Joint discrete universality for $L$-functions from the Selberg class and periodic Hurwitz zeta-functions

A. Balčiūnas ; R. Macaitienė ; D. Šiaučiūnas

Čebyševskij sbornik, Tome 20 (2019) no. 1, pp. 46-65

Résumé

The Selberg class $\mathcal{S}$ contains Dirichlet series $$ \mathcal{L}(s)= \sum_{m=1}^\infty \frac{a(m)}{m^s}, \quad s=\sigma+it, $$ such that, for every $\varepsilon>0$, $a(m)\ll_\varepsilon m^\varepsilon$; there exists an integer $k\geqslant 0$ such that $(s-1)^k \mathcal{L}(s)$ is an entire function of finite order; the functions $\mathcal{L}$ satisfy a functional equation connecting $s$ with $1-s$, and have a product representation over prime numbers. Steuding introduced a subclass $\widetilde{\mathcal{S}}$ of $\mathcal{S}$ with additional condition $$ \lim_{x\to\infty} \left(\sum_{p\leqslant x} 1\right)^{-1} \sum_{p\leqslant x}|a(p)|^2=\kappa>0, $$ where $p$ runs prime numbers. Let $\alpha$, $0\alpha\leqslant 1$, be a fixed parameter, and $\mathfrak{a}=\{a_m: m\in \mathbb{N}_0\}$ be a periodic sequence of complex numbers. The second object of the paper is the periodic Hurwitz zeta-function $\zeta(s,\alpha;\mathfrak{a})$ which is defined, for $\sigma>1$, by the Dirichlet series $$ \zeta(s,\alpha; \mathfrak{a})=\sum_{m=0}^\infty \frac{a_m}{(m+\alpha)^s}, $$ and is meromorphically continued to the whole complex plane. The paper is devoted to the discrete universality of the collection $$ \left(\mathcal{L}(\widetilde{s}), \zeta(s,\alpha_1; \mathfrak{a}_{11}), \dots,\zeta(s,\alpha_1; \mathfrak{a}_{1l_1}), \dots, \zeta(s,\alpha_r; \mathfrak{a}_{r1}), \dots, \zeta(s,\alpha_r; \mathfrak{a}_{rl_r})\right), $$ where $\mathcal{L}(\widetilde{s})\in \widetilde{S}$, and $\zeta(s,\alpha_j; \mathfrak{a}_{jl_j})$ are periodic Hurwitz zeta-functions, i. e., to the simultaneous approximation of a collection $$ \left(f(\widetilde{s}), f_{11}(s),\dots, f_{1l_1}(s), \dots, f_{r1}(s), \dots, f_{rl_r}(s)\right) $$ of analytic functions from a wide class by a collection of shifts \begin{align*} \big(\mathcal{L}(\widetilde{s}+ikh), \zeta(s+ikh_1,\alpha_1; \mathfrak{a}_{11}), \dots,\zeta(s+ikh_1,\alpha_1; \mathfrak{a}_{1l_1}), \dots, \\ \zeta(s+ikh_r,\alpha_r; \mathfrak{a}_{r1}), \dots, \zeta(s+ikh_r,\alpha_r; \mathfrak{a}_{rl_r})\big), \end{align*} where $h, h_1, \dots, h_r$ are positive numbers, is considered. For this, the linear independence over the field of rational numbers for the set $$ \left\{\left(h\log p: p\in \mathbb{P}\right), \left( h_j\log(m+\alpha_j): m\in \mathbb{N}_0,\, j=1,\dots, r\right), 2\pi\right\}, $$ where $\mathbb{P}$ denotes the set of all prime numbers, is applied.

Détail

Keywords: Dirichlet series, Hurwitz zeta-function, periodic Hurwitz zeta-function, Selberg class, universality, weak convergence.