{"title":"The circle method and shifted convolution sums involving the divisor function","authors":"Guangwei Hu, Huixue Lao","doi":"10.1016/j.jnt.2024.03.007","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>Q</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> be a positive definite integral quadratic form with the determinant <em>D</em> being squarefree, and <span><math><mi>r</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span> denote the number of representations of <em>n</em> by the quadratic form <em>Q</em>. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function <span><math><mi>d</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> and Fourier coefficients <span><math><mi>r</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>Q</mi><mo>)</mo></math></span>. With more efforts, our method should have a number of applications for other multiplicative functions.</p></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"262 ","pages":"Pages 1-27"},"PeriodicalIF":0.6000,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000775","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a positive definite integral quadratic form with the determinant D being squarefree, and denote the number of representations of n by the quadratic form Q. In this paper, we apply the Hardy-Littlewood-Kloosterman circle method to derive the asymptotic formula for the shifted convolution sum of the divisor function and Fourier coefficients . With more efforts, our method should have a number of applications for other multiplicative functions.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.