A Tauberian characterization of the Riemann hypothesis through the floor function

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.