Limit theorems for random Dirichlet series: boundary case

Buraczewski et al (2023) proved a functional limit theorem (FLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series $\sum_{k\geq 2}(\log k)^\alpha k^{-1/2-s}\eta_k$ as $s\to 0+$, where $\alpha>-1/2$ and $\eta_1$, $\eta_2,\ldots$ are independent identically distributed random variables with zero mean and finite variance. We prove a FLT and a LIL in a boundary case $\alpha=-1/2$. The boundary case is more demanding technically than the case $\alpha>-1/2$.