Remarks on Ramanujan’s inequality concerning the prime counting function

Abstract In this paper we investigate Ramanujan’s inequality concerning the prime counting function, asserting that ? ( x 2 ) < e log \pi \left( {{x^2}} \right) < {{ex} \over {\log x}}\pi {{x e}} for x sufficiently large. First, study its sharpness by giving full asymptotic expansions of left and right hand sides expressions. Then, discuss structure inequality, replacing factor {x x}} on side - h - h}} a given h , numerical e positive ?. Finally, introduce inequalities analogous to inequality.