Law of the iterated logarithm for a random Dirichlet series

Abstract

Let $(X_n{)}_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb{P}(X_1=1)=\mathbb{P}(X_1=-1)=1/2$. Let $F(\sigma )={\sum }_{n=1}^{\infty }{X}_n{n}^{-\sigma }$. We prove that the following holds almost surely \begin{equation*} \limsup_{σ\to 1/2^+}\frac{F(σ)}{\sqrt{2\mathbb E F(σ)^2\log\log \mathbb E F(σ)^2}}=1. \end{equation*}

19 pages, accepted version. To appear in ECP