A complete Riemann zeta distribution and the Riemann hypothesis

Abstract

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