{"title":"Shifted convolution sums of divisor functions with Fourier coefficients","authors":"Miao Lou","doi":"10.1016/j.jnt.2024.02.014","DOIUrl":null,"url":null,"abstract":"<div><p>Let <span><math><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo></math></span> be a holomorphic cusp form of weight <em>κ</em> for the full modular group <span><math><mi>S</mi><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>Z</mi><mo>)</mo></math></span>. Denote its <em>n</em>-th normalized Fourier coefficient by <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. Let <span><math><msub><mrow><mi>τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote that <em>k</em>-th divisor function with <span><math><mi>k</mi><mo>≥</mo><mn>4</mn></math></span>. In this paper, we consider the shifted convolution sum<span><span><span><math><munder><mo>∑</mo><mrow><mi>n</mi><mo>≤</mo><mi>X</mi></mrow></munder><msub><mrow><mi>τ</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>f</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>+</mo><mi>h</mi><mo>)</mo><mo>.</mo></math></span></span></span> We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter <em>h</em>.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000659","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let be a holomorphic cusp form of weight κ for the full modular group . Denote its n-th normalized Fourier coefficient by . Let denote that k-th divisor function with . In this paper, we consider the shifted convolution sum We succeed in obtaining a non-trivial upper bound, which is uniform in the shift parameter h.