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Article Dans Une Revue Communications in Mathematics Année : 2021

Remarks on Ramanujan’s inequality concerning the prime counting function

Mehdi Hassani
  • Fonction : Auteur correspondant

Résumé

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.
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Dates et versions

hal-03665004 , version 1 (11-05-2022)

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Paternité - Pas d'utilisation commerciale - Pas de modification

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Mehdi Hassani. Remarks on Ramanujan’s inequality concerning the prime counting function. Communications in Mathematics, 2021, Volume 29 (2021), Issue 3 (3), pp.473 - 482. ⟨10.2478/cm-2021-0014⟩. ⟨hal-03665004⟩
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