Hardy–Littlewood-type k-tuples conjecture for prime patterns

Establish that for any integer k ≥ 1 and any k-tuple of distinct positive even integers a = (a1, ..., ak) that does not cover all residue classes to any prime modulus, the count Lk(N, a) of integers m with 0 < m ≤ N such that m + a1, ..., m + ak are all prime numbers satisfies the asymptotic formula Lk(N, a) ≈ C(k) N (log N)^{-k} as N → ∞, where C(k) is a positive constant depending on k.

Background

The paper studies the complexity and randomness of sequences associated with primes by defining local and information entropies for individual trajectories. To estimate the number of distinct binary patterns occurring in the prime indicator sequence, the authors invoke a k-tuples conjecture that gives an asymptotic for the number of shifts where k specified positions are simultaneously prime.

Under the validity of this conjecture, the authors deduce the exact value of the information entropy for the prime indicator sequence. They note that this conjecture is closely related to the classical Hardy–Littlewood conjectures and that only partial results are currently known, highlighting its unresolved status in analytic number theory.