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A conjecture on the asymptotic behaviour of a polynomial satisfying the Bunyakovskii condition (cf. also [[Bunyakovskii conjecture|Bunyakovskii 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
+
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
  
<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>
+
$$
 +
f = f _ {1} \dots f _ {r}  $$
  
 
be their product.
 
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]]).
+
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
 
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>
+
$$
 +
C ( f ) = \prod _ {p  \textrm{ a  prime  } } \left ( 1 - {
 +
\frac{1}{p}
 +
} \right ) ^ {- r } \left ( 1 - {
 +
\frac{N _ {f} ( p ) }{p}
 +
} \right ) ,
 +
$$
  
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
+
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
  
<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>
+
$$
 +
\pi _ {f} ( x ) \sim {
 +
\frac{C ( f ) }{d _ {1} \dots d _ {r} }
 +
} \int\limits _ { 2 } ^ { x }  { {
 +
\frac{1}{( { \mathop{\rm log} } t )  ^ {r} }
 +
} }  {dt } ,
 +
$$
  
where <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.
+
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 <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]]].
+
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]].
 
See also [[Distribution of prime numbers|Distribution of prime numbers]].

Latest revision as of 10:15, 29 May 2020


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)
How to Cite This Entry:
Bateman-Horn conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bateman-Horn_conjecture&oldid=19202
This article was adapted from an original article by S. Lang (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article