When I perform a contour integral of
$\oint \dfrac{z}{e^z -1}~dz$
on a contour (as shown in Fig)
The integral has six parts and when done in this manner described here
I obtain:
$\zeta(1) = \int \limits_{0}^{\infty} \dfrac{1}{e^x - 1}~dx = \dfrac{\pi}{2}$
(along with a finite value for another divergent integral).
This feels like a $1=2$ result (which essentially means I am making a mistake in my assumptions) since $\zeta(1)$ should diverge.
Could somebody help me find the mistake in my assumptions?
Thanks a ton.
I think I figured out my error above. My assumption that the area within the contour is simply connected is not correct as $\dfrac{1}{e^z - 1} \approx \dfrac{1}{z}~~{\rm for}~~ z \ll 1$ and this causes the divergence. For higher powers of $z$ (in the numerator) the region will be simply connected (and hence the method will hold for $\zeta(n)~~{\rm for}~~n>1$).
I would like to acknowledge my reference to this source.
Will close this post - thank you all for looking through and thanks to @AHusain and @GEdgar for your efforts.