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 $a\ge 1/2$.

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)

References in corpus (1)

• Any non-monomial polynomial of the Riemann zeta-function has complex zeros off the critical line