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

Abstract

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 $x\ge \exp (43.5102147)$. 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 $\ddot{I}€(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.

Research Article. arXiv admin note: substantial text overlap with arXiv:2407.12052