  We extend the classical Euler-Maclaurin expansion to sums over
multidimensional lattices that involve functions with algebraic singularities.
This offers a tool for the precise quantification of the effect of microscopic
discreteness on macroscopic properties of a system. First, the Euler-Maclaurin
summation formula is generalised to lattices in higher dimensions, assuming a
sufficiently regular summand function. We then develop this new expansion
further and construct the singular Euler-Maclaurin (SEM) expansion in higher
dimensions, an extension of our previous work in one dimension, which remains
applicable and useful even if the summand function includes a singular function
factor. We connect our method to analytical number theory and show that all
operator coefficients can be efficiently computed from derivatives of the
Epstein zeta function. Finally we demonstrate the numerical performance of the
expansion and efficiently compute singular lattice sums in infinite
two-dimensional lattices, which are of high relevance in solid state and
quantum physics. An implementation in Mathematica is provided online along with
this article.
