Mixed joint universality for $L$-functions from Selberg’s class and periodic Hurwitz zeta-functions

In 1975, a Russian mathematician S. M. Voronin discovered the universality property of the Riemann zeta-function $\zeta(s)$, $s=\sigma+it$. Roughly speaking, this means that analytic functions from a wide class can be approximated uniformly on compact subsets of the strip $\{s\in \mathbb{C}: 1/2 < \sigma< 1\}$ by shifts $\zeta(s+i\tau)$, $\tau\in \mathbb{R}$. Later, it turned out that other classical zeta and $L$-functions are also universal in the Voronin sense. Moreover, some zeta and $L$-functions have a joint universality property. In this case, a given collection of analytic functions is approximated simultaneously by shifts of zeta and $L$-functions.
In the paper, we present our extended report given at the Conference dedicated to the memory of the famous number theorist Professor A. A. Karacuba. The paper contains the basic universality results on the so-called mixed joint universality initiated by H. Mishou who in 2007 obtained the joint universality for the Riemann zeta and Hurwitz zeta-functions. In a wide sense the mixed joint universality is understood as a joint universality for zeta and $L$-functions having and having no Euler product.
In 1989, A. Selber introduced a famous class $\mathcal{S}$ of Dirichlet series satisfying certain natural hypotheses including the Euler product. Periodic Hurwitz zeta-functions are a generalization of classical Hurwitz zeta-functions, and have no Euler product. In the paper, a new result on mixed joint universality for $L$-functions from the Selberg clas and periodic Hurwitz zeta-functions is presented. For the proof a probabilistic method can be applied.
Bibliography: 24 titles.