A modification of the mixed joint universality theorem for a class of zeta-functions

TL;DR

This paper extends the mixed joint universality theorem for zeta-functions, showing that the shifts approximating these functions have positive density for most accuracy levels, with implications for more general zeta-function tuples.

Contribution

It proves that the set of shifts achieving approximation has positive density for all but countably many accuracy levels, advancing the understanding of universality properties of zeta-functions.

Findings

Set of shifts with positive density for approximation

Extension of universality to more general zeta-function tuples

Refinement of previous universality results

Abstract

The property of zeta-functions on mixed joint universality in the Voronin's sense states that any two holomorphic functions can be approximated simultaneously with accuracy $\epsilon>0$ by suitable vertical shifts of the pair consisting from the Riemann zeta- and Hurwitz zeta-functions. In [1], it was shown rather general result, i.e., an approximating pair was composed of the Matsumoto zeta-functions' class and the periodic Hurwitz zeta-function. In this paper, we prove that this set of shifts has a strict positive density for all but at most countably many $\epsilon>0$. Also, we give the concluding remarks on certain more general mixed tuple of zeta-functions.