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?

$\begingroup$ Do you mean "similar statements" for the sum of divisors function $\sigma(n)=\sum_{d\mid n}d$? Because there are plenty of other multiplicative function for which similar asymptotics are known. $\endgroup$ – Wadim Zudilin Jun 15 '10 at 11:29

$\begingroup$ "that have been proven here," Where? $\endgroup$ – Andrés E. Caicedo Jun 15 '10 at 13:47

$\begingroup$ I fixed the typos. Theorem 9 in the cited preprint contains 5 more similar asymptotics. I wonder what is wanted. $\endgroup$ – Wadim Zudilin Jun 15 '10 at 14:50

$\begingroup$ Maybe a statement with $\limsup_{\begin{smallmatrix}n\to \infty \cr n\in S\end{smallmatrix}}(\cdots)=d_S e^{\gamma}$. Where $d_S$ is the density of $S$. $\endgroup$ – Gjergji Zaimi Jun 15 '10 at 15:02

$\begingroup$ I'm sorry for not being clear enough, it's my first question here, though. I mean similar statments for the $\sigma(n)$ function, not necessarily asymptotics, but anything that involves limit points of the function $\frac{\sigma(n)}{n \log \log n}$. For example, is there an important sequence $a_n$ such that $\frac{\sigma{a_n}}{n \log \log n}$ converges, besides the sequence of primes? A result that establishes the connection between the density and the limit superior? Etc. Nothing particular. $\endgroup$ – nikmil Jun 15 '10 at 22:53
One example possessing a limit is the colossally abundant numbers of Alaoglu and Erdos,
http://en.wikipedia.org/wiki/Colossally_abundant_number
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.