Limit theorems for random Dirichlet series with summation over primes, with an application to Rademacher random multiplicative functions

It is shown that two conjectures put forward in the recent article Iksanov and Kostohryz (2025) are true. Namely, we prove a functional central limit theorem (FCLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series ${\sum }_p\frac{{\eta }_p}{p^{1\mathrm{/}2+s}}$ as $s\to 0+$, where ${\eta }_1$, ${\eta }_2,\dots$ are independent identically distributed random variables with zero mean and finite variance, and ${\sum }_p$ denotes the summation over the prime numbers. As a consequence, an FCLT and an LIL are obtained for $\log {\sum }_{n\ge 1}\frac{f(n)}{n^{1\mathrm{/}2+s}}$ as $s\to 0+$, where $f$ is a Rademacher random multiplicative function.