Vol. 11, No. 5, 2017

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An explicit bound for the least prime ideal in the Chebotarev density theorem

Jesse Thorner and Asif Zaman

Vol. 11 (2017), No. 5, 1135–1197

We prove an explicit version of Weiss’ bound on the least norm of a prime ideal in the Chebotarev density theorem, which is a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. As an application, we prove the first explicit, nontrivial, and unconditional upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also consider applications to elliptic curves and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

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Chebotarev density theorem, least prime ideal, Linnik's theorem, binary quadratic forms, elliptic curves, modular forms, log-free zero density estimate
Mathematical Subject Classification 2010
Primary: 11R44
Secondary: 11M41, 14H52
Received: 12 May 2016
Revised: 25 October 2016
Accepted: 10 March 2017
Published: 12 July 2017
Jesse Thorner
Department of Mathematics
Stanford University
Building 380
Sloan Mathematical Center
Stanford, CA 94305
United States
Asif Zaman
Department of Mathematics
University of Toronto
Room 6290
40 St. George St.
Toronto, ON M5S 2E4