  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 $a \ge 1/2$.
