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 $Z(σ+ it):= \sum_{n=1}^\infty a(n) n^{-σ- it}$ where $σ, t \in {\mathbb{R}}$. In the present paper, we show that any zeta distribution whose characteristic function is defined by ${\mathcal{Z}}_σ(t) :=Z(σ+ it)/Z(σ)$ is pretended infinitely divisible if $σ>1$ is sufficiently large. Moreover, we prove that if ${\mathcal{Z}}_σ(t)$ is an infinitely divisible characteristic function for some $σ_{id} >1$, then ${\mathcal{Z}}_σ(t)$ is infinitely divisible for all $σ>1$. 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.