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In this paper we use refined approximations for Chebyshev's$\vartheta$-function to establish new explicit estimates for the prime countingfunction $\pi(x)$, 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 isdefined to be the smallest positive integer so that Ramanujan's prime countinginequality holds for every $x \geq H_0$.

Estimates for $\pi(x)$ for large values of $x$ and Ramanujan's prime counting inequality