%0 Journal Article
%T On the limit distributions of some sums of a random multiplicative function
%A Adam J. Harper
%J Mathematics
%D 2010
%I arXiv
%X 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.
%U http://arxiv.org/abs/1012.0207v1