On Approximation of Analytic Functions by Periodic Hurwitz Zeta-Functions

Abstract

The periodic Hurwitz zeta-function ζ(s, α; a), s = σ +it, with parameter 0 < α ≤ 1 and periodic sequence of complex numbers a = {am } is defined, for σ > 1, by series sum from m=0 to ∞ am / (m+α)s, and can be continued moromorphically to the whole complex plane. It is known that the function ζ(s, α; a) with transcendental orrational α is universal, i.e., its shifts ζ(s + iτ, α; a) approximate all analytic functions defined in the strip D = { s ∈ C : 1/2 σ < 1. In the paper, it is proved that, for all 0 < α ≤ 1 and a, there exists a non-empty closed set Fα,a of analytic functions on D such that every function f ∈ Fα,a can be approximated by shifts ζ(s + iτ, α; a).

Joint Discrete Approximation of a Pair of Analytic Functions by Periodic Zeta-Functions

“…If the above quantities are zero, then the corresponding zetafunctions are entire. The approximation of analytic functions by the functions ζ(s; a) and ζ(s, α; b) was studied in [8,26,28,29] and [2,7,18,22,24,25,27], respectively. The first joint results for a pair of functions ζ(s; a), ζ(s, α; b) has been obtained in [9].…”

A Discrete Version of the Mishou Theorem Related to Periodic Zeta-Functions

“…Discrete universality for ζ(s; a) can be found in [3,13]. Universality of ζ(s, α; b) with various types of the parameter α was considered in [8,11,28]. A version of the Mishou theorem for periodic zeta-functions ζ(s; a) and ζ(s, α; b) was obtained in [12].…”