Gronwall’s theorem

The function

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

in which $\sigma(n)$ means the sum of the positive divisors of $n$ , satisfies the equation

\limsup_{n\to\infty}\,G(n)\;=\;e^{\gamma}

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

References

1 T. H. Gronwall: Some asymptotic expressions in the theory of numbers. $-$ Trans. Amer. Math. Soc. 14 (1913) 113–122.