A short proof of Helson's conjecture

Abstract

Let $\hat{I}\pm :\mathbb{N}\to S^1$ be the Steinhaus multiplicative function: a completely multiplicative function such that $(\hat{I}\pm (p){)}_{p\text{ prime}}$ are i.i.d.~random variables uniformly distributed on the complex unit circle $S^1$. Helson conjectured that $\mathbb{E}|{\sum }_{n\le x}\hat{I}\pm (n)|=o(\sqrt{x})$ 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.

9 pages. Minor mistakes and typos fixed and discussion of previous works added; journal version