There is a nice survey on fairly elementary methods here by J. L. Nicolas, in a 1988 book called "Ramanujan Revisited." $$ $$ Meanwhile, there is an unconditional result which has not been mentioned, for $N \geq 3$ we have $$ \sigma(N) < e^\gamma \; N \log \log N + \frac{ 0.6482 N}{\log \log N} $$ I hope I am reporting this correctly, it is from a secondary source, attribution is to G.Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. (9) 63 (1984) 187-213. $$ $$ The overall methodology is to consider the colossally abundant numbers of Alaoglu and Erdos (1944), http://en.wikipedia.org/wiki/Colossally_abundant_number
which were eventually discovered to have also been present in the original version of Ramanujan's paper Highly Composite Numbers (1915). There is some history about why that section was initially omitted, evidently a paper shortage. $$ $$ Here is a link to the first page of a related recent article, also apparently a survey, by Nicolas: http://www.springerlink.com/content/p8311481mh32145v/

LATER EDIT: The very nice survey article by Steuding that David Speyer mentions in his comment actually refers for greater detail to the book by Mark Kac, an M.A.A. Carus Monograph, called "Statistical Independence in Probability, Analysis and Number Theory." Chapter 4 is called "Primes play a game of chance" and section 2 is called "The statistics of the Euler $\phi$-function." That begins on page 54. In the section Problems, pages 62-64, we learn that $$ \frac{\sigma(n)}{n} $$ does in fact have a limiting distribution (proved by Davenport, methods improved by Erdos), and this density $$ D \left\{ \frac{\sigma(n)}{n} < \omega \right\} = \tau(\omega)$$ is a continuous function of $\omega$. There is not much more to hope for in details, as Erdos showed that the analogous density for $$ \log \frac{\phi(n)}{n} $$ is continuous but "singular," that is has derivative 0 almost everywhere. $$ $$ However, Davenport's result does show that the abundant and deficient numbers both have densities, while the perfect numbers have density 0. While no "variance" is mentioned, a mean for the distribution is given, $$ M \left\{ \frac{\sigma(n)}{n} \right\} = \frac{\pi^2}{6}$$ ORIGINAL: There is a nice survey on fairly elementary methods here by J. L. Nicolas, in a 1988 book called "Ramanujan Revisited." $$ $$ Meanwhile, there is an unconditional result which has not been mentioned, for $N \geq 3$ we have $$ \sigma(N) < e^\gamma \; N \log \log N + \frac{ 0.6482 N}{\log \log N} $$ I hope I am reporting this correctly, it is from a secondary source, attribution is to G.Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. (9) 63 (1984) 187-213. $$ $$ The overall methodology is to consider the colossally abundant numbers of Alaoglu and Erdos (1944), http://en.wikipedia.org/wiki/Colossally_abundant_number
which were eventually discovered to have also been present in the original version of Ramanujan's paper Highly Composite Numbers (1915). There is some history about why that section was initially omitted, evidently a paper shortage. $$ $$ Here is a link to the first page of a related recent article, also apparently a survey, by Nicolas: http://www.springerlink.com/content/p8311481mh32145v/

There is a nice survey on fairly elementary methods here by J. L. Nicolas, in a book called "Ramanujan Revisited." $$ $$ Meanwhile, there is an unconditional result which has not been mentioned, for $N \geq 3$ we have $$ \sigma(N) < e^\gamma \; N \log \log N + \frac{ 0.6482 N}{\log \log N} $$ I hope I am reporting this correctly, it is from a secondary source, attribution is to G.Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. (9) 63 (1984) 187-213. $$ $$ The overall methodology is to consider the colossally abundant numbers of Alaoglu and Erdos (1944), which were eventually discovered to have also been present in the original version of Ramanujan's paper Highly Composite Numbers (1915). There is some history about why that section was omitted, I forget.

There is a nice survey on fairly elementary methods here by J. L. Nicolas, in a 1988 book called "Ramanujan Revisited." $$ $$ Meanwhile, there is an unconditional result which has not been mentioned, for $N \geq 3$ we have $$ \sigma(N) < e^\gamma \; N \log \log N + \frac{ 0.6482 N}{\log \log N} $$ I hope I am reporting this correctly, it is from a secondary source, attribution is to G.Robin, Grandes valeurs de la fonction somme des diviseurs et hypothese de Riemann, J. Math. Pures Appl. (9) 63 (1984) 187-213. $$ $$ The overall methodology is to consider the colossally abundant numbers of Alaoglu and Erdos (1944), http://en.wikipedia.org/wiki/Colossally_abundant_number
which were eventually discovered to have also been present in the original version of Ramanujan's paper Highly Composite Numbers (1915). There is some history about why that section was initially omitted, evidently a paper shortage. $$ $$ Here is a link to the first page of a related recent article, also apparently a survey, by Nicolas: http://www.springerlink.com/content/p8311481mh32145v/