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arXiv · 1407.1901

On Solving a Curious Inequality of Ramanujan

Dave Platt·Adrian Dudek
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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.

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Dave Platt, Adrian Dudek. 2014-07-07. On Solving a Curious Inequality of Ramanujan. https://arxiv.org/abs/1407.1901

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