The distribution of partial sums of random multiplicative functions with a large prime factor

2025/03/08 by Hardy, Seth · 4 citations
#11K65 #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR)

paper · doi:10.48550/arxiv.2503.06256

Abstract

For f a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \frac(log log x)^1/4√(x) ∑_\substackn ≤ x
P(n) gt; √(x) f(n), where P(n) denotes the largest prime factor of n. We find that the limiting distribution is given by the square root of an integral with respect to a critical Gaussian multiplicative chaos measure multiplied by an independent standard complex normal random variable.

Cited by

• Helson's conjecture for smooth numbers
• Limit theorems for random Dirichlet series with summation over primes, with an application to Rademacher random multiplicative functions
• On the limiting distribution of sums of random multiplicative functions
• A sharp almost sure upper bound for partial sums of random multiplicative functions

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