{"title":"稀疏整数集上的傅立叶系数振荡","authors":"Lalit Vaishya","doi":"10.1007/s00605-024-01989-5","DOIUrl":null,"url":null,"abstract":"<p>Let <span>\\(f \\in S_{k}(\\Gamma _{0}(N))\\)</span> be a normalized Hecke eigenforms of integral weight <i>k</i> and level <span>\\(N \\ge 1\\)</span>. In the article, we establish the asymptotics of power moment associated to the sequences <span>\\(\\{\\lambda _{f \\otimes f \\otimes f}(\\mathcal {Q}(\\underline{x}))\\}_{\\mathcal {Q} \\in \\mathcal {S}_{D}, \\underline{x} \\in \\mathbb {Z}^{2}}\\)</span> and <span>\\(\\{\\lambda _{f \\otimes \\mathrm{sym^{2}}f}(\\mathcal {Q}(\\underline{x}))\\}_{\\mathcal {Q} \\in \\mathcal {S}_{D}, \\underline{x} \\in \\mathbb {Z}^{2}}\\)</span> where <span>\\(\\mathcal {S}_{D}\\)</span> denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant <span>\\(D < 0.\\)</span> As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Oscillations of Fourier coefficients over the sparse set of integers\",\"authors\":\"Lalit Vaishya\",\"doi\":\"10.1007/s00605-024-01989-5\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <span>\\\\(f \\\\in S_{k}(\\\\Gamma _{0}(N))\\\\)</span> be a normalized Hecke eigenforms of integral weight <i>k</i> and level <span>\\\\(N \\\\ge 1\\\\)</span>. In the article, we establish the asymptotics of power moment associated to the sequences <span>\\\\(\\\\{\\\\lambda _{f \\\\otimes f \\\\otimes f}(\\\\mathcal {Q}(\\\\underline{x}))\\\\}_{\\\\mathcal {Q} \\\\in \\\\mathcal {S}_{D}, \\\\underline{x} \\\\in \\\\mathbb {Z}^{2}}\\\\)</span> and <span>\\\\(\\\\{\\\\lambda _{f \\\\otimes \\\\mathrm{sym^{2}}f}(\\\\mathcal {Q}(\\\\underline{x}))\\\\}_{\\\\mathcal {Q} \\\\in \\\\mathcal {S}_{D}, \\\\underline{x} \\\\in \\\\mathbb {Z}^{2}}\\\\)</span> where <span>\\\\(\\\\mathcal {S}_{D}\\\\)</span> denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant <span>\\\\(D < 0.\\\\)</span> As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences.</p>\",\"PeriodicalId\":18913,\"journal\":{\"name\":\"Monatshefte für Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte für Mathematik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-024-01989-5\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01989-5","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Oscillations of Fourier coefficients over the sparse set of integers
Let \(f \in S_{k}(\Gamma _{0}(N))\) be a normalized Hecke eigenforms of integral weight k and level \(N \ge 1\). In the article, we establish the asymptotics of power moment associated to the sequences \(\{\lambda _{f \otimes f \otimes f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q} \in \mathcal {S}_{D}, \underline{x} \in \mathbb {Z}^{2}}\) and \(\{\lambda _{f \otimes \mathrm{sym^{2}}f}(\mathcal {Q}(\underline{x}))\}_{\mathcal {Q} \in \mathcal {S}_{D}, \underline{x} \in \mathbb {Z}^{2}}\) where \(\mathcal {S}_{D}\) denotes the set of inequivalent primitive integral positive-definite binary quadratic forms (reduced forms) of fixed discriminant \(D < 0.\) As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences.