View source for Bateman-Horn conjecture

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

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101601.png" /> be polynomials (cf. [[Polynomial|Polynomial]]) with integer coefficients, of degrees <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101602.png" />, irreducible (cf. [[Irreducible polynomial|Irreducible polynomial]]), and with positive leading coefficients. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101603.png" /></td> </tr></table>

be their product.

V. Bunyakovskii considered the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101604.png" /> and asked whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101605.png" /> could represent infinitely many prime numbers as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101606.png" /> ranges over the positive integers. An obvious necessary condition is that all coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101607.png" /> be relatively prime. However, that is not sufficient. He conjectured that, in addition, the following Bunyakovskii condition is sufficient: there is no prime number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101608.png" /> dividing all the values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b1101609.png" /> for the positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016010.png" /> (cf. [[Bunyakovskii conjecture|Bunyakovskii conjecture]]).

Assuming the Bunyakovskii condition, let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016011.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016012.png" /> is the number of solutions of the [[Congruence equation|congruence equation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016013.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016014.png" /> prime). The Bateman–Horn conjecture asserts that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016015.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016016.png" /> is the number of positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016017.png" /> such that all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016018.png" /> are prime.

This formula gives the density of primes in an arithmetic progression (cf. [[Dirichlet theorem|Dirichlet theorem]]), using the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016019.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016020.png" />. It also gives the Hardy–Littlewood conjecture for the behaviour of the number of twin primes, by applying the formula to the polynomials <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110160/b11016022.png" /> (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>