A supplement to Chebotarev’s density theorem

Abstract

Let L/K be a Galois extension of number fields with Galois group G. We show that if the density of prime ideals in K that split totally in L tends to 1/∣G∣ with a power saving error term, then the density of prime ideals in K whose Frobenius is a given conjugacy class C ⊂ G tends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of ζL(s)/ζK(s).