On the limit distributions of some sums of a random multiplicative function

TL;DR

This paper investigates the distributional limits of sums of random multiplicative functions, establishing normal approximation results for certain regimes and demonstrating the failure of such limits when the number of prime factors grows too quickly.

Contribution

It introduces martingale-based methods to analyze the asymptotic distribution of sums of random multiplicative functions, extending to generalized functions and different prime factor regimes.

Findings

Normal approximation holds when the number of prime factors k = o(log log x).

Normal limit theorems fail when k is on the order of log log x.

Methods extend to sums over functions with at most k prime factors.

Abstract

We study sums of a random multiplicative function; this is an example, of number-theoretic interest, of sums of products of independent random variables (chaoses). Using martingale methods, we establish a normal approximation for the sum over those n \leq x with k distinct prime factors, provided that k = o(log log x) as x \rightarrow \infty. We estimate the fourth moments of these sums, and use a conditioning argument to show that if k is of the order of magnitude of log log x then the analogous normal limit theorem does not hold. The methods extend to treat the sum over those n \leq x with at most k distinct prime factors, and in particular the sum over all n \leq x. We also treat a substantially generalised notion of random multiplicative function.