Norm forms represent few integers but relatively many primes

TL;DR

Norm forms represent very few integers overall but still represent a positive proportion of primes, with the paper proving these density results using class field theory tools.

Contribution

The paper establishes the zero density of integers represented by norm forms and the positive density of primes they represent, using class field theory techniques.

Findings

Integers represented by norm forms have zero natural density.

Primes represented by norm forms have positive relative density.

The results are proved using class field theory and the Chebotarev density theorem.

Abstract

Norm forms, examples of which include $x^2 + y^2$, $x^2 + x y - 57 y^2$, and $x^3 + 2 y^3 + 4 z^3 - 6 x y z$, are integral forms arising from norms on number fields. We prove that the natural density of the set of integers represented by a norm form is zero, while the relative natural density of the set of prime numbers represented by a norm form exists and is positive. These results require tools from class field theory, including the Artin-Chebotarev density theorem and the Hilbert class field. We introduce these tools as we need them in the course of the main arguments. This article is expository in nature and assumes only a first course in algebraic number theory.