Faithful Inductions Theorems and the Chebotarev Density Theorem

The Chebotarev density theorem is a powerful and ubiquitous tool in number theory used to guarantee the existence of infinitely many primes satisfying splitting conditions in a Galois extension of number fields. In many applications, however, it is necessary to know not just that there are many such primes in the limit, but to know that there are many such primes up to a given finite point. This is the domain of so-called effective Chebotarev density theorems. In forthcoming joint work with Alex Smith that extends previous joint work of the author with Thorner and Zaman and earlier work of Pierce, Turnage-Butterbaugh, and Wood, we prove that in any family of irreducible complex Artin representations, almost all are subject to a very strong effective prime number theorem. This implies that almost all number fields with a fixed Galois group are subject to a similarly strong effective form of the Chebotarev density theorem. Under the hood, the key result is a new theorem in the character theory of finite groups that is similar in spirit to classical work of Artin and Brauer on inductions of one-dimensional characters.