Average Bateman--Horn for Kummer polynomials

TL;DR

This paper proves that for most small shifts, Kummer polynomials generate the expected number of primes, and applies this to arithmetic geometry problems involving norm equations over quadratic fields.

Contribution

It introduces a new large sieve inequality for Dirichlet characters of exact order r and applies it to prime values of Kummer polynomials and related Hasse principle variants.

Findings

Almost all Kummer polynomials of the form n^r + k have the expected prime count.

Established a new large sieve inequality for Dirichlet characters of order r.

Derived results on the Hasse principle for certain norm form varieties.

Abstract

For any $r \in \mathbb{N}$ and almost all $k \in \mathbb{N}$ smaller than $x^r$, we show that the polynomial $f(n) = n^r + k$ takes the expected number of prime values as $n$ ranges from 1 to $x$. As a consequence, we deduce statements concerning variants of the Hasse principle and of the integral Hasse principle for certain open varieties defined by equations of the form $N_{K/\mathbb{Q}}(\mathbf{z}) = t^r +k \neq 0$ where $K/\mathbb{Q}$ is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order $r$.