## Electronic Communications in Probability

### Real zeros of random Dirichlet series

Marco Aymone

#### 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

#### 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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