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 Chebyshevskii Sb., 2016, Volume 17, Issue 1, Pages 148–159 (Mi cheb460)

A discrete universality theorem for periodic Hurwitz zeta-functions

A. Laurinčikas, D. Mokhov

Faculty of Mathematics and Informatics, Vilnius University, Naugarduko str. 24, LT-03225 Vilnius, Lithuania

Abstract: 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 $k=2,3,…$, where $h>0$ and $0<\beta_1<1$, $\beta_2>0$ 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.

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

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UDC: 519.14
Received: 11.12.2015
Accepted:10.03.2016
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Citation: A. Laurinčikas, D. Mokhov, “A discrete universality theorem for periodic Hurwitz zeta-functions”, Chebyshevskii Sb., 17:1 (2016), 148–159

Citation in format AMSBIB
\Bibitem{LauMok16} \by A.~Laurin{\v{c}}ikas, D.~Mokhov \paper A discrete universality theorem for periodic Hurwitz zeta-functions \jour Chebyshevskii Sb. \yr 2016 \vol 17 \issue 1 \pages 148--159 \mathnet{http://mi.mathnet.ru/cheb460} \elib{https://elibrary.ru/item.asp?id=25795077} 

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