Definitions

# Dedekind zeta function

In mathematics, the Dedekind zeta-function is a Dirichlet series defined for any algebraic number field $K$, and denoted $zeta_K \left(s\right)$ where $s$ is a complex variable. It is the infinite sum

$zeta_K \left(s\right) = sum_\left\{I subseteq O_K\right\} \left(N_\left\{K/mathbb\left\{Q\right\}\right\} \left(I\right)\right)^\left\{-s\right\}$

where $I$ ranges through the non-zero ideals of the ring of integers $O_K$ of $K$. Here $N_\left\{K/mathbb\left\{Q\right\}\right\} \left(I\right) = \left[O_K : I\right]$ denotes the norm of $I$ (to the rational field $mathbb\left\{Q\right\}$). It is equal to the cardinality of $O_K / I$, in other words, the number of residue classes modulo $I$. This sum converges absolutely for all complex numbers $s$ with real part $Re\left(s\right) > 1$. In the case $K = mathbb\left\{Q\right\}$ this definition reduces to the Riemann zeta function.

The properties of $zeta_K\left(s\right)$ as a meromorphic function turn out to be of considerable significance in algebraic number theory. It has an Euler product, which is a product over all the prime ideals $P$ of $O_K$

$zeta_K \left(s\right) = prod_\left\{P subseteq O_K\right\} frac\left\{1\right\}\left\{1 - \left(N_\left\{K/mathbb\left\{Q\right\}\right\}\left(P\right)\right)^\left\{-s\right\}\right\}.$

This is the expression in analytic terms of the uniqueness of prime factorization of the ideals $I$.

It is known (proved first in general by Erich Hecke) that $zeta_K\left(s\right)$ does have an analytic continuation to the whole complex plane as a meromorphic function, having a simple pole only at s = 1. The residue at that pole is an important quantity, involving invariants of the unit group and class group of K; details are at class number formula. There is a functional equation for the Dedekind zeta-function, relating its values at s and 1−s.

For the case in which K is an abelian extension of Q, its Dedekind zeta-function can be written as a product of Dirichlet L-functions. For example, when K is a quadratic field this shows that the ratio

$frac\left\{zeta_K\left(s\right)\right\}\left\{zeta_\left\{mathbb\left\{Q\right\}\right\}\left(s\right)\right\}$

is an L-function L(s,χ); where $chi$ is a Jacobi symbol as Dirichlet character. That the zeta-function of a quadratic field is a product of the Riemann zeta-function and a certain Dirichlet L-function is an analytic formulation of the quadratic reciprocity law of Gauss.

In general if K is a Galois extension of Q with Galois group G, its Dedekind zeta-function has a comparable factorization in terms of Artin L-functions. These are attached to linear representations of G.

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