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Gronwall's Theorem

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GronwallsTheorem

Let sigma(n) be the divisor function. Then

limsup_(n->infty)(sigma(n))/(nlnlnn)=e^gamma,

where gamma is the Euler-Mascheroni constant. Ramanujan independently discovered a less precise version of this theorem (Berndt 1985).

See also

Divisor Function, Robin's Theorem

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References

Berndt, B. C. Ramanujan's Notebooks: Part I. New York: Springer-Verlag, p. 94, 1985.Gronwall, T. H. "Some Asymptotic Expressions in the Theory of Numbers." Trans. Amer. Math. Soc. 14, 113-122, 1913.Nicolas, J.-L. "On Highly Composite Numbers." In Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987 (Ed. G. E. Andrews, B. C. Berndt, and R. A. Rankin). Boston, MA: Academic Press, pp. 215-244, 1988.Robin, G. "Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann." J. Math. Pures Appl. 63, 187-213, 1984.

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Gronwall's Theorem

Cite this as:

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

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