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/
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.