The first Hardy-Littlewood Conjecture asserts:

Conjecture 1: Fix integers $0< a_1 < \ldots < a_k$. The density of those primes $p\le x$ such that $p + 2a_1,\ldots, p+2a_k$ are also primes, is:

$$\pi_{a_j, j=1,\ldots, k}(x) \sim C_{a_j, j=1,\ldots, k}\cdot\int_2^{x}\frac{\text{d}t}{\log(t)}$$

for an explicit constant $C_{a_j, j=1,\ldots, k}$ (that I’m going to omit here).

Remarks:

• Conj. 1 is believed to be true, although wide open. It is known as the “$k$-tuple conjecture” or the “prime constellations conjecture”.

• When $k = 1$ and $a_1 = 1$, it recovers the “twin primes conjecture”.

It feels this conjecture should amount to some nontrivial relations among the zeros of the completed Riemann zeta function $\widehat{\zeta}(s)$ (i.e. the nontrivial zeros of the Riemann zeta function $\zeta(s)$).

By “nontrivial relations” I mean relations that are more involved than just “they all lie on the critical line” (which means, among the other things, that the conjecture ought to lie deeper than RH).

Question: Is there, in the literature, a conjecture about “nontrivial relations” (in the above sense) among the zeta zeros, that is proved to imply the first Hardy-Littlewood Conjecture, or is even logically equivalent to it?