Robin’s theorem

Let $\sigma(n)$ be the sum of the positive divisors of an integer $n$ and

G(n)\;:=\;\frac{\sigma(n)}{n\ln(\ln{n})}\qquad(n\;=\;2,\,3,\,4,\,\ldots).

The Riemann Hypothesis is true if and only if

G(n)\;<\;e^{\gamma}\quad\mbox{for all }\,n\;>\;7!

where $\gamma$ is the Euler–Mascheroni constant.

References

1 G. Robin: Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann. $-$ J. Math. Pures Appl. 14 (1984) 187–213.
2 J. C. Lagarias: An elementary problem equivalent to the Riemann hypothesis. $-$ Amer. Math. Monthly 109 (2002), 534–543. Available at http://arxiv.org/abs/math/0008177