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:
$$\pi_{a_j, j=1,\ldots, k}(x) \sim C_{a_j, j=1,\ldots, k}\cdot\int_2^{x}\frac{\text{d}t}{(\log t)^k}$$
for an explicit constant $C_{a_j, j=1,\ldots, k}$ (omitted, 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”.
There is a “second” Hardy-Littlewood Conjecture, that is less trusted than Conjecture 1 and proved to contradict Conjecture 1, if true.
There is a more general statement, that is actually the one I’m curious about, omitted here.
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 $\zeta(s)$. Abbreviated, “zeta zeros").
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 Conjecture 1 ought to lie (much) deeper than RH).
Question: Is there, in the literature, a conjecture about “nontrivial relations” (in the above sense) among the zeta zeros, that satisfies the following requirements:
(1) it is expected to be true, and the literature provides evidence towards it, to some extent;
(2) it is proved or expected to imply Conjecture 1, or to even be logically equivalent to it.
In other words, is there a conjecture, in the literature, that translates Conjecture 1 into a conjecture about "nontrivial relations" among the zeta zeros?
