Limit theorems for random Dirichlet series: boundary case

Abstract

Buraczewski et al (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series ${\sum }_{k\ge 2}(\log k{)}^{\hat{I}}\pm k^{-1/2-s}\hat{I}{\cdot }_k$ as $s\to 0+$, where $\hat{I}\pm >-1/2$ and $\hat{I}{\cdot }_1$, $\hat{I}{\cdot }_2,\dots$ are independent identically distributed random variables with zero mean and finite variance. We prove a FLT and a LIL in a boundary case $\hat{I}\pm =-1/2$. The boundary case is more demanding technically than the case $\hat{I}\pm >-1/2$.

submitted for publication