Mehdi Hassani - Remarks on Ramanujan’s inequality concerning the prime counting function

cm:9533 - Communications in Mathematics, December 23, 2021, Volume 29 (2021), Issue 3 - https://doi.org/10.2478/cm-2021-0014
Remarks on Ramanujan’s inequality concerning the prime counting functionArticle

Authors: Mehdi Hassani ORCID

    In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that π ( x 2 ) < e x log x π ( x e ) \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 x log x {x \over {\log x}} on its right hand side by the factor x log x - 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.


    Volume: Volume 29 (2021), Issue 3
    Published on: December 23, 2021
    Imported on: May 11, 2022
    Keywords: General Mathematics,[MATH]Mathematics [math]

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