On the Bateman-Horn conjecture. (Q1864871)

The Bateman-Horn conjecture is a quantitative version of the famous Schinzel conjecture about prime values attained simultaneously by a finite collection of irreducible integer-valued polynomials satisfying some natural constraints. Suppose \(f_1,\dots, f_r\), \(r\geq 2\), are such polynomials and let \(1\leq s< r\) be an integer. The author assumes certain uniformity in the Bateman-Horn conjecture for suitably shifted polynomials \(f_1,\dots, f_s\) and \(f_{s+1},\dots, f_r\) and proves that then the original Bateman-Horn conjecture for the whole collection \(f_1,\dots, f_r\) is equivalent to another conjecture concerning minor arcs in the circle method.