A complete Riemann zeta distribution and the Riemann hypothesis

TL;DR

This paper introduces a new probability distribution related to the Riemann zeta function, linking the Riemann hypothesis to properties of certain characteristic functions and exploring their divisibility characteristics.

Contribution

It establishes that the complete Riemann zeta distribution's characteristic functions are pretended-infinitely divisible and connects the Riemann hypothesis to these properties.

Findings

$ ext{Xi}_ ext{sigma}(t)$ is a characteristic function for all real $ ext{sigma}$.

The Riemann hypothesis holds if and only if $ ext{Xi}_ ext{sigma}(t)$ is pretended-infinitely divisible for $1/2< ext{sigma}<1$.

$ ext{Xi}_ ext{sigma}(t)$ is quasi-infinitely divisible for $ ext{sigma}>1$.

Abstract

Let $\sigma,t\in{\mathbb{R}}$, $s=\sigma+\mathrm{{i}}t$, $\Gamma (s)$ be the Gamma function, $\zeta(s)$ be the Riemann zeta function and $\xi(s):=s(s-1)\pi ^{-s/2}\Gamma(s/2)\zeta(s)$ be the complete Riemann zeta function. We show that $\Xi_{\sigma}(t):=\xi (\sigma-\mathrm{{i}}t)/\xi(\sigma)$ is a characteristic function for any $\sigma\in{\mathbb{R}}$ by giving the probability density function. Next we prove that the Riemann hypothesis is true if and only if each $\Xi_{\sigma}(t)$ is a pretended-infinitely divisible characteristic function, which is defined in this paper, for each $1/2<\sigma<1$. Moreover, we show that $\Xi_{\sigma}(t)$ is a pretended-infinitely divisible characteristic function when $\sigma=1$. Finally we prove that the characteristic function $\Xi_{\sigma}(t)$ is not infinitely divisible but quasi-infinitely divisible for any $\sigma>1$.