Except the original Grönwall's theorem that $$\limsup_{n \to \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma},$$ and the two variants $$\limsup_{\begin{smallmatrix} n\to\infty\cr n\ \text{is square free}\end{smallmatrix}} \frac{\sigma(n)}{n \log \log n} = \frac{6e^{\gamma}}{\pi^2}$$ and $$\limsup_{\begin{smallmatrix} n\to\infty\cr n\ \text{is odd}\end{smallmatrix}} \frac{\sigma(n)}{n \log \log n} = \frac{e^{\gamma}}{2}$$ that have been proven here, are there any other similar statements known?
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One example possessing a limit is the colossally abundant numbers of Alaoglu and Erdos, where the limit of the Choie, Lichiardopol, Moree and Sole's $$f_1(a_n) = \frac{\sigma(a_n)}{a_n \log \log a_n}$$ is the same $$ e^\gamma .$$ That is, the limit for these numbers is the lim sup for all numbers. These are more natural than people realize. There is a simple recipe that takes some $ \epsilon > 0$ and gives an explicit factorization for the best value $n_\epsilon;$ see page 7 in the Briggs pdf "Notes on the Riemann hypothesis and abundant numbers" at the bottom of the Wikipedia entry. The exponent of a prime $p$ in the factorization of $n_\epsilon$ is $$ \left\lfloor \log_p \left( \frac{p^{1 + \epsilon} - 1}{p^\epsilon -1} \right) \right\rfloor - 1 $$ The process of making a sequence of "champion" numbers this way was invented by Ramanujan. |
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