# Real zeros of random Dirichlet series

## TL;DR

This paper investigates the distribution of real zeros of a random Dirichlet series with random signs, establishing conditions under which the series almost surely has no or infinitely many real zeros.

## Contribution

It provides a rigorous analysis of the zero distribution of random Dirichlet series based on the convergence of the sum of reciprocals of the sequence.

## Key findings

- If the sum of reciprocals converges, the series has no real zeros with positive probability.
- If the sum diverges, the series almost surely has infinitely many real zeros.

## 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.