Average Bateman--Horn for Kummer polynomials

Francesca Balestrieri, Nick Rome

Published 2020-05-24Version 1

For any $r \in \mathbb{N}$ and almost all $k \in \mathbb{N}$, we show that the polynomial $f(n) = n^r + k$ takes infinitely many prime values. 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 $r$.