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?