I am currently self reading " Spectral theory of Riemann zeta function" by Yoichi Motohashi. The example is in first chapter and of poincare series. I want to know how the Poisson sum formula on the last sum led to the final equation. I dont need any exact answer. Any help/hint would be appreciated.
Edit 1:
Confusion in proof for writing Poincares series in terms of Kloosterman sum
This question I asked earlier today related to this topic.
Please type your questions in mathjax, more readable and easier for us to copy/paste. $$f(x+iy)=\sum_n |x+iy+h/l+n|^{-2s}\exp(\frac{-2i\pi m}{ l^2(x+iy+h/l+n)})$$ is smooth $1$-periodic, it is equal to its Fourier series $$\sum_k \exp(2i\pi k x) c_k(y)$$ where $$c_k(y)=\int_0^1 f(x+iy)\exp(-2i\pi k x) dx$$ $$=\int_{-\infty}^\infty |x+iy+h/l|^{-2s}\exp(\frac{-2i\pi m}{ l^2(x+iy+h/l)})\exp(-2i\pi k x) dx$$
What do you get then?