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 Sigma(n <= N) f (n) exhibits "more than square-root cancellation," and in particular 1/root N Sigma(n <= N) f (n) does not have a (complex) Gaussian distribution. This paper studies Sigma(n is an element of A) f(n), where A is a subset of the integers in [1, N], and produces several new examples of sets A where a central limit theorem can be established. We also consider more general sums such as Sigma(n <= N) f (n)e(2 pi in)theta, where we show that a central limit theorem holds for any irrational theta that does not have extremely good Diophantine approximations.