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?
As a comment above indicates, the problem is that the sequence $\sigma(N)$ does not have an asymptotic behavior: as $N$ goes to infinity through the subsequence of primes, $\sigma(N)=N+1\sim N$. On the other hand, as $N$ goes to infinity through the subsequence of powers of $2$, $N=2^k$, $\sigma(N)=(2^{k+1}-1)/(2-1)$ which is $2\cdot 2^k-1\sim 2N$.