{"title":"有傅里叶系数的除数函数的移位卷积和","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":50110,"journal":{"name":"Journal of Number Theory","volume":"261 ","pages":"Pages 1-21"},"PeriodicalIF":0.6000,"publicationDate":"2024-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"261 \",\"pages\":\"Pages 1-21\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-03-20\",\"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/S0022314X24000659\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24000659","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
摘要
设 f(z) 是全模态群 SL2(Z) 权重为 κ 的全形尖顶形式。用 λf(n) 表示其 n 次归一化傅里叶系数。让 τk(n) 表示 k≥4 的第 k 个除数函数。本文考虑的是移位卷积和∑n≤Xτk(n)λf(n+h)。我们成功地得到了一个非微妙的上界,它与移位参数 h 一致。
Shifted convolution sums of divisor functions with Fourier coefficients
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.
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.
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