Hurwitz zeta and Euler-Zagier-Hurwitz type of double zeta distributions and real zeros of these zeta functions

Abstract

In this paper, we give Hurwitz zeta distributions with $0 < σ\ne 1$ by using the Gamma function. During the proof process, we show that the Hurwitz zeta function $ζ(σ,a)$ does not vanish for all $0 <σ<1$ if and only if $a \ge 1/2$. Next we define Euler-Zagier-Hurwitz type of double zeta distributions not only in the region of absolute convergence but also the outside of the region of absolute convergence. Moreover, we show that the Euler-Zagier-Hurwitz type of double zeta function $ζ_2 (σ_1,σ_2\,;a)$ does not vanish when $0<σ_1<1$, $σ_2>1$ and $1<σ_1+σ_2<2$ if and only if $a \ge 1/2$.

This paper arXiv:1312.4712 `Hurwitz zeta and Euler-Zagier-Hurwitz type of double zeta distributions and real zeros of these zeta functions' by Takashi Nakamura was divided into the following two papers (some new results are added): (1) Number theoretical part (arXiv:1405.1504), (2) Probability theoretical part (arXiv:1405.1799)