Bateman-Horn conjecture

A conjecture on the asymptotic behaviour of a polynomial satisfying the Bunyakovskii condition (cf. also Bunyakovskii conjecture).

Let $ f _ {1} ( x ) \dots f _ {r} ( x ) $ be polynomials (cf. Polynomial) with integer coefficients, of degrees $ d _ {1} \dots d _ {r} \geq 1 $, irreducible (cf. Irreducible polynomial), and with positive leading coefficients. Let

$$ f = f _ {1} \dots f _ {r} $$

be their product.

V. Bunyakovskii considered the case $ r = 1 $ and asked whether $ f ( n ) $ could represent infinitely many prime numbers as $ n $ ranges over the positive integers. An obvious necessary condition is that all coefficients of $ f $ be relatively prime. However, that is not sufficient. He conjectured that, in addition, the following Bunyakovskii condition is sufficient: there is no prime number $ p $ dividing all the values $ f ( n ) $ for the positive integers $ n $( cf. Bunyakovskii conjecture).

Assuming the Bunyakovskii condition, let

$$ C ( f ) = \prod _ {p \textrm{ a prime } } \left ( 1 - { \frac{1}{p} } \right ) ^ {- r } \left ( 1 - { \frac{N _ {f} ( p ) }{p} } \right ) , $$

where $ N _ {f} ( p ) $ is the number of solutions of the congruence equation $ f ( n ) \equiv0 ( { \mathop{\rm mod} } p ) $( for $ p $ prime). The Bateman–Horn conjecture asserts that

$$ \pi _ {f} ( x ) \sim { \frac{C ( f ) }{d _ {1} \dots d _ {r} } } \int\limits _ { 2 } ^ { x } { { \frac{1}{( { \mathop{\rm log} } t ) ^ {r} } } } {dt } , $$

where $ \pi _ {f} ( x ) $ is the number of positive integers $ n \leq x $ such that all $ f _ {1} ( n ) \dots f _ {r} ( n ) $ are prime.

This formula gives the density of primes in an arithmetic progression (cf. Dirichlet theorem), using the polynomial $ f ( x ) = ax + b $. After some computations, it gives the asymptotic behaviour conjectured by G.H. Hardy and J.E. Littlewood for the number of primes representable by the polynomial $ x ^ {2} + 1 $. It also gives the Hardy–Littlewood conjecture for the behaviour of the number of twin primes, by applying the formula to the polynomials $ x $ and $ x + 2 $( cf. also Twins). Similarly, it implies many other conjectures of Hardy and Littlewood stated in [a2].

See also Distribution of prime numbers.

References

[a1]	P.T. Bateman, R. Horn, "A heuristic formula concerning the distribution of prime numbers" Math. Comp. , 16 (1962) pp. 363–367
[a2]	G.H. Hardy, J.E. Littlewood, "Some problems of Partitio Numerorum III" Acta Math. , 44 (1922) pp. 1–70
[a3]	H. Halberstam, H.-E. Richert, "Sieve methods" , Acad. Press (1974)