A short proof of Helson's conjecture

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-02-07 DOI:10.1112/blms.70015

Ofir Gorodetsky, Mo Dick Wong

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引用次数: 0

Abstract

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

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Helson猜想的简短证明

设α: N→s1 $\alpha \colon \mathbb {N}\rightarrow S^1$为斯坦豪斯乘法函数：一个完全乘法函数，使得（α (p)） p ' $(\alpha (p))_{p\text{ prime}}$为i.d个均匀分布在复单位圆s1上的随机变量$S^1$。Helson推测E |∑n≥x α (n) | = 0 (x) $\mathbb {E}|\sum _{n\leqslant x}\alpha (n)|=o(\sqrt {x})$ as x→∞$x \rightarrow \infty$，这个问题被哈珀以强形式解决了。我们利用Saksman和Webb在zeta函数随机模型上的一个结果，给出了这个猜想的简短证明。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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