Hecke's integral formula for relative quadratic extensions of algebraic number fields

Abstract

Let $K/F$ be a quadratic extension of number fields. After developing a theory of the Eisenstein series over $F$, we prove a formula which expresses a partial zeta function of $K$ as a certain integral of the Eisenstein series. As an application, we obtain a limit formula of Kronecker's type which relates the $0$-th Laurent coefficients at $s=1$ of zeta functions of $K$ and $F$.