Remarks on Ramanujan’s inequality concerning the prime counting function

Q3 Mathematics
M. Hassani
{"title":"Remarks on Ramanujan’s inequality concerning the prime counting function","authors":"M. Hassani","doi":"10.2478/cm-2021-0014","DOIUrl":null,"url":null,"abstract":"Abstract In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π(x2)<exlogxπ(xe)\\pi \\left( {{x^2}} \\right) < {{ex} \\over {\\log x}}\\pi \\left( {{x \\over e}} \\right) for x sufficiently large. First, we study its sharpness by giving full asymptotic expansions of its left and right hand sides expressions. Then, we discuss the structure of Ramanujan’s inequality, by replacing the factor xlogx{x \\over {\\log x}} on its right hand side by the factor xlogx-h{x \\over {\\log x - h}} for a given h, and by replacing the numerical factor e by a given positive α. Finally, we introduce and study inequalities analogous to Ramanujan’s inequality.","PeriodicalId":37836,"journal":{"name":"Communications in Mathematics","volume":"29 1","pages":"473 - 482"},"PeriodicalIF":0.0000,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"4","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/cm-2021-0014","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 4

Abstract

Abstract In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π(x2)
关于素数计数函数的拉马努金不等式的注解
摘要在本文中,我们研究了关于素数计数函数的Ramanujan不等式,断言对于足够大的x,π(x2)
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
Communications in Mathematics
Communications in Mathematics Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
26
审稿时长
45 weeks
期刊介绍: Communications in Mathematics publishes research and survey papers in all areas of pure and applied mathematics. To be acceptable for publication, the paper must be significant, original and correct. High quality review papers of interest to a wide range of scientists in mathematics and its applications are equally welcome.
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信