Draft:Hardy-Littlewood constant

The Hardy-Littlewood constant of a given polynomial with integer coefficient is a constant related to how often this polynomial gives prime numbers, for example, the Hardy-Littlewood constant of the polynomial $x^{2}+x+41$ is 3.319773177471… (sequence A221712 in the OEIS), and the Hardy-Littlewood constant of the polynomial $x^{2}+1$ is 0.686406731409… (sequence A331941 in the OEIS), and the Hardy-Littlewood constant of the polynomial $x^{4}+1$ is 0.669740969937… (sequence A337607 in the OEIS).

Records

The numbers k > 0 such that the polynomial x^2 + x + k produces a record of its Hardy-Littlewood Constant is

1, 11, 17, 41, 21377, 27941, 41537, 55661, 115721, 239621, 247757, … (sequence A331940 in the OEIS)

The numbers k > 0 such that the polynomial x^3 + x^2 + k produces a record of its Hardy-Littlewood Constant is

1, 17, 101, 1487, 13301, 19421, 91127, … (sequence A331950 in the OEIS)

The numbers k > 0 such that the polynomial k*x^2 + 1 produces a record of its Hardy-Littlewood constant is

1, 2, 3, 4, 12, 18, 28, 58, 190, 462, 708, 5460, 10602, 39292, 141100, 249582, 288502, … (sequence A331945 in the OEIS)

Generalization

The Hardy-Littlewood constant for more than one polynomial (that give prime numbers simultaneously), for example, the Hardy-Littlewood constant for x^4+1 and (x+2)^4+1 both primes is 0.792208238167… (sequence A337608 in the OEIS)

See also

• Bunyakovsky conjecture
• Dickson's conjecture
• Schinzel's hypothesis H
• Bateman–Horn conjecture

External links

• Keith Conrad, Hardy-Littlewood Constants
• Michael J. Jacobson Jr. and Hugh C. Williams, New Quadratic Polynomials With High Densities Of Prime Values

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Categories:

• Analytic number theory