How many real zeros does a random Dirichlet series have?

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 $T\to 1/2^{+}$. 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.

21 pages. V3: Accepted (EJP) version