A DISCRETE UNIVERSALITY THEOREM FOR PERIODIC HURWITZ ZETA-FUNCTIONS

Abstract

In 1975, Sergei Mikhailovich Voronin discovered the universality of the Riemann zetafunction ζ(s), s = σ + it , on the approximation of a wide class of analytic functions by shifts ζ(s + iτ ), τ ∈ R. Later, it turned out that also some other zeta-functions are universal in the Voronin sense. If τ takes values from a certain descrete set, then the universality is called discrete. In the present paper, the discrete universality of periodic Hurwitz zeta-functions is considered. The periodic Hurwitz zeta-function ζ(s, α; a) is defined by the series with terms am(m + α)−s, where 0 < α ≤ 1 is a fixed number, and a = {am} is a periodic sequence of complex numbers. It is proved that a wide class of analytic functions can be approximated by 
shifts ζ(s+ihkβ1 logβ2 k, α; a) with k = 2, 3, ..., where h > 0 and 0 < β1 < 1, β2 > 0 are fixed numbers, and the set {log(m+α) : m = 0, 1, 2} is linearly independent over the field of rational numbers. It is obtained that the set of such k has a positive lower density. For the proof, properties of uniformly distributed modulo 1 sequences of real numbers are applied.