Dedekind's zeta function

The Dedekind zeta function of a number field is defined as ${\displaystyle K}$

${\displaystyle \zeta _{K}(s):=\sum _{\mathfrak {a}}{{\mathfrak {N}}({\mathfrak {a}})}^{-s}}$

where the ideals of the integral ring of the number field runs through and is its absolute norm . The series is absolutely and uniformly convergent in the area for all and the product presentation applies ${\displaystyle {\mathfrak {a}}}$ ${\displaystyle O(K)}$ ${\displaystyle K}$ ${\displaystyle {\mathfrak {N}}({\mathfrak {a}})}$ ${\displaystyle \zeta _{K}(s)}$ ${\displaystyle \Re (s)\geq 1+\delta }$ ${\displaystyle \delta >0}$

${\displaystyle \zeta _{K}(s)=\prod _{\mathfrak {p}}{\frac {1}{1-{{\mathfrak {N}}({\mathfrak {p}})}^{-s}}}}$,

where the prime ideals of passes through. The zeta function has an analytical continuation on and a pole in . ${\displaystyle {\mathfrak {p}}}$ ${\displaystyle O(K)}$ ${\displaystyle \mathbb {C} \setminus \{1\}}$ ${\displaystyle s=1}$

The Dedekind zeta function thus represents a generalization of the Riemann zeta function , which corresponds to the field of rational numbers (whose wholeness ring is even ). ${\displaystyle \mathbb {Z} }$

See also

• L function

literature

• Jürgen Neukirch: Algebraic Number Theory , Springer-Verlag Berlin Heidelberg, 1992, ISBN 2-540-54273-5
• Wolfgang Schwarz: From the history of number theory , supplemented elaboration of a one-hour lecture in the winter semester 2000/2001, Frankfurt am Main
• Stavros Garoufalidis, James E. Pommersheim: Values ​​of zeta functions at negative integers, Dedekind sums and toric geometry , Department of Mathematics, Harvard University, Cambridge, MA, USA.