manuscripta mathematica

, Volume 124, Issue 4, pp 551–560

Evaluation of the Dedekind zeta functions of some non-normal totally real cubic fields at negative odd integers

Article

Abstract

Let {Km}m ≥  4 be the family of non-normal totally real cubic number fields defined by the irreducible cubic polynomial fm(x)  =  x3  −  mx2  −  (m  +  1)x  −  1, where m is an integer with m ≥  4. In this paper, we will apply Siegel’s formula for the values of the zeta function of a totally real algebraic number field at negative odd integers to Km, and compute the values of the Dedekind zeta function of Km.

Mathematics Subject Classification (2000)

Primary 11R42 Secondary 11R16 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Kim H.K. and Hwang H.J. (2000). Values of zeta functions and class number 1 criterion for the simplest cubic fields. Najoya Math. J. 160: 161–180 MATHMathSciNetGoogle Scholar
  2. 2.
    Kim H.K. and Kim J.S. (2002). Evaluation of zeta function of simplest cubic field at negative odd integers. Math. Comput. 71: 1243–1262 MATHCrossRefGoogle Scholar
  3. 3.
    Lang S. (1986). Algebraic Number Theory. Graduate Texts in Mathematics 110. Springer, New York Google Scholar
  4. 4.
    Louboutin S. (2001). Class number and class group problems for some non-normal totally real cubic number fields. Manuscripta Math. 106: 411–427 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Louboutin, S.: Numerical evaluations at negative integers of the Dedekind zeta function of totally real cubic number fields. Algorithmic Number Theory (University of Vermont, 2004). Lecture Notes in Computer Science, vol. 3076, pp. 318–326 (2004)Google Scholar
  6. 6.
    Ribenboim P. (2001). Classical Theory of Algebraic Numbers. Universitext. Springer, New York Google Scholar
  7. 7.
    Siegel C.L. (1969). Berechnung von Zetafuncktionen an ganzzahligen Stellen. Nachr. Akad. Wiss. Göttingen Math. Phys. Kl.II: 87–102 Google Scholar
  8. 8.
    Zagier D.B. (1976). On the values at negative integers of the zeta function of a real quadratic field. Enseign. Math. 22: 55–95 MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsPohang University of Science and TechnologyPohangSouth Korea

Personalised recommendations