Zeta distributions generated by Dirichlet series and their (quasi) infinite divisibility

Abstract

Let $a(1)>0$, $a(n)\ge 0$ for $n\ge 2$ and $a(n)=O(n^{\epsilon })$ for any $\epsilon >0$, and put $Z(\sigma +it):={\sum }_{n=1}^{\infty }a(n)n^{-\sigma -it}$ where $\sigma ,t\in \mathbb{R}$. In the present paper, we show that any zeta distribution whose characteristic function is defined by ${\mathcal{Z}}_{\sigma }(t):=Z(\sigma +it)/Z(\sigma )$ is pretended infinitely divisible if $\sigma >1$ is sufficiently large. Moreover, we prove that if ${\mathcal{Z}}_{\sigma }(t)$ is an infinitely divisible characteristic function for some ${\sigma }_{id}>1$, then ${\mathcal{Z}}_{\sigma }(t)$ is infinitely divisible for all $\sigma >1$. Note that the corresponding Lévy or quasi-Lévy measure can be given explicitly. A key of the proof is a corrected version of Theorem 11.14 in Apostol's famous textbook.

arXiv comment12 pages