A short proof of Helson's conjecture

Gorodetsky, Ofir; Wong, Mo Dick

Abstract

Let α : N → S 1 $\alpha \colon \mathbb {N}\rightarrow S^1$ be the Steinhaus multiplicative function: a completely multiplicative function such that ( α ( p ) ) p prime $(\alpha (p))_{p\text{ prime}}$ are i.i.d. random variables uniformly distributed on the complex unit circle S 1 $S^1$ . Helson conjectured that E | ∑ n ⩽ x α ( n ) | = o ( x ) $\mathbb {E}|\sum _{n\leqslant x}\alpha (n)|=o(\sqrt {x})$ as x → ∞ $x \rightarrow \infty$ , 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.

Citation

Gorodetsky, O., & Wong, M. D. (2025). A short proof of Helson's conjecture. Bulletin of the London Mathematical Society, 57(4), 1065-1076. https://doi.org/10.1112/blms.70015

• Journal Article Type: Article
• Acceptance Date: Jan 15, 2025
• Online Publication Date: Feb 7, 2025
• Publication Date: 2025-04
• Deposit Date: Jan 21, 2025
• Publicly Available Date: Feb 12, 2025
• Journal: Bulletin of the London Mathematical Society
• Print ISSN: 0024-6093
• Electronic ISSN: 1469-2120
• Publisher: Wiley
• Peer Reviewed: Peer Reviewed
• Volume: 57
• Issue: 4
• Pages: 1065-1076
• DOI: https://doi.org/10.1112/blms.70015
• Public URL: https://durham-repository.worktribe.com/output/3344048