On Kronecker limit formulas for real quadratic fields

Shuji Yamamoto

Research output: Contribution to journalArticle

9 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)426-450
Number of pages25
JournalJournal of Number Theory
Volume128
Issue number2
DOIs
Publication statusPublished - 2008 Feb
Externally publishedYes

Fingerprint

Real Quadratic Fields
Riemann zeta function
Laurent Expansion
Constant term
Continued fraction
Half line
Signature
Partial
Derivative
Invariant
Class
Generalization

Keywords

  • Double sine functions
  • Kronecker limit formula
  • Real quadratic fields
  • Zeta and L-functions

ASJC Scopus subject areas

  • Algebra and Number Theory

Cite this

On Kronecker limit formulas for real quadratic fields. / Yamamoto, Shuji.

In: Journal of Number Theory, Vol. 128, No. 2, 02.2008, p. 426-450.

Research output: Contribution to journalArticle

Yamamoto, Shuji. / On Kronecker limit formulas for real quadratic fields. In: Journal of Number Theory. 2008 ; Vol. 128, No. 2. pp. 426-450.
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