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 $0 < \sigma \ne 1$ by using the Gamma function. During the proof process, we show that the Hurwitz zeta function $\zeta (\sigma,a)$ does not vanish for all $0 <\sigma <1$ 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 $\zeta_2 (\sigma_1,\sigma_2\,;a)$ does not vanish when $0<\sigma_1<1$, $\sigma_2>1$ and $1<\sigma_1+\sigma_2<2$ if and only if $a \ge 1/2$.