A unified and improved Chebotarev density theorem

Abstract

We establish an unconditional effective Chebotarev density theorem that improves uniformly over the well-known result of Lagarias and Odlyzko. As a consequence, we give a new asymptotic form of the Chebotarev density theorem that can count much smaller primes with arbitrary log-power savings, even in the case where a Landau-Siegel zero is present. Our main theorem interpolates the strongest unconditional upper bound for the least prime ideal with a given Artin symbol as well as the Chebotarev analogue of the Brun-Titchmarsh theorem proved by the authors. We also present a new application of our main result that exhibits considerable gains over earlier versions of the Chebotarev density theorem. If $f$ is a positive definite primitive binary quadratic form then we count lattice points $(u,v)\in {\mathbb{Z}}^2$ such that $f(u,v)$ is prime and $u,v$ have no prime factors $\le z$ with uniformity in $z$ and the discriminant of $f$.

arXiv comment26 pages; v3 intro revised and application added in Sections 6 and 7; typos corrected

References in corpus (2)

• An explicit bound for the least prime ideal in the Chebotarev density theorem
• A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures

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• On Malle's conjecture for nilpotent groups, I
• Highly Uniform Prime Number Theorems