On Solving a Curious Inequality of Ramanujan

Abstract

Ramanujan proved that the inequality $\pi (x{)}^2<\frac{ex}{\log x}\pi (\frac{x}{e})$ 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 $x\ge \exp (9658)$. Furthermore, we solve the inequality completely on the Riemann Hypothesis, and show that $x=38,358,837,682$ is the largest integer counterexample.

11 pages, 1 figure; feedback welcome