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On the Bateman–Horn Conjecture

✍ Scribed by Stephan Baier

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
156 KB
Volume
96
Category
Article
ISSN
0022-314X

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✦ Synopsis

If f 1 ðnÞ; . . . ; f r ðnÞ are all prime for infinitely many n; then it is necessary that the polynomials f i are irreducible in Z½X ; have positive leading coefficients, and no prime p divides all values of the product f 1 ðnÞ Á Á Á f r ðnÞ; 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 n4x such that f 1 ðnÞ; . . . ; f r ðnÞ are all primes. In the present paper, we apply the Hardy-Littlewood circle method to study the Bateman-Horn conjecture when r52: We consider the Bateman-Horn conjecture for the polynomials in any partition ff 1 ; . . . ; f s g; ff sþ1 ; . . . ; f r g 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 1 ; . . . ; f r is equivalent to a plausible error term conjecture for the minor arcs in the circle method.

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