A Tauberian characterization of the Riemann hypothesis through the floor function

TL;DR

This paper introduces a novel Tauberian framework connecting the floor function to the Riemann hypothesis, providing a new equivalence and extending classical theorems beyond prime number distribution.

Contribution

It establishes a new Tauberian characterization of the Riemann hypothesis using regular arithmetic functions and links to combinatorial number theory.

Findings

New Tauberian equivalence of the Riemann hypothesis

Extension of classical Tauberian theorems beyond prime number theorem

Connections to combinatorial number theory

Abstract

We introduce a new Tauberian framework through the theory of "regular arithmetic functions". This allows us to establish a characterization of the Riemann hypothesis by linking the floor function to the distribution of nontrivial zeros of the Riemann zeta function. We thereby obtain a novel Tauberian equivalence of the Riemann hypothesis, extending classical Tauberian theorems beyond their traditional confinement to the prime number theorem. We further uncover connections to combinatorial number theory and set the groundwork for a "combinatorial Tauberian theory", highlighting the broader applicability of regular arithmetic functions.