“棘手的算术难题”
用英语怎么说？

Robert Langlands launched this monumental inquiry in 1967 by conjecturing a startling link between number theory and harmonic analysis. At its core, the framework scrutinizes the symmetries governing solutions to algebraic equations, known as Galois groups. In complicated non-abelian settings, these permutations can be represented as numeric matrices. Remarkably, Langlands realized that the traces of these matrix representations correspond precisely to the spectral coefficients of modular forms-periodic harmonic functions defined by their infinite rotational symmetries. By transmuting intractable arithmetic conundrums into the analytical language of continuous waves, this dictionary establishes an uncanny bridge between the discrete behaviors of prime numbers and smooth harmonic waveforms.

罗伯特·朗兰兹于1967年发起了这项不朽的探索，提出了数论与调和分析之间令人惊叹的关联。其核心在于审视支配代数方程解的对称性，即伽罗瓦群。在复杂的非阿贝尔情形下，这些置换可由数值矩阵来表示。引人注目的是，朗兰兹意识到这些矩阵表示的迹恰好对应于模形式的谱系数--模形式是由无限旋转对称性定义的周期调和函数。通过将棘手的算术难题转化为连续波动的分析语言，这部词典在素数的离散性质与平滑谐波之间架起了不可思议的桥梁。