Estimates for $\pi (x)$ 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 $x\ge H_0$.

10 pages; v2: updated references