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2019 Real zeros of random Dirichlet series
Marco Aymone
Electron. Commun. Probab. 24: 1-8 (2019). DOI: 10.1214/19-ECP260

Abstract

Let $F(\sigma )$ be the random Dirichlet series $F(\sigma )=\sum _{p\in \mathcal{P} } \frac{X_{p}} {p^{\sigma }}$, where $\mathcal{P} $ is an increasing sequence of positive real numbers and $(X_{p})_{p\in \mathcal{P} }$ is a sequence of i.i.d. random variables with $\mathbb{P} (X_{1}=1)=\mathbb{P} (X_{1}=-1)=1/2$. We prove that, for certain conditions on $\mathcal{P} $, if $\sum _{p\in \mathcal{P} }\frac{1} {p}<\infty $ then with positive probability $F(\sigma )$ has no real zeros while if $\sum _{p\in \mathcal{P} }\frac{1} {p}=\infty $, almost surely $F(\sigma )$ has an infinite number of real zeros.

Citation

Download Citation

Marco Aymone. "Real zeros of random Dirichlet series." Electron. Commun. Probab. 24 1 - 8, 2019. https://doi.org/10.1214/19-ECP260

Information

Received: 6 April 2019; Accepted: 30 July 2019; Published: 2019
First available in Project Euclid: 12 September 2019

zbMATH: 1422.60072
MathSciNet: MR4003128
Digital Object Identifier: 10.1214/19-ECP260

Subjects:
Primary: 60G50
Secondary: 11M41

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