Average Bateman--Horn for Kummer polynomials

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 $N_{K/\mathbb{Q}}(\mathbf{z})=t^r+k\ne 0$ 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 $r$.

arXiv commentFinal version, to appear in Acta Arithmetica