How many real zeros does a random Dirichlet series have?

Abstract

Let F(σ) = P∞ n=1 Xn nσ be a random Dirichlet series where (Xn)n∈N are independent standard Gaussian random variables. We compute in a quantitative form the expected number of zeros of F(σ) in the interval [T, ∞), say EN(T, ∞), as T → 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, ∞). And finally, we also prove results for another class of random Dirichlet series, e.g., when the summation is restricted to prime numbers.