On the Bateman–Horn Conjecture

AbstractLet r be a positive integer and f1,…,fr be distinct polynomials in Z[X]. If f1(n),…,fr(n) are all prime for infinitely many n, then it is necessary that the polynomials fi are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f1(n)···fr(n), as n runs over Z. Assuming these necessary conditions, Bateman and Horn (Math. Comput.16 (1962), 363–367) proposed a conjectural asymptotic estimate on the number of positive integers n⩽x such that f1(n),…,fr(n) are all primes. In the present paper, we apply the Hardy–Littlewood circle method to study the Bateman–Horn conjecture when r⩾2. We consider the Bateman–Horn conjecture for the polynomials in any partition {f1,…,fs}, {fs+1,…,fr} with a linear change of variables. Our main result is as follows: If the Bateman–Horn conjecture on such a partition and change of variables holds true with some conjectural error terms, then the Bateman–Horn conjecture for f1,…,fr is equivalent to a plausible error term conjecture for the minor arcs in the circle method