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

2025/08/20 by Congzao Dong, Dong, Congzao, Alexander Iksanov +1
Decision Sciences · Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT) #Probability (math.PR) #Probability and Risk Models

paper · pdf · doi:10.48550/arxiv.2508.15032

openalex publication_date 2025/08/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 ∑_p \fracη_pp^1/2+s as s→ 0+, where η₁, η₂,… are independent identically distributed random variables with zero mean and finite variance, and ∑_p denotes the summation over the prime numbers. As a consequence, an FCLT and an LIL are obtained for log ∑_{n≥ 1} \fracf(n)n^1/2+s as s→ 0+, where f is a Rademacher random multiplicative function.

Citations

• On the limiting distribution of sums of random multiplicative functions
• Brownian behaviour of the Riemann zeta function around the critical line
• The distribution of partial sums of random multiplicative functions with a large prime factor
• Limit theorems for random Dirichlet series: boundary case
• Sign changes of short character sums and real zeros of Fekete polynomials
• Counting sign changes of partial sums of random multiplicative functions
• Almost sure upper bound for random multiplicative functions
• Limit theorems for random Dirichlet series
• Sign changes of the partial sums of a random multiplicative function
• Almost sure large fluctuations of random multiplicative functions
• On the limit distributions of some sums of a random multiplicative\n function

Related

• Limit theorems for random Dirichlet series: boundary case
• Limit theorems for random Dirichlet series
• On the limit distributions of some sums of a random multiplicative\n function
• Central limit theorems for random multiplicative functions
• Almost sure bounds for weighted sums of Rademacher random multiplicative functions