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.