Are the ratios of Hardy Littlewood constants for prime constellations already well understood? If so, can you please point me to a relevant paper(s)?
I came upon an Euler-inspired formula that yields the ratios of Hardy Littlewood constants for prime constellations (consistently for the ratios I’ve checked based on lists of published constants), using the number of elements in the constellations one is comparing and the prime factors of the pair wise differences between the elements (the absolute differences between elements are irrelevant). But I don’t know whether it’s a significant finding; perhaps this is well-trodden ground already. The connection isn’t random; I bothered to check because intuition told me there was a connection.
For example, the H-L constants for constellations (0,2,6,8) and (0,4,6,10) are 4.15... and 8.30..., respectively (not intuitive that the ratio should be 1/2).
If this is novel territory, I’m happy to share more. I haven’t yet found any published work on this specific topic that I recognize as such (due to my math illiteracy).