Joint universality and simple a-points of Lerch zeta-functions

摘要

For $0 alpha le 1$ and $0 lambda le 1$, let $L(s, alpha, lambda)$ denote the associated Lerch zeta-function, which is defined to be smash{$sum_{n=0}^{infty} e^{2 pi i n lambda} (n+alpha)^{-s}$} for $operatorname{Re}s 1$. We obtain the joint universality theorem for the collection of Lerch zeta-functions smash{$bigcup_{j=1}^J { L(s, alpha_j, lambda) : 0 lambda le 1 }$}, when $alpha_1, dots, alpha_J$ satisfy a certain condition. This theorem is an improvement of several previous joint universality theorems for Lerch zeta-functions. We also investigate the distribution of simple $a$-points of Lerch zeta-functions and their derivatives.