Squarefrees are Gaussian in short intervals

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-12-22 DOI:10.1515/crelle-2022-0066

O. Gorodetsky, Alexander P. Mangerel, B. Rodgers

引用次数: 2

Abstract

Abstract We show that counts of squarefree integers up to X in short intervals of size H tend to a Gaussian distribution as long as H → ∞ {H\to\infty} and H = X o ⁢ ( 1 ) {H=X^{o(1)}} . This answers a question posed by R. R. Hall in 1989. More generally, we prove a variant of Donsker’s theorem, showing that these counts scale to a fractional Brownian motion with Hurst parameter 1 / 4 {1/4} . In fact, we are able to prove these results hold in general for collections of B-free integers as long as the sieving set B satisfies a very mild regularity property, for Hurst parameter varying with the set B.

查看原文本刊更多论文

在短间隔内，无平方是高斯分布

摘要我们证明了只要H→∞H {\to\infty}和H=X o¹(1){H{=X}}^{o(1)，在大小为H的短间隔内，小于X的无平方整数的计数趋向于高斯分布}。这回答了r.r.提出的一个问题。Hall, 1989年。更一般地说，我们证明了Donsker定理的一个变体，表明这些计数缩放到Hurst参数为1/4 /4的分数布朗运动。事实上，我们能够证明这些结果一般适用于无B整数的集合，只要筛选集B满足一个非常温和的正则性，对于Hurst参数随集合B变化。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.