A short proof of Helson's conjecture

Abstract

Let $\alpha \colon \mathbb{N} \to S^1$ be the Steinhaus multiplicative function: a completely multiplicative function such that $(\alpha(p))_{p\text{ prime}}$ are i.i.d.~random variables uniformly distributed on the complex unit circle $S^1$. Helson conjectured that $\mathbb{E}|\sum_{n\le x}\alpha(n)|=o(\sqrt{x})$ as $x \to \infty$, and this was solved in a strong form by Harper. We give a short proof of the conjecture using a result of Saksman and Webb on a random model for the zeta function.