{"title":"Diophantine approximation with prime denominator in quadratic number fields under GRH","authors":"Stephan Baier, Sourav Das, Esrafil Ali Molla","doi":"10.1007/s11139-024-00942-2","DOIUrl":null,"url":null,"abstract":"<p>Matomäki proved that if <span>\\(\\alpha \\in {\\mathbb {R}}\\)</span> is irrational, then there are infinitely many primes <i>p</i> such that <span>\\(|\\alpha -a/p|\\le p^{-4/3+\\varepsilon }\\)</span> for a suitable integer <i>a</i>. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke <i>L</i>-functions.</p>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"44 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00942-2","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
Matomäki proved that if \(\alpha \in {\mathbb {R}}\) is irrational, then there are infinitely many primes p such that \(|\alpha -a/p|\le p^{-4/3+\varepsilon }\) for a suitable integer a. In this paper, we extend this result to all quadratic number fields under the condition that the Grand Riemann Hypothesis holds for their Hecke L-functions.
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