A complete Riemann zeta distribution and the Riemann hypothesis

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