I'm familiar with the integral. But why is that sum a particularly good approximation to the integral? It seems better than one has a right to expect, although it's been a while since I did any numerical analysis.
Replying to my own comment. See (4.4) in the article and the discussion after it. The error in this approximation is like exp(-pi^2 c). (4.10) in the article translates this to that we should expect about 4.2c digits of accuracy, since log_10 exp(-pi^2) is about 4.2. This falls out of Poisson's summation formula, according to the OP.
Generally I'd expect a numerical integration scheme to have an error like h^k for some constant k - for example for Riemann sums the error goes like h^2 and for Simpson's rule like h^5. h is the distance between the sampling points and so is analogous to 1/c. So this doesn't just follow from Riemann summation.
Trapezoidal rule quadrature is better than you would expect (2nd order) for periodic (even better for analytic and indeed entire, in this case) integrands. See this paper: http://eprints.maths.ox.ac.uk/1734/
Riemann sums. EDIT: The short answer is that it's a good approximation because e^(-n^2/c) drops off very vast as n approaches zero, meaning most of the contributions will come from small (on the order of sqrt(c)).
Other than that, it's just a Riemann sum presented in an aesthetically pleasing manner.
Do you mean 'approaches infinity'? Because at n=0 the expression is maximal :-)
But the point is not to drop small terms at the extremes - it is still an infinite sum. The point is to approximate sections of the function by rectangles, and those rectangles are a bad approximation around 0 too.
