Functional distribution for a collection of Lerch zeta functions

Abstract

Let 0 < $\alpha$ < 1 be a transcendental real number and $\lambda_1,\ldots,\lambda_r$ be real numbers with $0 \le \lambda_j$ < 1. It is conjectured that a joint universality theorem for a collection of Lerch zeta functions $\{L(\lambda_j,\alpha,s)\}$ will hold for every numbers $\lambda_j$'s which are different each other. In this paper we will prove that the joint universality theorem for the set $\{L(\lambda_j,\alpha,s)\}$ holds for almost all real numbers $\lambda_j$'s.