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

TL;DR

This paper proves a functional central limit theorem and a law of the iterated logarithm for a random Dirichlet series over primes, with applications to the behavior of Rademacher random multiplicative functions near the critical point.

Contribution

It establishes the first rigorous proofs of FCLT and LIL for these specific random Dirichlet series and their implications for multiplicative functions.

Findings

Proved FCLT for the Dirichlet series over primes.

Established LIL for the same series.

Derived FCLT and LIL for the logarithm of the sum involving Rademacher functions.

Abstract

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 $\sum_p \frac{\eta_p}{p^{1/2+s}}$ as $s\to 0+$, where $\eta_1$, $\eta_2,\ldots$ 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 $\log \sum_{n\geq 1} \frac{f(n)}{n^{1/2+s}}$ as $s\to 0+$, where $f$ is a Rademacher random multiplicative function.