What are the applications of Sigma Function? I read about the Sigma Function today.It tells that- 

The $\sigma(n)$ is the sum of all the positive divisors of $n$.

But I had no idea how they can be useful.What are the practical applications or uses of the sigma function? Are there some seemingly big mathematical problems that can be solved in seconds by applying the Sigma Function?
Thanks for any reply!!

As far as I know the main properties of the sigma function are that


*

*its Dirichlet series $\zeta (s) \zeta(s-1) = \sum \sigma(n)n^{-s}$ is related to the Riemann zeta function  

*the sum of the $\sigma$ function on intervals is the famous problem of lattice point counting in a hyperbola  
I don't think there are direct applications, but 1-2 provide a relation between some difficult counting problems in analytic number theory and the easier problem of sums over lattice points.
