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.
Keywords
Cite
@article{arxiv.2407.18859,
title = {A Tauberian characterization of the Riemann hypothesis through the floor function},
author = {Benoit Cloitre},
journal= {arXiv preprint arXiv:2407.18859},
year = {2024}
}
Comments
36 pages. Major revisions: revised title and abstract to focus on Tauberian-floor function approach; restructured introduction with RAF framework; added combinatorial applications; streamlined presentation (reduced from 44 to 36 pages). All essential results preserved