A two-dimensional limit theorem for Lerch zeta-functions. II

. We prove a two-dimensional limit theorem for Lerch zeta-functions with transcendental and rational parameters. σ + it be a complex variable, and 0 < λ < 1 and 0 < α 6 1 be ﬁxed parameters. The Lerch-zeta function L ( λ, α, s ) is deﬁned, for σ > 1 , by the Dirichlet series L λ, α, of analytically continued to an entire function.

Probabilistic limit theorems for the function L(λ, α, s) with transcendental and rational parameter α were proved in [2] while the case of algebraic irrational parameter α was considered in [3,4,6,7].
Let P denote the set of all prime numbers. Since α 1 is transcendental, the set {log(m + α 1 ): m ∈ N 0 } is linearly independent over the field of rational numbers Q. The set {log p: p ∈ P} is also linearly independent over Q. Therefore, it is not difficult to prove that the set is linearly independent as well. This leads to the following lemma. [8].) Suppose that the number α 1 is transcendental. Then the probability measure

Lemma 1. (See
converges weakly to the Haar measure m H as T → ∞.

Lemma 2.
Suppose that min(σ 1 , σ 2 ) > 1 2 . Then on (C 2 , B(C 2 )), there exists a probability measure P n such that both the measures P T,n and P T,n converge weakly to P n as T → ∞.
Proof. Define the function h n : Ω → C 2 by the formula h n (ω) = L(λ, α, ω, σ). Then the function is continuous, and Therefore, P T,n = Q T h −1 n . This, the continuity of h n , Lemma 1, and Theorem 5.1 of [1] show that P T,n converges weakly to m H h −1 n as T → ∞. By the same arguments, using the invariance of the Haar measure m H , we obtain that the measure P T,n also converges weakly to m H h −1 n as T → ∞. ⊓ ⊔ For z 1 = (z 11 , z 12 ), z 2 = (z 21 , z 22 ) ∈ C 2 , let ρ 2 (z 1 , z 2 ) = ( Proof. The lemma follows from corresponding one-dimensional relations, see [2], and from the definition of the metric ρ 2 . ⊓ ⊔ Define one more probability measure Liet. mat. rink. LMD darbai, 52:7-12, 2011. i i 10 D.R. Genienė Lemma 4. Suppose that min(σ 1 , σ 2 ) > 1 2 . Then on (C 2 , B(C 2 )), there exists a probability measure P such that both the measures P T and P T converge weakly to P as T → ∞.
Proof. We remind that P n is the limit measure in Lemma 2. First we observe that the family of probability measures {P n : n ∈ N 0 } is tight. This is obtained by using Lemma 2 and the fact that By the Prokhorov theorem, the tightness implies a relative compactness. Therefore, there exists a subsequence {P n k } ⊂ {P n } such that P n k converges weakly to a certain probability measure P on (C 2 , B(C 2 )) as k → ∞.
Let A be a continuity set of the measure P . Then by Lemma 4 we have that On the probability space (Ω, B(Ω), m H ), define the random variable ξ by Then, clearly, the expectation where P L is the distribution of the random element L. Let, for t ∈ R, a t = (((m + α 1 ) −it : m ∈ N 0 ), (p −it : p ∈ P)), and ϕ t (ω) = ωa t , ω ∈ Ω. Then {ϕ t : t ∈ R} is a one-parameter group of measurable measure preserving transformations on Ω. Since the set L(α 1 ) is linearly independent over Q, by a standard method can be proved that the group {ϕ t : t ∈ R} is ergodic. Hence, the random process ξ(ϕ t (ω)) is ergodic as well. Therefore, the Birkhoff-Khintchine theorem shows that On the other hand, by the definition of ξ and ϕ t we find that 1 T T 0 ξ ϕ t (ω) dt = ν T L(λ, α, ω, σ + it) ∈ A .