Vol. 13, No. 5, 2019

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A unified and improved Chebotarev density theorem

Jesse Thorner and Asif Zaman

Vol. 13 (2019), No. 5, 1039–1068
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.

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