### 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 language | English |
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Pages (from-to) | 426-450 |

Number of pages | 25 |

Journal | Journal of Number Theory |

Volume | 128 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2008 Feb 1 |

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

Yamamoto, S. (2008). On Kronecker limit formulas for real quadratic fields.

*Journal of Number Theory*,*128*(2), 426-450. https://doi.org/10.1016/j.jnt.2007.05.010