A short proof of Helson's conjecture

TL;DR

This paper provides a concise proof of Helson's conjecture, which predicts that the expected magnitude of sums of Steinhaus multiplicative functions grows slower than the square root of x, using a random zeta function model.

Contribution

The paper introduces a short proof of Helson's conjecture leveraging a recent result on a random model for the zeta function, simplifying previous approaches.

Findings

Proof confirms Helson's conjecture in a concise manner.

Utilizes a random zeta function model to establish the result.

Simplifies the understanding of the behavior of Steinhaus multiplicative functions.

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.