On the bateman-horn conjecture

Abstract

Let 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}, {s+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. © 2002 Elsevier Science (USA).