A supplement to Chebotarev's density theorem

Let $L\mathrm{/}K$ 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 $1\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$ with a power saving error term, then the density of prime ideals in $K$ whose Frobenius is a given conjugacy class $C\subset G$ tends to $\mathrm{\mid }C\mathrm{\mid }\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros of ${\zeta }_L(s)\mathrm{/}{\zeta }_K(s)$.