Given an integer N, and two non-negative integer valued variables x,y which take values in ${0,1,...,N-1}$. Is it possible to obtain a close form for the following summation?
$$f(x,y)=\frac{1}{N}\sum_{z=0}^{N-1} e^{-i \frac{2\pi z}{N}\left( x + \frac{y}{N} \right)}$$
Clearly when y=0, $f(x,0)=\delta_{x,0}$, but I am not sure how to evaluate the other limiting case $f(0,y)$, not even mention the general case. If I take $N\rightarrow\infty$, I think I should have roughly $\delta(x+y/N)$, but numerically it does not seem to be case.
Any help is greatly appreciated.