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.

中文翻译：

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多维格上的奇异Euler-Maclaurin展开

我们将经典的Euler-Maclaurin扩展扩展为涉及涉及代数奇异函数的多维晶格的求和。这提供了一种工具，用于精确量化微观离散对系统宏观特性的影响。首先，假设一个足够规则的求和函数，将欧拉-麦克劳伦求和公式推广到更高维度的晶格。然后，我们进一步开发此新扩展，并在更高的维度上构造奇异的Euler-Maclaurin（SEM）扩展，这是我们先前工作在一个维度上的扩展，即使summand函数包含奇异函数因子，它仍然适用且有用。我们将我们的方法与解析数论联系起来，并表明可以从Epstein zeta函数的导数有效地计算所有算子系数。最后，我们证明了展开的数值性能，并有效地计算了无限二维晶格中的奇异晶格和，这在固态和量子物理学中具有很高的相关性。与本文一起在线提供了Mathematica中的实现。