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 ${\sum }_{k\ge 2}(\log k{)}^{\alpha }{k}^{-1\mathrm{/}2-s}{\eta }_k$ as $s\to 0+$, where $\alpha >-1\mathrm{/}2$ and ${\eta }_1$, ${\eta }_2,\dots$ are independent identically distributed random variables with zero mean and finite variance. We prove a FLT and a LIL in a boundary case $\alpha =-1\mathrm{/}2$. The boundary case is more demanding technically than the case $\alpha >-1\mathrm{/}2$.