  This paper introduces a new method for the efficient computation of
oscillatory multidimensional lattice sums in geometries with boundaries. Such
sums are ubiquitous in both pure and applied mathematics, and have immediate
applications in condensed matter physics and topological quantum physics. The
challenge in their evaluation results from the combination of singular
long-range interactions with the loss of translational invariance caused by the
boundaries, rendering standard tools ineffective. Our work shows that these
lattice sums can be generated from a generalization of the Riemann zeta
function to multidimensional non-periodic lattice sums. We put forth a new
representation of this zeta function together with a numerical algorithm that
ensures exponential convergence across an extensive range of geometries.
Notably, our method's runtime is influenced only by the complexity of the
considered geometries and not by the number of particles, providing the
foundation for efficient simulations of macroscopic condensed matter systems.
We showcase the practical utility of our method by computing interaction
energies in a three-dimensional crystal structure with $3\times 10^{23}$
particles. Our method's accuracy is demonstrated through extensive numerical
experiments. A reference implementation is provided online along with this
article.
