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

Let $K$ be a real quadratic number field of discriminant $D_{K}$ and let $\zeta(s,K)$ be the Dedekind zeta function associated to $K$ . By the Siegel-Klingen Theorem, if $n>0$ then $\zeta(-n,K)$ 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:

$\zeta(s,K)=\zeta(s)L(s,\chi)$

where $\zeta(s)$ is the famous Riemann zeta function and $L(s,\chi)$ is the Dirichlet L-function associated to $\chi$ , where $\chi$ is the unique Dirichlet character with conductor $D_{K}$ such that the group of characters of $K/\mathbb{Q}$ is $\{\chi_{0},\chi\}$ and $\chi_{0}$ is the trivial character. In fact, the values of $\chi$ are simply given by

$\chi(a)=\left(\frac{D_{K}}{a}\right)$

where the parentheses denote the Kronecker symbol.

Furthermore, if $k$ is a positive integer then:

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

$\zeta(1-k)=-\frac{B_{k}}{k}$

where $B_{k}$ is the $k$ th Bernoulli number;

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

$L(1-k,\chi)=-\frac{B_{k,\chi}}{k}$

where $B_{k,\chi}$ is the $k$ th generalized Bernoulli number associated to $\chi$ .

Therefore:

$\zeta(1-k,K)=\zeta(1-k)L(1-k,\chi)=\frac{B_{k}\cdot B_{k,\chi}}{k^{2}}.$

The interested reader can find tables of values at the http://www.math.cornell.edu/ 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