A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures

TL;DR

This paper advances bounds related to the distribution of primes and Frobenius traces by refining the Chebotarev variant of the Brun-Titchmarsh theorem using recent zero density estimates and zero repulsion results for Hecke L-functions.

Contribution

It introduces improved unconditional bounds for Lang-Trotter conjectures and new results on prime distribution in quadratic forms, leveraging recent advances in zero density estimates.

Findings

Enhanced bounds for Frobenius trace distributions.

Improved unconditional bounds for Lang-Trotter conjectures.

New results on primes represented by quadratic forms.

Abstract

We improve the Chebotarev variant of the Brun-Titchmarsh theorem proven by Lagarias, Montgomery, and Odlyzko using the log-free zero density estimate and zero repulsion phenomenon for Hecke L-functions that were recently proved by the authors. Our result produces an improvement for the best unconditional bounds toward two conjectures of Lang and Trotter regarding the distribution of traces of Frobenius for elliptic curves and holomorphic cuspidal modular forms. We also obtain new results on the distribution of primes represented by positive-definite integral binary quadratic forms.