# Euler–Maclaurin formula in $\mathbb{Z}^d$

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.