Title: Estimates for $π(x)$ for large values of $x$ and Ramanujan's prime counting inequality,
Abstract: In this paper we use refined approximations for Chebyshev's
$\vartheta$-function to establish new explicit estimates for the prime counting
function $\pi(x)$, which improve the current best estimates for large values of
$x$. As an application we find an upper bound for the number $H_0$ which is
defined to be the smallest positive integer so that Ramanujan's prime counting
inequality holds for every $x \geq H_0$.