Discrete universality, continuous universality and hybrid universality are equivalent

TL;DR

This paper establishes the unconditional equivalence among discrete, continuous, and hybrid universality theorems for zeta-functions, extending previous results and removing reliance on the Riemann hypothesis.

Contribution

It proves the unconditional equivalence of different universality notions for zeta-functions, including hybrid universality, without assuming the Riemann hypothesis.

Findings

Unconditional proof of equivalence among universality types

Extension of universality results beyond Dirichlet series

General proof applicable to broader classes of functions

Abstract

Recently Sourmelidis proved that the discrete universality theorem is equivalent to the continuous universality theorem for zeta-functions. He treats both the zero-free universality theorem and the strong universality theorem. Unfortunately in the zero-free case his result is conditional on a Riemann hypothesis, and in the strong universality case he only proves the implication in one direction. We prove this equivalence unconditionally, and also prove an equivalence with hybrid universality. While the main application of our result is on Dirichlet series and zeta-functions, our proof is general and does not use the fact that the functions in question can be represented by Dirichlet series.