The least prime ideal in the Chebotarev density theorem

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 Turan 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 Fejer kernel.