A short proof of Helson's conjecture

TL;DR

This work proves Helson's conjecture for a Steinhaus random multiplicative function by deriving sharp uniform bounds on the moments of $S_x = \frac{1}{\sqrt{x}}\sum_{n\le x} \alpha(n)$. The authors combine a number-theoretic decomposition using a truncated Euler product $A_y(s)$ with a y-smooth/rough factorization, and a multiplicative chaos framework that couples $\log A_y$ to a log-correlated Gaussian field. A key result is that for $\delta\in(0,1)$ and $q\in[0,1-\delta]$, $\mathbb{E}[|S_x|^{2q}] \ll (\log\log x)^{-q/2}$, which implies $\mathbb{E}|S_x|=o(1)$ and hence Helson’s conjecture follows. The approach leverages a Saksman–Webb coupling to a GMC model and employs GMC moment criteria and Kahane’s convexity inequality to control the critical moments, providing a concise alternative to previous heavy-technical proofs and highlighting the role of GMC in random multiplicative models.

Abstract

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

Theorems & Definitions (14)

• Theorem 1.1
• Lemma 1.2
• Lemma 1.3
• Remark 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• Theorem 3.3: cf. SaksmanWebb and SW
• Lemma 3.4
• proof : Sketch of proof