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First Published
16 Jun 2010
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2 times per year
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English
access type Open Access

Remarks on Ramanujan’s inequality concerning the prime counting function

Published Online: 27 Apr 2021
Page range: -
Received: 08 Sep 2019
Accepted: 01 Oct 2019
Journal Details
License
Format
Journal
First Published
16 Jun 2010
Publication timeframe
2 times per year
Languages
English
Abstract

In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π(x)2<exlogxπ(xe)\pi {\left( x \right)^2} < {{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.

Keywords

MSC 2010

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