The Steinhaus function is a random, completely multiplicative function on the integers, whose values on primes are i.i.d random variables uniformly distributed on the complex unit circle. Its study is motivated by the study of deterministic multiplicative functions such as the Möbius function and Dirichlet characters. We'll describe recent joint work with Mo Dick Wong, where the limiting distribution of the partial sums of the Steinhaus function was determined. The limiting distribution is Gaussian with random variance; the variance is given by the total mass of a random measure. This measure is an instance of critical multiplicative chaos. We'll highlight key ideas in the proof and explain why and how critical multiplicative chaos arises in this problem. No prior background is required.
