Explicit counting of ideals and a Brun-Titchmarsh inequality for the\n Chebotarev Density Theorem

2016/11/30 by Korneel Debaene, Debaene, Korneel
Mathematics · #11H16 #11N35 #11R44 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1611.10103

openalex publication_date 2016/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a bound on the number of primes with a given splitting behaviour in\na given field extension. This bound generalises the Brun-Titchmarsh bound on\nthe number of primes in an arithmetic progression. The proof is set up as an\napplication of Selberg's Sieve in number fields. The main new ingredient is an\nexplicit counting result estimating the number of integral elements with\ncertain properties up to multiplication by units. As a consequence of this\nresult, we deduce an explicit estimate for the number of ideals of norm up to\nx.\n

Related

• An effective version of Chebotarev's density theorem
• An explicit upper bound for the least prime ideal in the Chebotarev\n density theorem
• Divisibility conditions on the order of the reductions of algebraic numbers
• Counting ideals in abelian number fields
• Unconditional Chebyshev biases in number fields