{"title":"Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.","authors":"Kostadinka Lapkova","doi":"10.1007/s00605-017-1061-y","DOIUrl":null,"url":null,"abstract":"<p><p>Consider the divisor sum <math><mrow><msub><mo>∑</mo><mrow><mi>n</mi><mo>≤</mo><mi>N</mi></mrow></msub><mi>τ</mi><mrow><mo>(</mo><msup><mi>n</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>b</mi><mi>n</mi><mo>+</mo><mi>c</mi><mo>)</mo></mrow></mrow></math> for integers <i>b</i> and <i>c</i>. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of <math><mrow><mi>D</mi><mo>(</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></math> -quadruples.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1007/s00605-017-1061-y","citationCount":"9","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00605-017-1061-y","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"2017/5/27 0:00:00","PubModel":"Epub","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 9
Abstract
Consider the divisor sum for integers b and c. We extract an asymptotic formula for the average divisor sum in a convenient form, and provide an explicit upper bound for this sum with the correct main term. As an application we give an improvement of the maximal possible number of -quadruples.
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