Limit theorems for random Dirichlet series

Abstract

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(\alpha ;z)={\sum }_{n\ge 2}(\log n{)}^{\alpha }({\eta }_n+\mathrm{i}{\theta }_n)/n^z$, properly scaled and normalized, where $({\eta }_n,{\theta }_n{)}_{n\in \mathbb{N}}$ is a sequence of independent copies of a centered ${\mathbb{R}}^2$-valued random vector $(\eta ,\theta )$ with a finite second moment and $\alpha >-1/2$ is a fixed real parameter. As a consequence, we show that the point processes of complex and real zeros of $D(\alpha ;z)$ converge vaguely, thereby obtaining a universality result. In the real case, that is, when $\mathbb{P}\{\theta =0\}=1$, we also prove a law of the iterated logarithm for $D(\alpha ;z)$, properly normalized, as $z\to (1/2)+$.

29 pages