I always really loved the derivation of the exact expression for the partition function, and I happened to recently stumble across this generating function for the sum of divisors function $\sigma(N)$:
$\hspace{7cm}\displaystyle\sum{\frac{Nz^N}{1-z^N}}=\sum{\sigma(N)z^N}$
I was wondering if it would be possible to use this generating function to derive some sort of analytic expression for $\sigma(N)$ in a similar way to how the exact formula for the partition function is derived.
I want to write something like:
$\hspace{6cm}\displaystyle \sigma(N)=\int_{\gamma}{\sum{\frac{Nz^N}{1-z^N}}e^{-2\pi iz}dz}$
and look at the asymptotic behavior of the integral for large values of N. Does anyone have any suggestions as to how to proceed?