A discrete universality theorem for periodic Hurwitz zeta-functions

A. Laurinčikas ; D. Mokhov

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 148-159

Résumé

In 1975, Sergei Mikhailovich Voronin discovered the universality of the Riemann zeta-function $\zeta(s)$, $s=\sigma+it$ , on the approximation of a wide class of analytic functions by shifts $\zeta(s+i\tau), \tau \in \mathbb{R}$. Later, it turned out that also some other zeta-functions are universal in the Voronin sense. If $\tau$ 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 $\zeta(s,\alpha;\mathfrak{a})$ is defined by the series with terms $a_m(m+\alpha)^{-s}$, where $0\alpha\leq1$ is a fixed number, and $\mathfrak{a}=\{a_m\}$ is a periodic sequence of complex numbers. It is proved that a wide class of analytic functions can be approximated by shifts $\zeta(s+ihk^{\beta_1} \log^{\beta_2}k, \alpha; \mathfrak{a})$ with $k=2,3,\dots$, where $h>0$ and $0\beta_11$, $\beta_2>0$ are fixed numbers, and the set $\{ \log(m+\alpha): m =0,1,2 \}$ 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. Bibliography: 15 titles.

Détail

Keywords: periodic Hurwitz zeta-function, space of analytic functions, limit theorem, universality.