values of Dedekind zeta functions of real quadratic number fields at negative integers
Let be a real quadratic number field of discriminant and let be the Dedekind zeta function associated to . By the Siegel-Klingen Theorem, if then is a rational number. On the other hand, is obviously an abelian number field, thus the factorization of the Dedekind zeta function of an abelian number field tells us that:
where is the famous Riemann zeta function and is the Dirichlet L-function associated to , where is the unique Dirichlet character with conductor such that the group of characters of is and is the trivial character. In fact, the values of are simply given by
where the parentheses denote the Kronecker symbol.
Furthermore, if is a positive integer then:
Putting the values of the Riemann zeta function in terms of Bernoulli numbers one gets:
where is the th Bernoulli number;
The values of Dirichlet L-series at negative integers can be written in terms of generalized Bernoulli numbers as follows:
where is the th generalized Bernoulli number associated to .
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|
|Date of creation||2013-03-22 16:01:27|
|Last modified on||2013-03-22 16:01:27|
|Last modified by||alozano (2414)|