Hardy-Littlewood Conjectures

The first Hardy-Littlewood conjecture is called the k-tuple conjecture. It states that the asymptotic number of prime constellations can be computed explicitly. A particular case gives the so-called strong twin prime conjecture

The second Hardy-Littlewood conjecture states that

for all , where  is the prime counting function.

The following table summarizes the first few values of  for integer  and , 2, .... The values of this function are plotted above.

	OEIS	for , 2, ...
1	A080545	1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, ...
2	A090405	2, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, ...
3	A090406	2, 2, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 0, ...

Although it is not obvious, Richards (1974) proved that the first and second conjectures are incompatible with each other.

REFERENCES:

Guy, R. K. §A9 in Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.

Hardy, G. H. and Littlewood, J. E. "Some Problems of 'Partitio Numerorum.' III. On the Expression of a Number as a Sum of Primes." Acta Math. 44, 1-70, 1923.

Richards, I. "On the Incompatibility of Two Conjectures Concerning Primes." Bull. Amer. Math. Soc. 80, 419-438, 1974.

Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 61-62 and 68-69, 1994.

Sloane, N. J. A. Sequences A080545, A090405, A090406, in "The On-Line Encyclopedia of Integer Sequences."