Joint Discrete Universality in the Selberg–Steuding Class

In the paper, we consider the approximation of analytic functions by shifts from the wide class $\tilde{S}$ of L-functions. This class was introduced by A. Selberg, supplemented by J. Steuding, and is defined axiomatically. We prove the so-called joint discrete universality theorem for the function $L\left(s\right)\in \tilde{S}$. Using the linear independence over $\mathbb{Q}$ of the multiset $\left\{(h_{j}logp:p\in \mathbb{P}),j=1,\dots ,r;2\pi \right\}$ for positive $h_j$, we obtain that there are many infinite shifts $\left(L(s+ik{h}_1),\dots ,L(s+ik{h}_r)\right)$, $k=0,1,\dots$, approximating every collection $\left(f_1\left(s\right),\dots ,f_r\left(s\right)\right)$ of analytic non-vanishing functions defined in the strip $\{s\in \mathbb{C}:{\sigma }_L<\sigma <1\}$, where ${\sigma }_L$ is a degree of the function $L\left(s\right)$. For the proof, the probabilistic approach based on weak convergence of probability measures in the space of analytic functions is applied.