Electronic Communications in Probability

Real zeros of random Dirichlet series

Marco Aymone

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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

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

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

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Zentralblatt MATH identifier

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}

random series zeros of random analytic functions Dirichlet series

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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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