# The least prime ideal in the Chebotarev Density Theorem

## TL;DR

This paper establishes a sharper bound for the least prime ideal in the Chebotarev density theorem by innovatively applying Harnack's inequality to enhance Turán's power sum method.

## Contribution

It introduces a novel approach using Harnack's inequality to improve the existing bounds in the Chebotarev density theorem.

## Key findings

- Improved bound for the least prime ideal by a factor of 5/2
- New application of Harnack's inequality in number theory
- Enhanced Turán's power sum method

## Abstract

In this article, we prove a new bound for the least prime ideal in the Chebotarev density theorem, which improves the main theorem of Zaman [Funct. Approx. Comment. Math. 57 (2017), no.1, 115-142] by a factor of $5/2$. Our main improvement comes from a new version of Tur\'an's power sum method. The key new idea is to use Harnack's inequality for harmonic functions to derive a superior lower bound for the generalised Fej\'er kernel.