• Corpus ID: 252545363

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

@inproceedings{Nakamura2022ZetaDG,
  title={Zeta distributions generated by Dirichlet series and their (quasi) infinite divisibility},
  author={Takashi Nakamura},
  year={2022}
}

• Takashi Nakamura
• Published 27 September 2022
• Mathematics

. 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 ) := P ∞ 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 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.