Real zeros of random Dirichlet series

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.