{"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}
引用次数: 0
Abstract
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.