Approximating a multiple sum with an integral

Hi,

I want to approximate a multiple sum of the form $$\sum_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)},$$ where each $x_i$ is an integer between $0$ and $n$, by an integral $$\int_{x_1+x_2+\cdots+x_m \leq n}e^{g(x_1,x_2,\ldots,x_m)}dx_1dx_2\cdots dx_m\,.$$ I know that the Euler-Maclaurin formula can be used to derive the error term when $m=1$ but often see sums of this form with $m > 1$ approximated by integrals, though with little justification. I do not have much of a background in mathematical analysis so am not sure where to look for a reference for this.

Any help will be much appreciated.

-
In fact Euler-Maclaurin applies to your case as well. See mathoverflow.net/questions/10667 – Steve Huntsman Apr 2 '10 at 17:51
First of all, for a "generic" function $g$ your approximation by sum can be really bad. There are different techniques in asymptotic analysis (including Euler-Maclaurin) which can be used for your sums. – Wadim Zudilin Jun 9 '10 at 11:49