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