On Joint Universality of the Riemann and Hurwitz Zeta-Functions

Abstract

In 2007, H. Mishou proved the universality theorem on the joint approximation of a pair of analytic functions by the shifts $(\zeta(s+i\tau),\zeta(s+i\tau,\alpha))$ of the Riemann zeta-function and the Hurwitz zeta-function with transcendental parameter $\alpha$. In this paper, we obtain a similar theorem on approximation by the shifts $(\zeta_{u_N}(s+ikh_1),\zeta_{u_N}(s+ikh_2,\alpha))$, $k\in\mathbb{N}\cup\{0\}$, $h_1,h_2>0$, where $\zeta_{u_N}(s)$ and $\zeta_{u_N}(s,\alpha)$ are absolutely convergent Dirichlet series, and, as $N\to\infty$, they tend in mean to $\zeta(s)$ and $\zeta(s,\alpha)$ respectively.