On proving an Inequality of Ramanujan using Explicit Order Estimates for the Mertens Function

TL;DR

This paper unconditionally proves Ramanujan's inequality involving the prime counting function for large x, using explicit estimates of the Mertens function and properties of the Riemann zeta function.

Contribution

It introduces a novel proof of Ramanujan's inequality leveraging explicit order estimates for the Mertens function and the analytic properties of the Riemann zeta function.

Findings

Proves the inequality for all x ≥ exp(547)

Establishes a new connection between Mertens function and prime counting

Improves approximation methods for π(x)

Abstract

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 $\pi(x)$.