Let $e(x):=e^{2\pi i x}$. I'd like to prove the following Dirichlet's formula with hints given below (From Iwaniec's Topics in Automorphic forms, page 5):
$$G(N)=\sum_{n\pmod{N}} e(\frac{n^2}{N})=\epsilon_N \sqrt{N}.$$
In the hint, we are asked to first prove the asymptotic formula for the above equation as $N\to \infty$
Take $f$ to be a smooth function on the real line supported on $[1-\delta, N+\delta]$ such that
$$G(N):=\sum_{n\in \mathbb Z} e(\frac{n^2}{N})f(n),$$
and we apply the Poisson summation formula to $$\sum_{n\in \mathbb Z} e(\frac{n^2}{N})f(n)=\sum_{n\in \mathbb Z} \widehat{e(\frac{x^2}{N})f(x)}[n] \\ =\sum_{n\in \mathbb Z}\int_{\mathbb R}e(\frac{x^2}{N})f(x)e(-nx)dx$$
But how to I continue from here? Should I try to approximate this Fourier transform somehow?
In below is the screenshot of the book.
