A complete Riemann zeta distribution and the Riemann hypothesis

Takashi Nakamura
2015 Bernoulli  
Let σ,t∈R, s=σ+it, Γ (s) be the Gamma function, ζ(s) be the Riemann zeta function and ξ(s):=s(s-1)π ^-s/2Γ(s/2)ζ(s) be the complete Riemann zeta function. We show that Ξ_σ(t):=ξ (σ-it)/ξ(σ) is a characteristic function for any σ∈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
more » ... etended-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.
doi:10.3150/13-bej581 fatcat:jj2rjoiwnzgvfgrohnm4uxvtky