The first Hardy-Littlewood conjecture is called the k-Tuple Conjecture. It states that the asymptotic number of Prime Constellations can be computed explicitly.
The second Hardy-Littlewood conjecture states that
See also Prime Constellation, Prime Counting Function
References
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.