论文标题
Dirichlet系列及其（准）无限划分产生的Zeta分布
Zeta distributions generated by Dirichlet series and their (quasi) infinite divisibility
论文作者
论文摘要
令$ a（1）> 0 $，$ a（n）\ ge 0 $ for $ n \ ge 2 $和$ a（n）= o（n^\ varepsilon）对于任何$ \ varepsilon> 0 $ for任何$ \ varepsilon> 0 $，然后将$ z（σ+ it）放在{\ mathbb {r}} $。在本文中，我们表明其特征函数的任何Zeta分布都由$ {\ Mathcal {Z}}_σ（t）_ = z（σ+ it）/z（σ）$定义，如果$σ> 1 $足够大。此外，我们证明，如果$ {\ Mathcal {z}}_σ（t）$是某些$σ_{id}> 1 $的无限分开的特征函数，则$ {\ Mathcal {z}}_σ（t）$是无限的，对于所有$ c $ c $ c $ c>> 1 $> 1 $。请注意，可以明确给出相应的Lévy或Quasi-Lévy措施。证明的关键是Apostol著名教科书中定理11.14的校正版本。
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.
