This work introduces a framework for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries, a problem intersecting pure and applied mathematics with immediate applications in condensed matter physics and topological quantum physics. The challenge in evaluating the arising sums 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 super-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 sheer 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 thoroughly assessed through a detailed error analysis that both uses analytical results and numerical experiments. A reference implementation is provided online along with the article

翻译：本文提出了一种框架，用于高效计算带有边界的几何结构中振荡多维格点求和，该问题横跨纯数学与应用数学，在凝聚态物理和拓扑量子物理中具有直接应用。评估这些求和所面临的挑战源于奇异长程相互作用与边界导致的平移对称性丧失的结合，这使得标准工具失效。我们的研究表明，这些格点求和可以通过将黎曼ζ函数推广至多维非周期格点求和来生成。我们提出了该ζ函数的新表示形式，并结合数值算法，确保在广泛的几何结构中实现超指数收敛。值得注意的是，我们方法的运行时间仅受所考虑几何结构复杂性的影响，而非粒子数量，这为宏观凝聚态系统的有效模拟奠定了基础。我们通过计算包含$3\times 10^{23}$个粒子的三维晶体结构中的相互作用能，展示了该方法的应用价值。通过详细的误差分析（包括解析结果和数值实验），我们全面评估了方法的精度。本文附有参考实现的在线链接。