that the Riemann hypothesis is equivalent to the assertion that the following inequality is true for all n > 5040: (where γ is the Euler–Mascheroni constant) This inequality is known to fail for 27 numbers : 2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040 Robin showed that if the Riemann hypothesis is true then n = 5040 is the last integer for which it fails. The inequality is now known as Robin's inequality after his work. It is known that Robin's inequality, if it ever fails to hold, will fail for a colossally abundant number n; thus the Riemann hypothesis is in fact equivalent to Robin's inequality holding for every colossally abundant number n > 5040. In 2001–2 Lagarias demonstrated an alternate form of Robin's assertion which requires no exceptions, using the harmonic numbers instead of log: Or, other than the 8 exceptions of n = 1, 2, 3, 4, 6, 12, 24, 60: References External links Keith Briggs on colossally abundant numbers and the Riemann hypothesis MathWorld entry Notes on the Riemann hypothesis and abundant numbers More on Robin's formulation of the RH Divisor function Integer sequences