{"title":"关于素数计数函数的拉马努金不等式的注解","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":"{\"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}","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}
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.