Gronwall’s theorem

The function

$$G(n):=\frac{\sigma (n)}{n\ln (\ln n)}\mathit{\hspace{1em}\hspace{1em}}(n=\mathrm{\hspace{0.33em}2},\mathrm{\hspace{0.17em}3},\mathrm{\hspace{0.17em}4},\mathrm{\dots }),$$

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

$$\underset{n\to \mathrm{\infty }}{lim\; sup}G(n)=e^{\gamma }$$

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