Formulas for Chebotarev densities of Galois extensions of number fields

TL;DR

This paper extends Chebotarev density formulas to arbitrary Galois extensions of number fields, providing a new limit formula involving ideals, the M"obius function, and conjugacy classes.

Contribution

It generalizes Chebotarev density formulas to all finite Galois extensions of number fields using ideal sums and duality concepts.

Findings

Established a limit formula for Chebotarev density involving ideals and the M"obius function.

Generalized classical number theory results to the setting of number field ideals.

Extended Alladi's duality concept to ideals in number fields.

Abstract

We generalize the Chebotarev density formulas of Dawsey (2017) and Alladi (1977) to the setting of arbitrary finite Galois extensions of number fields $L/K$. In particular, if $C \subset G = \textrm{Gal}(L/K)$ is a conjugacy class, then we establish that the Chebotarev density is the following limit of partial sums of ideals of $K$: \[ -\lim_{X\rightarrow\infty} \sum_{\substack{2\leq N(I)\leq X \\ I \in S(L/K; C)}} \frac{\mu_K(I)}{N(I)} = \frac{|C|}{|G|}, \] where $\mu_K(I)$ denotes the generalized M\"obius function and $S(L/K;C)$ is the set of ideals $I\subset \mathcal{O}_K$ such that $I$ has a unique prime divisor $\mathfrak{p}$ of minimal norm and the Artin symbol $\left[\frac{L/K}{\mathfrak{p}}\right]$ is $C$. To obtain this formula, we generalize several results from classical analytic number theory, as well as Alladi's concept of duality for minimal and maximal prime divisors, to the setting of ideals in number fields.