On Solving a Curious Inequality of Ramanujan

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