Average Bateman–Horn for Kummer polynomials
Volume 207 / 2023
Acta Arithmetica 207 (2023), 315-350
MSC: Primary 14G05; Secondary 11N37, 11N32, 11N36, 11P55.
DOI: 10.4064/aa220921-20-2
Published online: 14 April 2023
Abstract
For any prime $r \in \mathbb N$ 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 $N_{K/\mathbb Q}(\textbf z) = t^r + k \ne 0$, 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 $r$.
