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A unified and improved Chebotarev density theorem
- Source :
- Alg. Number Th. 13 (2019) 1039-1068
- Publication Year :
- 2018
-
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 $\leq z$ with uniformity in $z$ and the discriminant of $f$.<br />Comment: 26 pages; v3 intro revised and application added in Sections 6 and 7; typos corrected
- Subjects :
- Mathematics - Number Theory
11R44
Subjects
Details
- Database :
- arXiv
- Journal :
- Alg. Number Th. 13 (2019) 1039-1068
- Publication Type :
- Report
- Accession number :
- edsarx.1803.02823
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.2140/ant.2019.13.1039