For any rβN and almost all kβN smaller than
xr, we show that the polynomial f(n)=nr+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
NK/Qβ(z)=tr+kξ =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.Comment: V2: Minor correction