OFFSET
1,2
COMMENTS
The Hardy and Littlewood Conjecture F provides an estimate of the number of primes generated by a quadratic polynomial P(x) for 0 <= x <= m in the form C * Integral_{x=2..m} 1/log(x) dx), with C given by an Euler product that is a function of the fundamental discriminant of the polynomial. Cohen describes an efficient method for the computation of C.
The following table provides the record values of C, together with the number of primes np generated by the polynomial x^2 + x + a(n) for x <= 10^8 and the actual ratio 2*np/Integral_{x=2..10^8} 1/log(x) dx.
a(n) C np C from ratio
1 2.24147 6456835 2.24110
11 3.25944 9389795 3.25910
17 4.17466 12027453 4.17460
41 6.63955 19132653 6.64073
21377 6.92868 19962992 6.92894
27941 7.26400 20931145 7.26497
41537 7.32220 21092134 7.32085
55661 7.45791 21483365 7.45664
115721 7.70935 22210771 7.70912
239621 7.72932 22268336 7.72909
247757 8.24741 23762118 8.24757
Jacobson and Williams found significantly larger values of C for very large addends k, e.g. C = 2*5.36708 = 10.73416 for k = 3399714628553118047.
REFERENCES
Keith Conrad, Hardy-Littlewood Constants. In: Mathematical Properties of Sequences and Other Combinatorial Structures, eds. Jong-Seon No, Hong-Yeop Song, Tor Helleseth, P. Vijay Kumar, Springer New York, 2003, pages 133-154.
LINKS
Karim Belabas, Henri Cohen, Computation of the Hardy-Littlewood constant for quadratic polynomials, PARI/GP script, 2020.
Henri Cohen, High precision computation of Hardy-Littlewood constants. [Cached pdf version, with permission]
Keith Conrad, Hardy-Littlewood Constants, (2003).
Michael J. Jacobson Jr. and Hugh C. Williams, New Quadratic Polynomials With High Densities Of Prime Values, Math. Comp., 72, 241, 499-519, 2002.
PROG
(PARI) \\ The function HardyLittlewood2 is provided at the Belabas, Cohen link.
hl2max=0; for(add=0, 100, my(hl=HardyLittlewood2(n^2+n+add)); if(hl>hl2max, print1(add, ", "); hl2max=hl))
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Hugo Pfoertner, Feb 02 2020
STATUS
approved