Law of the iterated logarithm for a random Dirichlet series

Abstract

Let $(X_{n})_{n\in \mathds {N}}$ 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 \[ \limsup _{\sigma \to 1/2^{+}}\frac {F(\sigma )}{\sqrt {2\mathbb {E} F(\sigma )^{2} \log \log \mathbb {E} F(\sigma )^{2}}}=1. \]