A random multiplicative function is a multiplicative function on the positive integers whose values on primes are i.i.d. random variables.

The simplest example is the Steinhaus function, which is a completely multiplicative function with alpha(p) uniformly distributed on the unit circle. A basic question in the field is finding the limiting distribution of the (normalized) sum of alpha(n) from n=1 to n=x, possibly restricted to a subset of integers of interest. This question is currently resolved only in a few cases.

We shall describe the existing approaches to tackle this question and then explain our recent work, joint with Mo Dick Wong (Durham University), where we are able to find the limiting distribution in many new instances of interest. The distribution we find is non-Gaussian, in contrast to previous works.

No background in number theory is required.