values of Dedekind zeta functions of real quadratic number fields at negative integers

Let K be a real quadratic number field of discriminantPlanetmathPlanetmathPlanetmath DK and let ζ(s,K) be the Dedekind zeta function associated to K. By the Siegel-Klingen Theorem, if n>0 then ζ(-n,K) is a rational number. On the other hand, K is obviously an abelian number field, thus the factorization of the Dedekind zeta function of an abelian number field tells us that:


where ζ(s) is the famous Riemann zeta functionDlmfDlmfMathworldPlanetmath and L(s,χ) is the Dirichlet L-function associated to χ, where χ is the unique Dirichlet characterDlmfMathworldPlanetmath with conductorPlanetmathPlanetmath DK such that the group of characters of K/ is {χ0,χ} and χ0 is the trivial character. In fact, the values of χ are simply given by


where the parentheses denote the Kronecker symbolMathworldPlanetmath.

Furthermore, if k is a positive integer then:

  1. 1.

    Putting the values of the Riemann zeta function in terms of Bernoulli numbersDlmfDlmfMathworldPlanetmathPlanetmath one gets:


    where Bk is the kth Bernoulli number;

  2. 2.

    The values of Dirichlet L-series at negative integers can be written in terms of generalized Bernoulli numbersDlmfPlanetmath as follows:


    where Bk,χ is the kth generalized Bernoulli number associated to χ.



The interested reader can find tables of values at the alozano/dedekind-values/index.htmlauthor’s personal website.

Title values of Dedekind zeta functions of real quadratic number fields at negative integers
Canonical name ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers
Date of creation 2013-03-22 16:01:27
Last modified on 2013-03-22 16:01:27
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 5
Author alozano (2414)
Entry type Application
Classification msc 11R42
Classification msc 11M06
Related topic FactorizationOfTheDedekindZetaFunctionOfAnAbelianNumberField