Robin's Theorem

Consider the inequality

for integer , where  is the divisor function and  is the Euler-Mascheroni constant. This holds for 7, 11, 13, 14, 15, 17, 19, ... (OEIS A091901), and is false for 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, and 5040 (OEIS A067698).

Robin's theorem states that the truth of the inequality for all  is equivalent to the Riemann hypothesis (Robin 1984; Havil 2003, p. 207).

REFERENCES:

Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, 2003.

Nicolas, J.-L. "Petites valeurs de la fonction d'Euler." J. Number Th. 17, 375-388, 1983.

Robin, G. "Grandes Valeurs de la fonction somme des diviseurs et hypothèse de Riemann." J. Math. Pures Appl. 63, 187-213, 1984.

Schoenfeld, L. "Sharper Bounds for the Chebyshev Functions  and . II." Math. Comput. 30, 337-360, 1976.

Sloane, N. J. A. Sequences A067698 and A091901 in "The On-Line Encyclopedia of Integer Sequences."