Relative Hecke's integral formula for an arbitrary extension of number fields

TL;DR

This paper generalizes Hecke's integral formula to arbitrary number field extensions, providing new relative residue and Kronecker limit formulas for partial zeta functions, unifying previous results by Hecke and Yamamoto.

Contribution

It introduces a generalized Hecke's integral formula for any extension of number fields and derives relative residue and Kronecker limit formulas for the associated partial zeta functions.

Findings

Generalized Hecke's integral formula for arbitrary extensions

Derived relative residue formulas for partial zeta functions

Established relative Kronecker limit formulas

Abstract

In this article, we present a generalized Hecke's integral formula for an arbitrary extension $E/F$ of number fields. As an application, we present relative versions of the residue formula and Kronecker's limit formula for the "relative" partial zeta function of $E/F$. This gives a simultaneous generalization of two different known results given by Hecke himself and Yamamoto.