On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $\ddot{I}€(x)$

Subham De

This article provides a proof that the Ramanujan's Inequality given by, $$\ddot{I}€(x{)}^2<\frac{ex}{\log x}\ddot{I}€(\frac{x}{e})$$ holds unconditionally for every $x\geq \exp(43.5102…