Algebra & Number Theory

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

Jesse Thorner and Asif Zaman

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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.

Article information

Algebra Number Theory, Volume 11, Number 5 (2017), 1135-1197.

Received: 12 May 2016
Revised: 25 October 2016
Accepted: 10 March 2017
First available in Project Euclid: 12 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R44: Distribution of prime ideals [See also 11N05]
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 14H52: Elliptic curves [See also 11G05, 11G07, 14Kxx]

Chebotarev density theorem least prime ideal Linnik's theorem binary quadratic forms elliptic curves modular forms log-free zero density estimate


Thorner, Jesse; Zaman, Asif. An explicit bound for the least prime ideal in the Chebotarev density theorem. Algebra Number Theory 11 (2017), no. 5, 1135--1197. doi:10.2140/ant.2017.11.1135.

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