Average Bateman--Horn for Kummer polynomials

Abstract

For any prime r ∈ N and almost all k ∈ 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/Q (z) = t r + k = 0, where K/Q is a quadratic extension. A key ingredient in our proof is a new large sieve inequality for Dirichlet characters of exact order r.