{"title":"On the divisor function over Piatetski-Shapiro sequences","authors":"Hui Wang, Yu Zhang","doi":"10.21136/CMJ.2023.0205-22","DOIUrl":null,"url":null,"abstract":"Let [x] be an integer part of x and d(n) be the number of positive divisor of n. Inspired by some results of M. Jutila (1987), we prove that for 1<c<65\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < c < {6 \\over 5}$$\\end{document}∑n≤xd([nc])=cxlogx+(2γ−c)x+O(xlogx),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sum\\limits_{n \\le x} {d([{n^c}]) = cx\\,\\log x + (2{\\rm{\\gamma }} - c)x + O\\left( {{x \\over {\\log x}}} \\right),} $$\\end{document} where γ is the Euler constant and [nc] is the Piatetski-Shapiro sequence. This gives an improvement upon the classical result of this problem.","PeriodicalId":50596,"journal":{"name":"Czechoslovak Mathematical Journal","volume":"73 1","pages":"613 - 620"},"PeriodicalIF":0.4000,"publicationDate":"2023-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Czechoslovak Mathematical Journal","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.21136/CMJ.2023.0205-22","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
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Abstract
Let [x] be an integer part of x and d(n) be the number of positive divisor of n. Inspired by some results of M. Jutila (1987), we prove that for 1
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