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

Let  $  f _ {1} ( x ) \dots f _ {r} ( x ) $
be polynomials (cf. [[Polynomial|Polynomial]]) with integer coefficients, of degrees  $  d _ {1} \dots d _ {r} \geq 1 $, 
irreducible (cf. [[Irreducible polynomial|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|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|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|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|Twins]]). Similarly, it implies many other conjectures of Hardy and Littlewood stated in [[#References|[a2]]].

See also [[Distribution of prime numbers|Distribution of prime numbers]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.T. Bateman,   R. Horn,   "A heuristic formula concerning the distribution of prime numbers"  ''Math. Comp.'' , '''16'''  (1962)  pp. 363–367</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.H. Hardy,   J.E. Littlewood,   "Some problems of Partitio Numerorum III"  ''Acta Math.'' , '''44'''  (1922)  pp. 1–70</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  H. Halberstam,   H.-E. Richert,   "Sieve methods" , Acad. Press  (1974)</TD></TR></table>