{"title":"不可约二次多项式的平均除数的显式上界。","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":"{\"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}","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}
Explicit upper bound for the average number of divisors of irreducible quadratic polynomials.
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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