Central limit theorems for random multiplicative functions

Abstract

A Steinhaus random multiplicative function f is a completely multiplicative function obtained by setting its values on primes f(p) to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that $\sum\nolimits_{n \le N} {f(n)}$ exhibits “more than square-root cancellation,” and in particular ${1 \over {\sqrt N }}\sum\nolimits_{n \le N} {f(n)}$ does not have a (complex) Gaussian distribution. This paper studies $\sum\nolimits_{n \in {\cal A}} {f(n)}$, where ${\cal A}$ is a subset of the integers in [1, N], and produces several new examples of sets ${\cal A}$ where a central limit theorem can be established. We also consider more general sums such as $\sum\nolimits_{n \le N} {f(n){e^{2\pi in\theta }}}$, where we show that a central limit theorem holds for any irrational θ that does not have extremely good Diophantine approximations.