Electronic Communications in Probability

Real zeros of random Dirichlet series

Marco Aymone

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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.

Article information

Source
Electron. Commun. Probab., Volume 24 (2019), paper no. 54, 8 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1568253716

Digital Object Identifier
doi:10.1214/19-ECP260

Zentralblatt MATH identifier
07107318

Subjects
Primary: 60G50: Sums of independent random variables; random walks
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72}

Keywords
random series zeros of random analytic functions Dirichlet series

Rights
Creative Commons Attribution 4.0 International License.

Citation

Aymone, Marco. Real zeros of random Dirichlet series. Electron. Commun. Probab. 24 (2019), paper no. 54, 8 pp. doi:10.1214/19-ECP260. https://projecteuclid.org/euclid.ecp/1568253716


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