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

Abstract

We prove an explicit version of Weiss' bound on the least norm of a prime ideal in the Chebotarev density theorem, which is itself a significant improvement on the work of Lagarias, Montgomery, and Odlyzko. In order to accomplish this, we prove an explicit log-free zero density estimate and an explicit version of the zero-repulsion phenomenon for Hecke $L$-functions. As an application, we prove the first explicit nontrivial upper bound for the least prime represented by a positive-definite primitive binary quadratic form. We also present applications to the group of ${\mathbb{F}}_p$-rational points of an elliptic curve and congruences for the Fourier coefficients of holomorphic cuspidal modular forms.

45 pages. v1 subsumes arXiv:1510.08086 but adds too much new material to be considered a revised version; v2 exposition tightened, minor corrections, slight improvement in Theorem 1.1

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• Primes represented by positive definite binary quadratic forms
• Pointwise Bound for $\ell$-torsion in Class Groups II: Nilpotent Extensions
• An explicit version of Bombieri's log-free density estimate and Sárközy's theorem for shifted primes
• Primes in the Chebotarev density theorem for all number fields
• Pointwise Bound for $\ell$-torsion in Class Groups: Elementary Abelian Extensions
• Arithmetic progressions in binary quadratic forms and norm forms
• The least prime ideal in a given ideal class
• Exceptional zeros of Rankin-Selberg $L$-functions and joint Sato-Tate distributions
• The Density of Numbers Represented by Diagonal Forms of Large Degree
• The least unramified prime which does not split completely
• The least prime ideal in the Chebotarev Density Theorem