a continuity theorem on compact groups. Now let σ 1 > 1 2 be fixed, and, for m, n ∈ N 0 , v n (m, α) = e −( m+α n+α ) σ 1 .
Then it is easy to show that the series converges absolutely for σ > 1 2 . Consider the probability measure P T,n,σ,w (A) = 1 u T T0 w(t)I {t:ζn (σ+i·t,α,a)∈A} dt, A ∈ B(C).

Lemma 2.
Suppose that α is transcendental and σ > 1 2 . Then, on (C, B(C)), there exists a probability measure P n,σ such that the measure P T,n,σ,w converges weakly to P n,σ as T → ∞.
Proof. Define the function h n,σ : Ω → C by the formula The absolute convergence of the series (4) implies the continuity of the function h n,σ . Since h n,σ (m + α) −it : m ∈ N 0 = ζ n (σ + it, α; a), hence, using Theorem 5.1 from [1] and Lemma 2 we obtain that the measure P T,n,σ,w converges weakly to m H h −1 n,σ as T → ∞. For the proof of Theorem 1, it remains to pass from the function ζ n (s, α; a) to ζ(s, α; a). For this, the following statement will be applied.
The tightness of {P n,σ : n ∈ N} implies its relative compactness. Therefore, there exists a sequence {P n k ,σ } ⊂ {P n,σ } such that P n k ,σ converges weakly to a certain probability measure P σ on (C, B(C)) as k → ∞. Let θ = θ T be a random variable on a certain probability space ( Ω, B( Ω), P) having the distribution Suppose that X n (σ) is a complex-valued random variable with the distribution P n,σ . Then the above remark implies the relation where D − → denotes the convergence in distribution. Define X T,n (σ) = ζ n (σ + iθ T , α; a).