As for how to apply Poisson sommation formula, everything is given in the text: they even give the function to use. The formula says:

,

provided the second sum is absolutely convergent. And if the Fourier transform is defined as .

The left-hand side is exactly what you want, so you have to check that the right-hand side matches the formula you're given. In other words, you have to compute the Fourier transform of .

This can be tedious, but there are tricks:

first, notice that , hence . You have to be careful here, since is not integrable, but only square-integrable.

Then you can notice that (by easy integration) where is the Heavyside function ( if and else). As a consequence, using Fourier inversion formula (in ), .

We deduce in (hence for almost every ), and hence for every because both functions are continuous (even for because ).

As a consequence, if and 0 otherwise.

The series is absolutely convergent because and . That's why we can apply the Poisson sommation formula. And it gives exactly what is expected.

By the way, if , the series would diverge.