On Solving a Curious Inequality of Ramanujan

Abstract

Ramanujan proved that the inequality 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.

2010 AMS Subject Classification::

• Primar 11N05
• Secondary 11Y99

Keywords:

• Ramanujan
• prime counting function
• Riemann hypothesis
• primesieve

Notes

¹It should be noted that [Mossinghoff and Trudgian 14] recently improved on the value of R in Lemma 2.2. Specifically, the authors show that one can take R = 6.315. This can be used with a = 3130 to prove Theorem 1.2 for all x ≥ exp (9394).

²Precisely 319 870 505 122 591 of them.

³At the same time, we double-checked Oliviera e Silva’s computations, and as expected, we found no discrepancies.

⁴Available at http://code.google.com/p/primesieve/.

⁵See http://www.acrc.bris.ac.uk/acrc/. Each node comprises two 8-core Intel Xeon E5-2670 CPUs running at 2.6 GHz, and we ran with one thread per core.

⁶Actually, we used the midpoint of the interval computed for −f(x).