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

2022/09/27 by Takashi Nakamura, Nakamura, Takashi · 1 citation
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.2209.13257

Abstract

Let a(1) >0, a(n) ≥ 0 for n ≥ 2 and a(n) = O(n^ε) for any ε >0, and put Z(σ+ it):= ∑_n=1^∞ a(n) n^{-σ- it} where σ, t ∈ ℝ. In the present paper, we show that any zeta distribution whose characteristic function is defined by Z_σ(t) :=Z(σ+ it)/Z(σ) is pretended infinitely divisible if σ>1 is sufficiently large. Moreover, we prove that if Z_σ(t) is an infinitely divisible characteristic function for some σ_id >1, then 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.

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