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[parent] values of Dedekind zeta functions of real quadratic number fields at negative integers (Application)

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:

$\displaystyle \zeta(s,K)=\zeta(s)L(s,\chi)$
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
$\displaystyle \chi(a)=\left(\frac{D_K}{a}\right)$
where the parentheses denote the Kronecker symbol.

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

  1. Putting the values of the Riemann zeta function in terms of Bernoulli numbers one gets:
    $\displaystyle \zeta(1-k)=-\frac{B_k}{k}$
    where $ B_k$ is the $ k$th Bernoulli number;
  2. The values of Dirichlet L-series at negative integers can be written in terms of generalized Bernoulli numbers as follows:
    $\displaystyle L(1-k,\chi)= -\frac{B_{k,\chi}}{k}$
    where $ B_{k,\chi}$ is the $ k$th generalized Bernoulli number associated to $ \chi$.

Therefore:

$\displaystyle \zeta(1-k,K)=\zeta(1-k)L(1-k,\chi)=\frac{B_k \cdot B_{k,\chi}}{k^2}.$
The interested reader can find tables of values at the author's personal website.



"values of Dedekind zeta functions of real quadratic number fields at negative integers" is owned by alozano.
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See Also: Factorization of the Dedekind zeta function of an abelian number field


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values of the Dedekind zeta function of $\mathbb{Q}(\sqrt{5})$ at negative integers (Example) by alozano
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Cross-references: generalized Bernoulli numbers, terms, negative, Bernoulli number, values of the Riemann zeta function in terms of Bernoulli numbers, integer, positive, Kronecker symbol, trivial character, characters, group, conductor, Dirichlet character, Dirichlet L-function, Riemann zeta function, Factorization of the Dedekind zeta function of an abelian number field, abelian number field, rational number, Siegel-Klingen Theorem, Dedekind zeta function, discriminant, quadratic number field, real
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This is version 2 of values of Dedekind zeta functions of real quadratic number fields at negative integers, born on 2006-06-20, modified 2006-07-19.
Object id is 8064, canonical name is ValuesOfDedekindZetaFunctionsOfRealQuadraticNumberFieldsAtNegativeIntegers.
Accessed 903 times total.

Classification:
AMS MSC: 11M06 (Number theory :: Zeta and $L$-functions: analytic theory :: $\zeta $)
 11R42 (Number theory :: Algebraic number theory: global fields :: Zeta functions and $L$-functions of number fields)
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Notation by mathcam on 2006-06-20 21:23:46
What do you think about $\zeta_K(s)$ intead of $\zeta(s,K)$? It matches with the parent entry, and seems philosophically more satisfying to me -- we (or at least I) rarely think about this as a function of K.

Cam
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