Joint discrete universality for Lerch zeta-functions

A. Laurinčikas ; A. Mincevič

Čebyševskij sbornik, Tome 19 (2018) no. 1, pp. 138-151

Résumé

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 $L(\lambda, \alpha, s)$, $s= \sigma+it $, which are defined, for $ \sigma > 1$, by the Dirichlet series with terms $ e^{2 \pi i \lambda m}(m+ \alpha)^{-s} $ with parameters $\lambda \in \mathbb{R} $ and $\alpha$, $0 \alpha \leqslant 1$, and by analytic continuation elsewhere. We obtain joint discrete universality theorems for Lerch zeta-functions. More precisely, a collection of analytic functions $ f_{1}(s), \dots, f_{r}(s) $ simultaneously is approximated by shifts $L(\lambda_{1},\alpha_{1},s+ikh),\dots, L(\lambda_{r},\alpha_{r},s+ikh)$, $k=0,1,2,\dots$, where $h>0$ is a fixed number. For this, the linear independence over the field of rational numbers for the set $\left \{ (\log (m+ \alpha_{j}): m \in \mathbb{N}_{0},\; j=1,\dots,r),\frac{2 \pi}{h} \right\}$ is required. For the proof, probabilistic limit theorems on the weak convergence of probability measures in the space of analytic function are applied.

Détail

Keywords: Lerch zeta-function, Mergelyan theorem, space of analytic functions, universality, weak convergence.