The Hecke-Landau Conjecture: A Proof and Its Architecture

A proof that for any unitary Hecke character $\chi$ of infinite order over a number field $K$, the angles $\theta_\chi(\mathfrak{p}) \in [0, 2\pi)$ defined by $\chi(\mathfrak{p}) = e^{i\theta_\chi(\mathfrak{p})}$ are equidistributed on the unit circle with respect to the uniform measure. $\lim_{x \to \infty} \frac{1}{\pi_K(x)} \sum_{N(\mathfrak{p}) \leq x} f(\chi(\mathfrak{p})) = \int_0^{2\pi} f(e^{i\theta}) \, \frac{d\theta}{2\pi}$ for every continuous $f: S^1 \to \mathbb{C}$, where $\pi_K(x)$ is the number of prime ideals of $\mathcal{O}_K$ with norm at most $x$.