In seminal work, Harper showed that the partial sums of random multiplicative functions enjoy "better than square-root cancellation". This surprising phenomenon gives rise to many questions; for example, recent research has focused on whether certain subsums possess this property. In this talk, I will introduce smooth numbers, discuss their "anatomy", and relate this to recent progress on random multiplicative functions over smooth numbers. This is based on joint work with Max Wenqiang Xu.

Selberg’s central limit theorem shows that the modulus of the Riemann zeta function is asymptotically log-normal. Recent work has extended the theorem to the large-value regime. In this talk, I will survey these results and discuss ongoing work on the small-value regime, where new behavior emerges.