On certain inequalities for the prime counting function – Part III

Abstract

As a continuation of [10] and [11], we offer some new inequalities for the prime counting function $\pi (x).$ Particularly, a multiplicative analogue of the Hardy–Littlewood conjecture is provided. Improvements of the converse of Landau's inequality are given. Some results on $\pi (p_n^2)$ are offered, $p_n$ denoting the $n$-th prime number. Results on $\pi (\pi (x))$ are also considered.