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

TL;DR

This paper investigates the (quasi) infinite divisibility of zeta distributions derived from Dirichlet series, establishing conditions under which these distributions are infinitely divisible and providing explicit measures.

Contribution

It demonstrates that zeta distributions are pretended infinitely divisible for large enough , and proves that infinite divisibility at one  implies it for all larger , with explicit measures given.

Findings

Zeta distributions are pretended infinitely divisible for sufficiently large .

Infinite divisibility at some  > 1 implies infinite divisibility for all larger .

Explicit Le9vy or quasi-Le9vy measures are derived.

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.