### 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 |
---|---|

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 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Journal of Number Theory*,

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

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

Research output: Contribution to journal › Article

*Journal of Number Theory*, vol. 128, no. 2, pp. 426-450. https://doi.org/10.1016/j.jnt.2007.05.010

}

TY - JOUR

T1 - On Kronecker limit formulas for real quadratic fields

AU - Yamamoto, Shuji

PY - 2008/2

Y1 - 2008/2

N2 - 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.

AB - 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.

KW - Double sine functions

KW - Kronecker limit formula

KW - Real quadratic fields

KW - Zeta and L-functions

UR - http://www.scopus.com/inward/record.url?scp=36749006625&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=36749006625&partnerID=8YFLogxK

U2 - 10.1016/j.jnt.2007.05.010

DO - 10.1016/j.jnt.2007.05.010

M3 - Article

AN - SCOPUS:36749006625

VL - 128

SP - 426

EP - 450

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

IS - 2

ER -