Law of the iterated logarithm for a random Dirichlet series

• Main Authors: Aymone, Marco, Frómeta, Susana, Misturini, Ricardo
• Format: Journal Article
• Language: English
• Published: 23-07-2020
• Subjects: Mathematics - Number Theory; Mathematics - Probability
• DOI: 10.48550/arxiv.2004.10559

Electron. Commun. Probab., Volume 25 (2020), paper no. 56, 14 pp 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*}