On Solving a Curious Inequality of Ramanujan

TL;DR

This paper proves Ramanujan's inequality for all sufficiently large x unconditionally and completely on the Riemann Hypothesis, identifying the largest counterexample and providing explicit bounds.

Contribution

It provides an explicit bound for the inequality's validity and solves it completely under the Riemann Hypothesis, identifying the largest counterexample.

Findings

Inequality holds for all x ≥ exp(9658) unconditionally.

Under the Riemann Hypothesis, the inequality is fully solved.

The largest counterexample is x=38,358,837,682.

Abstract

Ramanujan proved that the inequality $\pi(x)^2 < \frac{e x}{\log x} \pi\Big(\frac{x}{e}\Big)$ holds for all sufficiently large values of $x$. Using an explicit estimate for the error in the prime number theorem, we show unconditionally that it holds if $x \geq \exp(9658)$. Furthermore, we solve the inequality completely on the Riemann Hypothesis, and show that $x=38, 358, 837, 682$ is the largest integer counterexample.