Law of the iterated logarithm for a random Dirichlet series

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 $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely \begin{equation*} \limsup_{\sigma\to 1/2^+}\frac{F(\sigma)}{\sqrt{2\mathbb E F(\sigma)^2\log\log \mathbb E F(\sigma)^2}}=1. \end{equation*}