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 $\pi(x)$.
Categories: math.GM