2006/02/27 by Shûji Yamamoto, Yamamoto, Shuji
Mathematics · #11M20 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.math/0602615
openalex publication_date 2006/02/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.