Explicit formula in a imaginary quadratic number field

Abstract

In [4], Haran, using Riesz potentials, presents a version of the classical explicit formula for the Riemann zeta function that treats all places equally. In this article, we extend Haran’s results to the case of an imaginary quadratic extension of $\mathbb{Q}$. To this end, we define Riesz kernels for both the totally ramified and complex cases. Although Haran implicitly addressed the unramified extension, we also include this case for the sake of completeness. Following Haran’s approach with the Riemann zeta function, we demonstrate that these Riesz kernels naturally arise in connection with the contributions of the totally ramified and complex places to the Dedekind zeta function of the imaginary quadratic extension.