{"title":"Asymptotic and non-asymptotic results for a binary additive problem involving Piatetski-Shapiro numbers","authors":"Yuuya Yoshida","doi":"10.1016/j.jnt.2024.06.012","DOIUrl":null,"url":null,"abstract":"<div><p>For all <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> with <span><math><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>></mo><mn>5</mn><mo>/</mo><mn>3</mn></math></span>, we show that the number of pairs <span><math><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of positive integers with <span><math><mi>N</mi><mo>=</mo><mo>⌊</mo><msubsup><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msubsup><mo>⌋</mo><mo>+</mo><mo>⌊</mo><msubsup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msubsup><mo>⌋</mo></math></span> is equal to <span><math><mi>Γ</mi><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mi>Γ</mi><mo>(</mo><mn>1</mn><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mi>Γ</mi><msup><mrow><mo>(</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>o</mi><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn><mo>/</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>N</mi><mo>→</mo><mo>∞</mo></math></span>, where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when <span><math><msub><mrow><mi>α</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span> lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.</p></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022314X24001562/pdfft?md5=f340a4f5d9777bfe3886facce83ff86f&pid=1-s2.0-S0022314X24001562-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001562","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
For all with , we show that the number of pairs of positive integers with is equal to as , where Γ denotes the gamma function. Moreover, we show a non-asymptotic result for the same counting problem when lie in a larger range than the above. Finally, we give some asymptotic formulas for similar counting problems in a heuristic way.