Dedekind series

Definition

The Dedekind series is a generalization of Dirichlet series from the ring of rational integers to the more general case of the ring of integers in a number field.

Suppose $K$ is a number field, i.e., a finite extension of the field of rational numbers $\mathbb {Q}$. Suppose ${\mathcal {O}}$ is the ring of integers in $K$. Suppose $f$ is a function from the set of nonzero ideals in ${\mathcal {O}}$ to $\mathbb {C}$.

The Dedekind series of $f$ is defined as:

$s\mapsto \sum _{I}{\frac {f(I)}{(N(I))^{s}}}$.

Here, the summation is over all nonzero ideals $I$ of ${\mathcal {O}}$, and $N(I)$ denotes the norm of the ideal, which is also equal to the index of $I$ in ${\mathcal {O}}$ as a subgroup.