How many real zeros does a random Dirichlet series have?

TL;DR

This paper analyzes the distribution and expected count of zeros of a random Dirichlet series with Gaussian coefficients, providing quantitative estimates, tail bounds, and results for variants involving primes.

Contribution

It offers the first detailed quantitative analysis of zeros of Gaussian random Dirichlet series, including moments, tail bounds, and bounds for prime-restricted series.

Findings

Expected number of zeros in [T,∞) as T→1/2+

Exponential tail bounds for the number of zeros in [T,1]

Almost sure bounds for zeros in [T,∞)

Abstract

Let $F(\sigma)=\sum_{n=1}^\infty \frac{X_n}{n^\sigma}$ be a random Dirichlet series where $(X_n)_{n\in\mathbb{N}}$ are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of $F(\sigma)$ in the interval $[T,\infty)$, say $\mathbb{E} N(T,\infty)$, as $T\to1/2^+$. We also estimate higher moments and with this we derive exponential tails for the probability that the number of zeros in the interval $[T,1]$, say $N(T,1)$, is large. We also consider almost sure lower and upper bounds for $N(T,\infty)$. And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers.