Geodesic
Remarks on Ramanujan's inequality concerning the prime counting function
Communications in Mathematics, Tome 29 (2021) no. 3, pp. 473-482

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In this paper we investigate Ramanujan's inequality concerning the prime counting function, asserting that $\pi (x)^2\frac {\mathrm{e} \,x}{\log x}\,\pi \left (\frac {x}{\mathrm{e} }\right )$ 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 $\frac {x}{\log x}$ on its right hand side by the factor $\frac {x}{\log x-h}$ for a given $h$, and by replacing the numerical factor $\mathrm{e} $ by a given positive $\alpha $. Finally, we introduce and study inequalities analogous to Ramanujan's inequality.
In this paper we investigate Ramanujan's inequality concerning the prime counting function, asserting that $\pi (x)^2\frac {\mathrm{e} \,x}{\log x}\,\pi \left (\frac {x}{\mathrm{e} }\right )$ 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 $\frac {x}{\log x}$ on its right hand side by the factor $\frac {x}{\log x-h}$ for a given $h$, and by replacing the numerical factor $\mathrm{e} $ by a given positive $\alpha $. Finally, we introduce and study inequalities analogous to Ramanujan's inequality.
Classification : 11A41
Keywords: Prime numbers; Ramanujan's inequality; Riemann hypothesis 
Hassani, Mehdi. Remarks on Ramanujan's inequality concerning the prime counting function. Communications in Mathematics, Tome 29 (2021) no. 3, pp. 473-482. http://geodesic.mathdoc.fr/item/COMIM_2021_29_3_a10/
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