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Robin's Theorem


RobinsTheorem

Consider the inequality

 sigma(n)<e^gammanlnlnn

for integer n>1, where sigma(n) is the divisor function and gamma 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 n>=5041 is equivalent to the Riemann hypothesis (Robin 1984; Havil 2003, p. 207).


See also

Divisor Function, Gronwall's Theorem, Riemann Hypothesis

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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 theta(x) and psi(x). II." Math. Comput. 30, 337-360, 1976.Sloane, N. J. A. Sequences A067698 and A091901 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Robin's Theorem

Cite this as:

Weisstein, Eric W. "Robin's Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RobinsTheorem.html

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