Almost sure bounds for a weighted Steinhaus random multiplicative function

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2024-08-22 DOI:10.1112/jlms.12979

Seth Hardy

{"title":"Almost sure bounds for a weighted Steinhaus random multiplicative function","authors":"Seth Hardy","doi":"10.1112/jlms.12979","DOIUrl":null,"url":null,"abstract":"We obtain almost sure bounds for the weighted sum <math>\n <semantics>\n <mrow>\n <msub>\n <mo>∑</mo>\n <mrow>\n <mi>n</mi>\n <mo>⩽</mo>\n <mi>t</mi>\n </mrow>\n </msub>\n <mfrac>\n <mrow>\n <mi>f</mi>\n <mo>(</mo>\n <mi>n</mi>\n <mo>)</mo>\n </mrow>\n <msqrt>\n <mi>n</mi>\n </msqrt>\n </mfrac>\n </mrow>\n <annotation>$\\sum _{n \\leqslant t} \\frac{f(n)}{\\sqrt {n}}$</annotation>\n </semantics></math>, where <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>(</mo>\n <mi>n</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$f(n)$</annotation>\n </semantics></math> is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated logarithm, giving sharp upper and lower bounds.","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 3","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.12979","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12979","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

Abstract

We obtain almost sure bounds for the weighted sum $\sum_{n ⩽ t} \frac{f (n)}{\sqrt{n}}$ , where $f (n)$ is a Steinhaus random multiplicative function. Specifically, we obtain the bounds predicted by exponentiating the law of the iterated logarithm, giving sharp upper and lower bounds.

查看原文本刊更多论文

加权斯坦豪斯随机乘法函数的几乎确定边界

我们得到了加权和 ∑ n ⩽ t f ( n ) n $\sum _{n \leqslant t} 的几乎确定的边界。\其中 f ( n ) $f(n)$ 是一个斯坦豪斯随机乘法函数。具体来说，我们通过迭代对数的指数化法则得到了预测的边界，给出了尖锐的上下限。

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

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.