Dedekind zeta function

In mathematics, the Dedekind zeta function of an algebraic number field K, usually denoted ${\displaystyle \zeta _{K}(s)}$, is an analytic function that represents information about the ideals in the corresponding number ring, generalizing how the Riemann zeta function ${\displaystyle \zeta (s)}$ represents information about the factorization of integers.

Dedekind zeta functions generalize many properties of the Riemann zeta function: they can be defined as a Dirichlet series, have an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, have an Euler product expansion, and satisfy a functional equation. Values of Dedekind zeta functions encode important arithmetic data of K.

The Dedekind zeta function is named for Richard Dedekind, who introduced it in his supplement to Peter Gustav Lejeune Dirichlet's Vorlesungen über Zahlentheorie.