Lower estimates for the prime ideal of degree one counting function in the Chebotarev density theorem
Volume 191 / 2019
Acta Arithmetica 191 (2019), 289-307
MSC: Primary 11R44, 11R42, 11M41; Secondary 11R45.
DOI: 10.4064/aa180427-18-12
Published online: 20 September 2019
Abstract
Let $K$ be a number field and $L$ a finite normal extension of $K$ with Galois group $G$. For a prime ideal $\mathfrak {p}$ of $K$ which is unramified in $L$ we let $\left [\frac {L/K}{\mathfrak {p}}\right ]$ be the conjugacy class of Frobenius automorphisms corresponding to the prime ideals $\mathfrak {P}$ of $L$ lying above $\mathfrak {p}$. For a given conjugacy class $C$ of $G$ we let $\widetilde {\pi }_C (x)$ be the number of prime ideals $\mathfrak {p}$ of $K$ unramified in $L$ such that $\left [\frac {L/K}{\mathfrak {p}}\right ]=C$ and $N_{K/{\mathbb Q}}\mathfrak {p}$ is a rational prime with $N_{K/{\mathbb Q}}\mathfrak {p}\leq x$. We give some lower bounds for $\widetilde {\pi }_C (x)$.
