Gronwall's theorem

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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.

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

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Theorem

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