{"title":"A note on Fourier coefficients of Hecke eigenforms in short intervals","authors":"Sanoli Gun, Sunil Naik","doi":"10.1007/s00605-024-01984-w","DOIUrl":null,"url":null,"abstract":"<p>In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in an interval of length <span>\\(\\frac{x}{(\\log x)^A}\\)</span> for any <span>\\(A>0\\)</span>, modifying an earlier work of Balog and Ono. Furthermore, we need to strengthen a work of Rouse-Thorner to derive a lower bound for the largest prime factor of Fourier coefficients in an interval of length <span>\\(x^{1/2 + \\epsilon }\\)</span> for any <span>\\(\\epsilon >0\\)</span>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-29","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/s00605-024-01984-w","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
In this article, we investigate large prime factors of Fourier coefficients of non-CM normalized cuspidal Hecke eigenforms in short intervals. One of the new ingredients involves deriving an explicit version of Chebotarev density theorem in an interval of length \(\frac{x}{(\log x)^A}\) for any \(A>0\), modifying an earlier work of Balog and Ono. Furthermore, we need to strengthen a work of Rouse-Thorner to derive a lower bound for the largest prime factor of Fourier coefficients in an interval of length \(x^{1/2 + \epsilon }\) for any \(\epsilon >0\).