Gronwall's Theorem

Let  be the divisor function. Then

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

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.