关于Piatetski-Shapiro序列上的除数函数

IF 0.4 4区数学 Q4 MATHEMATICS Czechoslovak Mathematical Journal Pub Date : 2023-03-06 DOI:10.21136/CMJ.2023.0205-22

Hui Wang, Yu Zhang

{"title":"关于Piatetski-Shapiro序列上的除数函数","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":"{\"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}","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}