Tauberian theory and the Riemann hypothesis

Benoit Cloitre

Published 2024-07-26Version 1

In this article, I present a Tauberian equivalence of the Riemann hypothesis within the framework of the theory of regular arithmetic functions, a branch of Tauberian theory that extends the theory of functions with good variation introduced in [Cloitre]. The central element of this study is the function $\Phi(x)=x\left\lfloor \frac{1}{x}\right\rfloor$, which allows for the extension of Ingham's summation method [Ingham] far beyond the prime number theorem by linking it to the Riemann hypothesis. I thus demonstrate the following equivalence $$RH\Longleftrightarrow\alpha\left(\Phi\right)=\frac{1}{2}$$ where $\alpha\left(\Phi\right)$ represents the index of good variation of $\Phi$, an essential characteristic of functions of good variation. This equivalence of the Riemann hypothesis is a potential new contribution absent from recent comprehensive catalogs of equivalent of the Riemann hypothesis in mathematical literature [Broughan1, Broughan2].