Bateman-Horn's conjecture on average

The Bateman-Horn conjecture predicts the density of primevalues assumed by integer polynomials. In this talk, we discuss recentprogress on an average version of this conjecture, achieved by employingHooley neutralizers in combination with standard sieving techniques. Inparticular, our findings suggest that a much worse version ofsquare-root cancellation for the Bateman-Horn conjecture should hold foralmost all polynomials.