I want to derive the form of the Poisson Summation Formula as in Equation (6) of this this paper
which is $$ \frac{1}{L^3} \sum_{\vec{k}} g(\vec{k}) = \int \frac{d^3 k}{(2\pi)^3} g(\vec{k}) + \sum_{\vec{l} \neq \vec{0} } \int \frac{d^3 k}{(2\pi)^3} e^{iL \vec{l} \cdot \vec{k}} g(\vec{k}) $$
where the summation on the left side is over all integer values of $\vec{n} = (n_1, n_2, n_3)$ and $\vec{k} = \frac{2\pi}{L} \vec{n}$ while that on the right is over integral values of $\vec{l} = (l_1, l_2, l_3)$ excluding $\vec{l} = (0,0,0)$
For context, the paper uses this formula to relate discrete sums over momenta of particles and integrals over a continuous spectrum in infinite volume.
I'm honestly a bit lost, and would appreciate a full derivation, but any hints on how this can be obtained would be greatly appreciated!