We provide a potential conceptual reason for the positivity of the Weil functional using the Hilbert space framework of the semi-local trace formula of Connes (Sel Math (NS) 5(1):29–106, 1999). We explore in great details the simplest case of the single archimedean place. The root of this result is the positivity of the trace of the scaling action compressed onto the orthogonal complement of the range of the cutoff projections associated to the cutoff in phase space, for \(\Lambda =1\). We express the difference between the Weil distribution and the trace associated to the above compression of the scaling action, in terms of prolate spheroidal wave functions, and use, as a key device, the theory of hermitian Toeplitz matrices to control that difference. All the concepts and tools used in this paper make sense in the general semi-local case, where Weil positivity implies RH.

我们使用 Connes (Sel Math (NS) 5(1):29–106, 1999) 的半局部迹公式的希尔伯特空间框架为 Weil 泛函的正性提供了一个潜在的概念原因。我们非常详细地探讨了单一阿基米德地方的最简单情况。这个结果的根源是压缩到与相空间中的截止相关的截止投影范围的正交补上的缩放动作的迹线的正性，对于\(\Lambda =1\). 我们用长球面波函数表达了 Weil 分布和与上述缩放作用压缩相关的迹线之间的差异，并使用厄米托普利茨矩阵理论作为关键设备来控制这种差异。本文中使用的所有概念和工具在一般半局部情况下都是有意义的，其中 Weil 正性意味着 RH。