Highly Influenced

Ramanujan proved that the inequality π(x) < ex 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… (More)