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Lower estimates for the prime ideal of degree one counting function in the Chebotarev density theorem

Volume 191 / 2019

Jeoung-Hwan Ahn, Soun-Hi Kwon 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 $\widetilde {\pi }_C (x)$.

Authors

  • Jeoung-Hwan AhnDepartment of Mathematics Education
    Korea University
    02841, Seoul, Korea
    e-mail
  • Soun-Hi KwonDepartment of Mathematics Education
    Korea University
    02841, Seoul, Korea
    e-mail

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