  This research article provides an unconditional proof of an inequality
proposed by Srinivasa Ramanujan involving the Prime Counting Function $\pi(x)$,
  \begin{align*}
  (\pi(x))^{2}<\frac{ex}{\log x}\pi\left(\frac{x}{e}\right)
  \end{align*} for every real $x\geq \exp(547)$, using specific order estimates
for the Mertens Function, $M(x)$. The proof primarily hinges upon investigating
the underlying relation between $M(x)$ and the Second Chebyshev Function,
$\psi(x)$, in addition to applying the meromorphic properties of the Riemann
Zeta Function, $\zeta(s)$ with an intention of deriving an improved
approximation for $\pi(x)$.
