Limit theorems for random Dirichlet series

Abstract

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series D(α; z) = ∑ n≥2 (log n) α (η n + iθ n )/n z , properly scaled and normalized, where (η n , θ n ) n∈N is a sequence of independent copies of a centered R 2 -valued random vector (η, θ ) with a finite second moment and α > −1/2 is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of D(α; z) converge vaguely, thereby obtaining a universality result. In the real case, that is, when P{θ = 0} = 1, we also prove a law of the iterated logarithm for D(α; z), properly normalized, as z → (1/2)+.