Editing Bateman–Horn conjecture (section)

==Definition==
The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of ''m'' distinct [[irreducible polynomial]]s ''ƒ''<sub>1</sub>,&nbsp;...,&nbsp;''ƒ''<sub>''m''</sub> with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy [[Bunyakovsky's property]], that there does not exist a prime number ''p'' that divides their product  ''f''(''n'') for every positive integer ''n''. For, if there were such a prime ''p'', having all values of the polynomials simultaneously prime for a given ''n'' would imply that at least one of them must be equal to ''p'', which can only happen for finitely many values of ''n'' or there would be a polynomial with infinitely many roots, whereas the conjecture is how to give conditions where the values are simultaneously prime for infinitely many ''n''.

An integer ''n'' is prime-generating for the given system of polynomials if every polynomial ''ƒ<sub>i</sub>''(''n'') produces a prime number when given ''n'' as its argument. If ''P''(''x'') is the number of prime-generating integers among the positive integers less than ''x'', then the Bateman–Horn conjecture states that

:<math>P(x) \sim \frac{C}{D} \int_2^x \frac{dt}{(\log t)^m},\,</math>

where ''D'' is the product of the degrees of the polynomials and where ''C'' is the product over primes ''p''

:<math>C = \prod_p \frac{1-N(p)/p}{(1-1/p)^m}\ </math>

with <math>N(p)</math> the number of solutions to

:<math>f(n) \equiv 0 \pmod p.\ </math>

Bunyakovsky's property implies  <math>N(p) < p</math> for all primes ''p'',
so each factor in the infinite product ''C'' is positive.
Intuitively one then naturally expects that the constant ''C'' is itself positive, and with some work this can be proved.
(Work is needed since some infinite products of positive numbers equal zero.)