Joint universality for Lerch zeta-functions (Q520633)

The Lerch zeta-function \(L(s;\alpha,\lambda)\), for complex \(s=\sigma+it\) and real numbers \(\alpha>0\) and \(\lambda \leq 1\), is given by the series \[ L(s;\alpha,\lambda)=\sum_{n=0}^{\infty}\frac{\exp(2 \pi i\lambda n)}{(n+\alpha)^s} \quad \text{as} \quad \sigma>1. \] In [J. Math. Soc. Japan 66, No. 4, 1105--1126 (2014; Zbl 1317.11089)], \textit{H. Mishou} conjectured that a joint universality theorem for a collection of Lerch zeta-functions \(\{L(s;\alpha, \lambda_j)\}\) holds for every numbers \(\lambda_j\)'s, \(0\leq \lambda_j<1\), which are different each other, and transcendental \(0<\alpha<1\). In the paper, this theorem is proved. More precisely, it is shown that following statement holds. Suppose that \(L(s;\alpha,\lambda_1),\dots,L(s;\alpha,\lambda_m)\) and \(\alpha\) satisfy Mishou's conjecture. Let \(K_j \subset \{s \in \mathbb{C}:\frac{1}{2}<\sigma<1\}\) be compact sets with connected complements, and \(f_j(s)\) be continuous function of \(K_j\) and analytic in the interior of \(K_j\), \(1 \leq j \leq m\). Then, for every \(\varepsilon >0\), \[ \liminf_{T \to \infty} \frac{1}{T}\operatorname{meas}\bigg\{ \tau \in [0,T]: \max_{1 \leq j \leq m}\max_{s \in K_j}|L(s+i\tau; \alpha,\lambda_j)-f_j(s)|< \varepsilon \bigg\}>0. \]