Exceptional Primes in Notions of Arithmetic Similarity

A notion of arithmetic similarity between number fields is defined by requiring equality of some arithmetic statistics over all but finitely many rational primes. The exceptional set is empty in all previously studied cases, but existing methods for proving this are eclectic. We give a group theoretic explanation for those cases and show that some notions of arithmetic similarity admit a non-empty exceptional set. Furthermore, we prove a formula for actions of finite cyclic groups on finite sets, which augments a classical result of Gassmann and might also be of independent interest.