I often run into double and triple sums in solid state physics, and there seems to be a definable suite of mostly analytic tools (Poisson summation, Mellin transforms, theta functions, modular forms etc.) that work in most cases. With the exception of parts of Bromwich and also Salekhov's Calculation of Series, I can't see any references that deal seriously with multiple series (i.e. adequately handles the multiple notions of convergence for conditionally convergent series and gives tools for their evaluation). Where can I find a short treatment of these things? Should I be looking at the Fourier analysis literature? I see some related results but very few directly useful theorems.
An approach to approximation of multiple sums by integrals, including summing possibly divergent multi-index series and sums over polytopes is presented in this paper. See also references there, especially [1, 6, 7].