Central limit theorems for random multiplicative functions

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 ∑n≤Nf(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\nolimits_{n \le N} {f(n)} $$\end{document} exhibits “more than square-root cancellation,” and in particular 1N∑n≤Nf(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${1 \over {\sqrt N }}\sum\nolimits_{n \le N} {f(n)} $$\end{document} does not have a (complex) Gaussian distribution. This paper studies ∑n∈Af(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\nolimits_{n \in {\cal A}} {f(n)} $$\end{document}, where A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal A}$$\end{document} is a subset of the integers in [1, N], and produces several new examples of sets A\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\cal A}$$\end{document} where a central limit theorem can be established. We also consider more general sums such as ∑n≤Nf(n)e2πinθ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\nolimits_{n \le N} {f(n){e^{2\pi in\theta }}} $$\end{document}, where we show that a central limit theorem holds for any irrational θ that does not have extremely good Diophantine approximations.