Helson猜想的简短证明

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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Helson conjectured that <math>\n <semantics>\n <mrow>\n <mrow>\n <mi>E</mi>\n <mo>|</mo>\n </mrow>\n <msub>\n <mo>∑</mo>\n <mrow>\n <mi>n</mi>\n <mo>⩽</mo>\n <mi>x</mi>\n </mrow>\n </msub>\n <mrow>\n <mi>α</mi>\n <mrow>\n <mo>(</mo>\n <mi>n</mi>\n <mo>)</mo>\n </mrow>\n <mo>|</mo>\n <mo>=</mo>\n <mi>o</mi>\n </mrow>\n <mrow>\n <mo>(</mo>\n <msqrt>\n <mi>x</mi>\n </msqrt>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathbb {E}|\\sum _{n\\leqslant x}\\alpha (n)|=o(\\sqrt {x})$</annotation>\n </semantics></math> as <math>\n <semantics>\n <mrow>\n <mi>x</mi>\n <mo>→</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$x \\rightarrow \\infty$</annotation>\n </semantics></math>, 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.","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 4","pages":"1065-1076"},"PeriodicalIF":0.8000,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70015","citationCount":"0","resultStr":"{\"title\":\"A short proof of Helson's conjecture\",\"authors\":\"Ofir Gorodetsky, Mo Dick Wong\",\"doi\":\"10.1112/blms.70015\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let <math>\\n <semantics>\\n <mrow>\\n <mi>α</mi>\\n <mo>:</mo>\\n <mi>N</mi>\\n <mo>→</mo>\\n <msup>\\n <mi>S</mi>\\n <mn>1</mn>\\n </msup>\\n </mrow>\\n <annotation>$\\\\alpha \\\\colon \\\\mathbb {N}\\\\rightarrow S^1$</annotation>\\n </semantics></math> be the Steinhaus multiplicative function: a completely multiplicative function such that <math>\\n <semantics>\\n <msub>\\n <mrow>\\n <mo>(</mo>\\n <mi>α</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>p</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>)</mo>\\n </mrow>\\n <mrow>\\n <mi>p</mi>\\n <mspace></mspace>\\n <mtext>prime</mtext>\\n </mrow>\\n </msub>\\n <annotation>$(\\\\alpha (p))_{p\\\\text{ prime}}$</annotation>\\n </semantics></math> are i.i.d. random variables uniformly distributed on the complex unit circle <math>\\n <semantics>\\n <msup>\\n <mi>S</mi>\\n <mn>1</mn>\\n </msup>\\n <annotation>$S^1$</annotation>\\n </semantics></math>. Helson conjectured that <math>\\n <semantics>\\n <mrow>\\n <mrow>\\n <mi>E</mi>\\n <mo>|</mo>\\n </mrow>\\n <msub>\\n <mo>∑</mo>\\n <mrow>\\n <mi>n</mi>\\n <mo>⩽</mo>\\n <mi>x</mi>\\n </mrow>\\n </msub>\\n <mrow>\\n <mi>α</mi>\\n <mrow>\\n <mo>(</mo>\\n <mi>n</mi>\\n <mo>)</mo>\\n </mrow>\\n <mo>|</mo>\\n <mo>=</mo>\\n <mi>o</mi>\\n </mrow>\\n <mrow>\\n <mo>(</mo>\\n <msqrt>\\n <mi>x</mi>\\n </msqrt>\\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation>$\\\\mathbb {E}|\\\\sum _{n\\\\leqslant x}\\\\alpha (n)|=o(\\\\sqrt {x})$</annotation>\\n </semantics></math> as <math>\\n <semantics>\\n <mrow>\\n <mi>x</mi>\\n <mo>→</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$x \\\\rightarrow \\\\infty$</annotation>\\n </semantics></math>, 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.\",\"PeriodicalId\":55298,\"journal\":{\"name\":\"Bulletin of the London Mathematical Society\",\"volume\":\"57 4\",\"pages\":\"1065-1076\"},\"PeriodicalIF\":0.8000,\"publicationDate\":\"2025-02-07\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70015\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Bulletin of the London Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/blms.70015\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the London Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/blms.70015","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

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。

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A short proof of Helson's conjecture

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.