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
$$
\lim_{x\to\infty} \left(\sum_{p\leqslant x} 1\right)^{-1} \sum_{p\leqslant x}|a(p)|^2=\kappa>0,
$$
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.
Keywords:
Dirichlet series, Hurwitz zeta-function, periodic Hurwitz zeta-function, Selberg class, universality, weak convergence.
UDC:
511.3
Received: 09.01.2019
Accepted: 10.04.2019
Language: English
DOI:
10.22405/2226-8383-2018-20-1-46-65