I would like gain detailed knowledge about properties of the divisor sum function $\sigma(n)$, special equation that have been studied (e.g. $\sigma(n) = 2n$ perfect numbers, ...) and progress that was made in the past centuries.

I am interested in arithmetic functions in general so the books mentioned in this question Good books on arithmetic functions? is already a good start.

However, it would be great if you could recommend a book or a comprehensive paper that introduces/summarizes knowledge about the $\sigma(n)$ function.