Hardy-Littlewood constants

Hardy-Littlewood constants and reciprocals

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 H_k, divided by log^kp, give us an estimate of the average frequency of prime k-tuples near p:

Frequency of k-tuples ≈ H_k / log^kp or, equivalently,

Total number of k-tuples below x

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

Expected average gap a = C_k log^kp, where C_k = 1/H_k.

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

Expected maximal gap = a(log(p/a) − b), with a = C_k log^kp, 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 H_k	Reciprocal C_k = 1/H_k
2-tuple, twin primes, twins	2	0 2	1.32032	0.757392
3-tuple, triplet (type A)	6	0 4 6	2.85825	0.349864
3-tuple, triplet (type B)	6	0 2 6	2.85825	0.349864
4-tuple, quadruplet	8	0 2 6 8	4.15118	0.240895
5-tuple, quintuplet (type A)	12	0 4 6 10 12	10.13179	0.09869924
5-tuple, quintuplet (type B)	12	0 2 6 8 12	10.13179	0.09869924
6-tuple, sextuplet	16	0 4 6 10 12 16	17.29861	0.05780811
7-tuple, septuplet (type A)	20	0 2 6 8 12 18 20	53.97195	0.01852814
7-tuple, septuplet (type B)	20	0 2 8 12 14 18 20	53.97195	0.01852814
8-tuple, octuplet (type A)	26	0 2 6 8 12 18 20 26	178.26195	0.005609722
8-tuple, octuplet (type B)	26	0 6 8 14 18 20 24 26	178.26195	0.005609722
8-tuple, octuplet (type C)	26	0 2 6 12 14 20 24 26	475.36521	0.002103646

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

Frequency of k-tuples ≈ H_k / log^kp or, equivalently,
Total number of k-tuples below x