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

Subham De

This article provides a proof that the Ramanujan's Inequality given by, $$\pi (x{)}^2<\frac{ex}{\log x}\pi (\frac{x}{e})$$ holds unconditionally for every $x\geq \exp(43.510214…