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).
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."
,
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).
is equivalent to the Riemann
hypothesis (Robin 1984; Havil 2003, p. 207).
and 
.
II." Math. Comput. 30, 337-360, 1976.Sloane, N. J. A.
Sequences