Cramer proved that under RH, the prime gap fulfills but he conjectured that much better is true, namely that , which holds almost surely in his random model of primes.
On the other hand, a famous question by Erdős and Turán was answered positively by Ford, Green, Konyagin, Maynard and Tao, namely that with a positive constant that we may take as large as we want. This suggests to consider the lower and upper bounds as -tuples of integers, denoted respectively by and , such that , with (and we may also require ) and .
This may be seen as some kind of atomic packing factor condition in a “prime crystal” made of “atoms” of positive radius (namely the quantity ) which equals on average but conjecturally can’t exceed some upper limit.
My idea for the norms of the -tuples comes from Beurling-Nyman criterion for RH relating the non vanishing of the Riemann zeta function in the half plane and the density of some function space in an space: could we prove that with and that with , we would get that RH implies Cramer’s conjecture.