Resolving a conjecture of Helson, Harper recently established that partial sums of random multiplicative functions typically exhibit more than square-root cancellation. Harper's work gives an example of a problem in number theory that is closely linked to ideas in probability theory connected with multiplicative chaos. In this talk, I plan to describe some of these connections and consider a problem that might be thought of as a simplified function field version of Helson's conjecture. This includes joint work with Soundararajan.