Gronwall's Theorem

Let $\sigma(n)$ be the Divisor Function. Then

$$\overline{\lim_{n\to\infty}}\, {\sigma(n)\over n\ln \ln n}=e^\gamma,$$

where $\gamma$ is the Euler-Mascheroni Constant. Ramanujan independently discovered a less precise version of this theorem (Berndt 1994). Robin (1984) showed that the validity of the inequality

$$\sigma(n)<e^\gamma n\ln\ln n$$

for $n\geq 5041$ is equivalent to the Riemann Hypothesis.

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. 37, 113-122, 1913.

Nicholas, J.-L. ``On Highly Composite Numbers.'' In Ramanujan Revisited: Proceedings of the Centenary Conference (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 hypothèse de Riemann.'' J. Math. Pures Appl. 63, 187-213, 1984.