Joint discrete universality for Lerch zeta-functions

Abstract

After Voronin’s work of 1975, it is known that some of zeta and L-functions are universal in the sense that their shifts approximate a wide class of analytic functions. Two cases of shifts, continuous and discrete, are considered.

The present paper is devoted to the universality of Lerch zeta-functions L(λ,α,s), s = σ+it, which are defined, for σ > 1, by the Dirichlet series with terms e^2πiλm(m+α)^−s with parameters λ ∈ R and α, 0 < α ≤ 1, and by analytic continuation elsewhere. We obtain joint discrete universality theorems for Lerch zeta-functions. More precisely, a collection of analytic functions f₁(s),...,f_r(s) simultaneously is approximated by shifts L(λ₁,α₁,s+ikh),...,L(λ_r,α_r,s+ikh), k = 0,1,2,..., where h > 0 is a fixed number. For this, the linear independence over the field of rational numbers for the set {(log(m + α_j) : m ∈N₀, j = 1,...,r), }is required. For the proof, probabilistic limit theorems on the weak convergence of probability measures in the space of analytic function are applied.