A unified and improved Chebotarev density theorem

Thorner, Jesse; Zaman, Asif

doi:10.2140/ant.2019.13.1039

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A unified and improved Chebotarev density theorem

Jesse Thorner and Asif Zaman

Vol. 13 (2019), No. 5, 1039–1068

DOI: 10.2140/ant.2019.13.1039

Abstract

We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau–Siegel zero is present. Our main theorem also interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun–Titchmarsh theorem proved by the authors.

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Keywords

distribution of primes, Chebotarev density theorem, effective, uniform, binary quadratic forms

Mathematical Subject Classification 2010

Primary: 11R44

Milestones

Received: 22 March 2018

Revised: 29 November 2018

Accepted: 30 January 2019

Published: 12 July 2019

Authors

Jesse Thorner
Department of Mathematics Stanford University Stanford, CA United States

Asif Zaman
Department of Mathematics Stanford University Stanford, CA United States

Vol. 13, No. 5, 2019