"Almost'' universality of the Lerch zeta-function

Abstract

The Lerch zeta-function $L(\lambda,\alpha,s)$ with transcendental parameter $\alpha$, or with rational parameters $\alpha$ and $\lambda$ is universal, i.e., a wide class of analytic functions is approximated by shifts $L(\lambda,\alpha,s+i\tau)$, $\tau \in \mathbb{R}$. The case of algebraic irrational $\alpha$ is an open problem. In the paper, it is proved that, for all parameters $\alpha$, $0<\alpha< 1$, and $\lambda$, $0<\lambda\leqslant 1$, including an algebraic irrational $\alpha$, there exists a closed non-empty set of analytic functions $F_{\alpha, \lambda}$ such that every function $f\in F_{\alpha, \lambda}$ can be approximated by shifts $L(\lambda,\alpha,s+i\tau)$.