Singular Euler–Maclaurin expansion on multidimensional lattices

Abstract We extend the classical Euler–Maclaurin (EM) expansion to sums over multidimensional lattices that involve functions with algebraic singularities. This offers a tool for precise and fast evaluation of singular appear in long-range interacting systems. find approximation error decays exponentially order band-limited runtime is independent number particles. First, EM summation formula generalised higher dimensions, assuming sufficiently regular summand function. then develop this new further construct an extension our previous work one dimension, which remains applicable useful even if function includes factor. connect method analytical theory show all operator coefficients can be efficiently computed from derivatives Epstein zeta Finally we demonstrate numerical performance compute lattice infinite two-dimensional lattices, are relevance condensed matter, statistical, quantum physics. An implementation mathematica provided online along article.