Originally Posted by

**Cairo** I'm trying to prove the reciprocity law for quadratic Gauss sums when m=1(mod 4) and need help showing something. Could anyone help? It's been a while since I done any residue calculus.

Let C be the rectangular contour with vertices, iR, -iR, 1/2m+iR and 1/2m-iR. It also has a semicircular indentation of radius epsilon (< 1) at 0.

Define

$\displaystyle f(z)=\frac{e^{2\pi iz^{2}/m}}{1-e^{2\pi iz}}$

Prove that

$\displaystyle \int_{C\:}f(z)dz=-\frac{1}{2}\overset{m-1}{\underset{r=1}{\Sigma}}e^{2\pi ir^{2}/m}$

Clearly f is analytic everywhere, except for a first order pole at each integer, but I am stumped at how to get the required result.