Weighted Discriminants and Mass Formulas for Number Fields

We define the notion of a weighted discriminant and corresponding counting function for number fields, and what it means for these counting functions to have a mass formula for a set of primes. We extend a result of Kedlaya to show that any proper counting function for a finite group $\Gamma$ has a mass formula for the set of primes not dividing $|\Gamma|$. We also prove that if $\Gamma$ is an $\ell$-group for some prime $\ell$, then there are only finitely many weighted discriminant counting functions for $\Gamma$-extensions of $\Q$ that have a mass formula for all primes. Finally, we enumerate all such counting functions for $\Gamma=D_4$ and $\Gamma=Q_8$.