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 $a(n) = O(n^\varepsilon)$ for any $\varepsilon >0$, 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 L\'evy or quasi-L\'evy measure can be given explicitly. A key of the proof is a corrected version of Theorem 11.14 in Apostol's famous textbook.