On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $\pi(x)$

TL;DR

This paper proves Ramanujan's inequality for the prime counting function (x) unconditionally for large x, using sharper bounds derived from estimates involving the Chebyshev Theta Function.

Contribution

It provides a new unconditional proof of Ramanujan's inequality for (x) for x ^{43.51}, employing improved bounds from Chebyshev Theta Function estimates.

Findings

Ramanujan's inequality holds for all x  e^{43.51}

Derived sharper bounds for (x) using (x) estimates

Improved the known conditions for the inequality to hold unconditionally

Abstract

This article provides a proof that the Ramanujan's Inequality given by, $$\pi(x)^2 < \frac{e x}{\log x} \pi\Big(\frac{x}{e}\Big)$$ holds unconditionally for every $x\geq \exp(43.5102147)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.