I recently come up with some wild ideas. Suppose we have a good enough (say, simple and smooth) closed curve in $\mathbb{R}^2$ defined by $(x(t), y(t))$, and we want to count the number of "integer points" inside this curve, denoted by $N$. (By integer point, I mean a point with integer abscissa and ordinate.)
Now let $M$ be an integer large enough, and
$$F(x) = \sum_{k,l=-M}^M \frac{1}{x-(k+l\cdot\mathrm{i})}$$
One can easily check that
$$2\pi\mathrm{i}\cdot N = \int_{\gamma} F(z) \mathrm{d}z$$
where $\gamma(t)=x(t)+y(t)\cdot\mathrm{i}$.
Is this relation considered useful? If so, where can I find some references? If not, (mainly) why is that?