On Solving a Curious Inequality of Ramanujan

2014/07/07 by Platt, Dave, Dudek, Adrian
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1407.1901

Abstract

Ramanujan proved that the inequality π(x)² < (e x)/(log x) π((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 ≥ 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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