How many real zeros does a random Dirichlet series have?

Abstract

Let $F(\mathit{\sigma })={\sum }_{n=1}^{\mathrm{\infty }}\frac{X_n}{n^{\mathit{\sigma }}}$ 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 $F(\mathit{\sigma })$ in the interval $[T,\mathrm{\infty })$, say $\mathbb{E}N(T,\mathrm{\infty })$, 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 $N(T,\mathrm{\infty })$. And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers.