### Hardy-Littlewood constants and reciprocals

© 2011 by Alexei Kourbatov, JavaScripter.net/math
Main article: Maximal gaps between prime k-tuples

The Hardy-Littlewood k-tuple conjecture allows one to predict the average frequencies of prime k-tuples near p, as well as the approximate total counts of prime k-tuples below x. Specifically, the Hardy-Littlewood k-tuple constants Hk, divided by logkp, give us an estimate of the average frequency of prime k-tuples near p:

 Frequency of k-tuples  ≈  Hk / logkp   or, equivalently, Total number of k-tuples below x

Reciprocal Hardy-Littlewood constants multiplied by logkp determine the average gap between prime k-tuples near p:

Expected average gap a  =  Ck logkp,     where  Ck = 1/Hk.

Using the expected average gap a, we can estimate maximal gaps gk between prime k-tuples near p:

Expected maximal gap  =  a(log(p/a) − b),     with  a = Ck logkp,  b ≈ 2/k.

Table. Hardy-Littlewood constants for prime k-tuples up to k = 8.

Name of prime k-tuple Width Pattern Hardy-Littlewood
constant Hk
Reciprocal
Ck = 1/Hk
2-tuple, twin primes, twins 2 0 2 1.320320.757392
3-tuple, triplet (type A) 6 0 4 6 2.858250.349864
3-tuple, triplet (type B) 6 0 2 6 2.858250.349864
4-tuple, quadruplet 8 0 2 6 8 4.151180.240895
5-tuple, quintuplet (type A) 12 0 4 6 10 12 10.131790.09869924
5-tuple, quintuplet (type B) 12 0 2 6 8 12 10.131790.09869924
6-tuple, sextuplet 16 0 4 6 10 12 16 17.298610.05780811
7-tuple, septuplet (type A) 20 0 2 6 8 12 18 20 53.971950.01852814
7-tuple, septuplet (type B) 20 0 2 8 12 14 18 20 53.971950.01852814
8-tuple, octuplet (type A) 26 0 2 6 8 12 18 20 26178.261950.005609722
8-tuple, octuplet (type B) 26   0 6 8 14 18 20 24 26  178.261950.005609722
8-tuple, octuplet (type C) 26   0 2 6 12 14 20 24 26  475.365210.002103646

Hardy-Littlewood constants can be computed with a high precision; see Tony Forbes k-tuplets pages for more information on this.