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 Chebyshevskii Sb., 2019, Volume 20, Issue 1, Pages 46–65 (Mi cheb717)

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

A. Balčiūnasa, R. Macaitienėbc, D. Šiaučiūnasb

a Vilnius University, Lithuania
b Research Institute, Šiauliai University, Lithuania
c Šiauliai State College, Lithuania

Abstract: 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
$$\lim_{x\to\infty} (\sum_{p\leqslant x} 1)^{-1} \sum_{p\leqslant x}|a(p)|^2=\kappa>0,$$
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
$$(\mathcal{L}(\widetilde{s}), \zeta(s,\alpha_1; \mathfrak{a}_{11}), …,\zeta(s,\alpha_1; \mathfrak{a}_{1l_1}), …, \zeta(s,\alpha_r; \mathfrak{a}_{r1}), …, \zeta(s,\alpha_r; \mathfrak{a}_{rl_r})),$$
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
$$(f(\widetilde{s}), f_{11}(s),…, f_{1l_1}(s), …, f_{r1}(s), …, f_{rl_r}(s))$$
of analytic functions from a wide class by a collection of shifts
\begin{align*} (\mathcal{L}(\widetilde{s}+ikh), &\zeta(s+ikh_1,\alpha_1; \mathfrak{a}_{11}), …,\zeta(s+ikh_1,\alpha_1; \mathfrak{a}_{1l_1}), …, & \zeta(s+ikh_r,\alpha_r; \mathfrak{a}_{r1}), …, \zeta(s+ikh_r,\alpha_r; \mathfrak{a}_{rl_r})), \end{align*}
where $h, h_1, …, h_r$ are positive numbers, is considered. For this, the linear independence over the field of rational numbers for the set
$$\{(h\log p: p\in \mathbb{P}), ( h_j\log(m+\alpha_j): m\in \mathbb{N}_0, j=1,…, r), 2\pi\},$$
where $\mathbb{P}$ denotes the set of all prime numbers, is applied.

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

DOI: https://doi.org/10.22405/2226-8383-2018-20-1-46-65

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UDC: 511.3
Accepted:10.04.2019
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Citation: A. Balčiūnas, R. Macaitienė, D. Šiaučiūnas, “Joint discrete universality for $L$-functions from the Selberg class and periodic Hurwitz zeta-functions”, Chebyshevskii Sb., 20:1 (2019), 46–65

Citation in format AMSBIB
\Bibitem{BalMacSia19} \by A.~Bal{\v{c}}i{\=u}nas, R.~Macaitien{\.e}, D.~{\v S}iau{\v{c}}i{\=u}nas \paper Joint discrete universality for $L$-functions from the Selberg class and periodic Hurwitz zeta-functions \jour Chebyshevskii Sb. \yr 2019 \vol 20 \issue 1 \pages 46--65 \mathnet{http://mi.mathnet.ru/cheb717} \crossref{https://doi.org/10.22405/2226-8383-2018-20-1-46-65}