Zeta functions for ideal classes in real quadratic fields, at s=0

摘要

Let K be a real quadratic field with discriminant d, and for a (fractional) ideal a of K, let Na be the norm of a. For a given fractional ideal I of K, and Dirichlet character χ of conductor q, we define where the sum is over all integral ideals a of K which are equivalent to I. We give a short, easily computable formula to evaluate ζ _I(0, χ), using familiar objects from considerations of K. We generalize our formula to ζ _I(1-k, χ) with k≥1, though the result obtained is not quite so satisfactory as that for k=1. We discuss connections between these formulae and small class numbers.