The Values of the Riemann Zeta-Function on Discrete Sets

TL;DR

This paper investigates the values of the Riemann zeta-function on discrete sets, establishing uniqueness, universality, and denseness properties related to these values and their implications for number theory.

Contribution

It introduces new results on the uniqueness and universality of the zeta-function values on discrete sets, extending classical theorems in the field.

Findings

Vertical arithmetic progressions are uniquely determined by zeta-values.

A joint discrete universality theorem for zeta is proven.

Generalizations of classical denseness theorems are established.

Abstract

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete universality theorem for $\zeta$ with respect to certain permutations of the set of positive integers. Finally, we study a generalization of the classical denseness theorems for $\zeta$.