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

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 
@article{COMIM_2021__29_3_a10,
     author = {Hassani, Mehdi},
     title = {Remarks on {Ramanujan's} inequality concerning the prime counting function},
     journal = {Communications in Mathematics},
     pages = {473--482},
     publisher = {mathdoc},
     volume = {29},
     number = {3},
     year = {2021},
     mrnumber = {4355420},
     zbl = {07484381},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a10/}
}
TY  - JOUR
AU  - Hassani, Mehdi
TI  - Remarks on Ramanujan's inequality concerning the prime counting function
JO  - Communications in Mathematics
PY  - 2021
SP  - 473
EP  - 482
VL  - 29
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a10/
LA  - en
ID  - COMIM_2021__29_3_a10
ER  - 
%0 Journal Article
%A Hassani, Mehdi
%T Remarks on Ramanujan's inequality concerning the prime counting function
%J Communications in Mathematics
%D 2021
%P 473-482
%V 29
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/COMIM_2021__29_3_a10/
%G en
%F COMIM_2021__29_3_a10
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/