Article ID Journal Published Year Pages File Type
4595030 Journal of Number Theory 2008 25 Pages PDF
Abstract

Let ζ(s,C) be the partial zeta function attached to a ray class C of a real quadratic field. We study this zeta function at s=1 and s=0, combining some ideas and methods due to Zagier and Shintani. The main results are (1) a generalization of Zagier's formula for the constant term of the Laurent expansion at s=1, (2) some expressions for the value and the first derivative at s=0, related to the theory of continued fractions, and (3) a simple description of the behavior of Shintani's invariant X(C), which is related to ζ′(0,C), when we change the signature of C.

Related Topics
Physical Sciences and Engineering Mathematics Algebra and Number Theory