I was wondering whether there is a Euler–Maclaurin formula of sorts for expressions such as 
$$
\sum\_{x \in [a,b]^d\cap \mathbb{Z}^d} f(x) - \int\_{[a,b]^d}f(x)
$$
where $d\ge 2$ is an integer, $a,b \in \mathbb{R}$ and $f:\mathbb{R}^d \longrightarrow \mathbb{R}$ is a smooth function in $[a,b]^d$. I am particularly interested in such expansion with a control of the error term.


I appreciate any reference or suggestions.



Euler–Maclaurin formula in $\mathbb{Z}^d$
See e.g. [Y. Karshon, S. Sternberg, and J.Weitsman. The Euler-Maclaurin formula for simple integral polytopes](https://www.ncbi.nlm.nih.gov/pubmed/12515861) and [Y. Karshon, S. Sternberg, and J. Weitsman. Euler-Maclaurin with remainder for a simple integral polytope](https://projecteuclid.org/euclid.dmj/1133447438). 


For an alternative summation formula -- in terms of integrals only, without using derivatives -- see [Approximating sums by integrals only: multiple sums and sums over lattice polytopes](https://arxiv.org/abs/1705.09159). 


In these papers, you will also find explicit expressions for and/or simple bounds on the remainder. 


