Abstract

For $0<\alpha \le 1$ and $0<\lambda \le 1$, let $L\left(s,\alpha ,\lambda \right)$ denote the associated Lerch zeta-function, which is defined to be ${\sum }_{n=0}^{\infty }{e}^{2\pi in\lambda }{\left(n+\alpha \right)}^{-s}$ for $Res>1$. We obtain the joint universality theorem for the collection of Lerch zeta-functions ${\bigcup }_{j=1}^J\left\{L\left(s,{\alpha }_j,\lambda \right):0<\lambda \le 1\right\}$, 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.

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