Exceptional Primes in Notions of Arithmetic Similarity

TL;DR

This paper explores the concept of arithmetic similarity between number fields, revealing cases with exceptional primes and providing a group theoretic explanation, along with a new formula for cyclic group actions.

Contribution

It introduces a group theoretic framework for understanding arithmetic similarity and demonstrates that some notions admit exceptional primes, unlike previous cases.

Findings

Some notions of arithmetic similarity have non-empty exceptional sets.

A new formula for cyclic group actions on finite sets is established.

The work extends classical results of Gassmann with potential independent interest.

Abstract

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.