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arXiv · 1604.01750

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

Jesse Thorner·Asif Zaman
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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.

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Jesse Thorner, Asif Zaman. 2016-08-17. An explicit bound for the least prime ideal in the Chebotarev density theorem. https://doi.org/10.2140/ant.2017.11.1135

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