On the bateman-horn conjecture

Let r be a positive integer and f,...,f, be distinct polynomials in Z[X]. If f (n), ...,f(n) are all prime for infinitely many n, then it is necessary that the polynomials f are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f(n)...f(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 f (n),...,f(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 {f,...,f}, {,...,f} 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 f,...,f, is equivalent to a plausible error term conjecture for the minor arcs in the circle method.