On mixed joint discrete universality for a class of zeta-functions II

TL;DR

This paper investigates the combined value distribution of a general zeta-function with an Euler product and a periodic Hurwitz zeta-function, establishing a universality theorem under specific arithmetic conditions.

Contribution

It proves a mixed joint discrete universality theorem for these functions with differing arithmetic progressions, extending previous results.

Findings

Established mixed joint discrete universality theorem

Identified necessary arithmetic conditions for the theorem

Extended previous universality results to broader classes

Abstract

We study analytic properties of the pair consisting of a rather general form of zeta-function with an Euler product and a periodic Hurwitz zeta-function with a transcendental parameter. We first survey briefly previous results, and then investigate the mixed joint discrete value distribution and prove the mixed joint discrete universality theorem for these functions, in the case when common differences of relevant arithmetic progressions are not necessarily the same. Also some generalizations are given. For this purpose, certain arithmetic conditions on the common differences are necessary.