Difference between revisions of "Bateman-Horn conjecture"
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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 | + | 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. | be their product. | ||
| − | V. Bunyakovskii considered the case | + | 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 | ||
| − | + | $$ | |
| + | C ( f ) = \prod _ {p \textrm{ a prime } } \left ( 1 - { | ||
| + | \frac{1}{p} | ||
| + | } \right ) ^ {- r } \left ( 1 - { | ||
| + | \frac{N _ {f} ( p ) }{p} | ||
| + | } \right ) , | ||
| + | $$ | ||
| − | where | + | 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 | + | 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 | + | 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) |
Bateman-Horn conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bateman-Horn_conjecture&oldid=19202