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

TL;DR

This paper establishes an explicit bound on the number of primes with specific splitting behavior in number fields, generalizing Brun-Titchmarsh inequality via Selberg's Sieve and counting integral elements.

Contribution

It introduces a new explicit counting method for integral elements in number fields, leading to bounds on prime splitting and ideals, extending classical inequalities.

Findings

Derived an explicit bound on primes with given splitting behavior

Generalized Brun-Titchmarsh inequality to number fields

Provided an explicit estimate for the number of ideals up to a certain norm

Abstract

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