Except the original Gronwall'sGrönwall's theorem that $\limsup_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma}$,$$\limsup_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma},$$ and the two variants $\limsup_{n\ is\ squarefree} \frac{\sigma(n)}{n \log \log n} = \frac{6e^{\gamma}}{\pi^2}$$$\limsup_{\begin{smallmatrix} n\to\infty\cr n\ \text{is square free}\end{smallmatrix}} \frac{\sigma(n)}{n \log \log n} = \frac{6e^{\gamma}}{\pi^2}$$ and $\limsup_{n\ is\ odd} \frac{\sigma(n)}{n \log \log n} = \frac{e^{\gamma}}{2}$$$\limsup_{\begin{smallmatrix} n\to\infty\cr n\ \text{is odd}\end{smallmatrix}} \frac{\sigma(n)}{n \log \log n} = \frac{e^{\gamma}}{2}$$ that have been proven herehere, are there any other similar statements known?
