OFFSET
1,2
COMMENTS
The method for calculating the Hardy-Littlewood constant for quadratic polynomials can be generalized to cubic polynomials (see the preprint by H. Cohen for the exact definition). In this case too, the constant is an estimate of which fraction (e.g. in relation to a random placement) of prime numbers the polynomial hits within its range of values. The following table shows that the ratio of the actual prime number hits for 1 <= x <= 10^8 for different addend values corresponds almost exactly to the ratio of the Hardy-Littlewood constants. The Hardy-Littlewood constant C and the number of prime hits np at offset = 1 are chosen as reference values.
k C np C(k)/C(1) np(k)/np(1)
1 3.075032 5907486 1.0000000 1.0000000
17 5.653199 10860984 1.8384196 1.8385120
101 6.035464 11594890 1.9627322 1.9627452
1487 6.783304 13030949 2.2059297 2.2058366
13301 6.890698 13236230 2.2408541 2.2405859
19421 6.967707 13380959 2.2658974 2.2650852
91127 7.121020 13682111 2.3157547 2.3160632
REFERENCES
Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
LINKS
Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for cubic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants, preprint, 1998. [pdf copy, with permission]
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial.
Wikipedia, Bateman-Horn conjecture.
PROG
(PARI) \\ The functions HardyLittlewood2 and HardyLittlewood3 are provided at the Belabas, Cohen links.
hl3max=0; for(add=0, 101, my(hl=HardyLittlewood3(n^3+n^2+add)); if(hl>hl3max, print1(add, ", "); hl3max=hl))
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Hugo Pfoertner, Feb 04 2020
STATUS
approved