Beurling-Nyman criterion for RH and Cramer’s conjecture

Cramer proved that under RH, the prime gap g_{n}:=p_{n+1}-p_{n} fulfills g_{n}=O(p_{n}^{\frac{1}{2}}\log p_{n}) but he conjectured that much better is true, namely that g_{n}=O(\log^{2} p_{n}), 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 g_{n}>c\frac{\log p_{n}\log_{2}p_{n}\log_{4}p_{n}}{(\log_{3}p_{n})^{2}} with a positive constant c that we may take as large as we want. This suggests to consider the lower and upper bounds as k-tuples of integers, denoted respectively by E_{k}:=(e_{1},\cdots,e_{k}) and F_{k}:=(f_{1},\cdots,f_{k}), such that \displaystyle{{C_{E_{k}}}\prod_{i=1}^{k}(\log_{i} p_{n})^{e_{i}}<g_{n}<D\prod_{i=1}^{k}(\log_{i} p_{n})^{f_{i}}}, with \vert\vert E_{k}\vert\vert_{1}:=\sup\{\vert e_{k}\vert\}=1 (and we may also require \displaystyle{\sum_{i=1}^{k}e_{i}=1}) and \vert\vert F_{k}\vert\vert_{2}:=(\sum_{i=1}^{k}f_{i}^{2})^{1/2}=2.

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 \frac{\log g_{n}}{\log_{2} p_{n}}) which equals 1 on average but conjecturally can’t exceed some upper limit.

My idea for the norms of the k-tuples comes from Beurling-Nyman criterion for RH relating the non vanishing of the Riemann zeta function in the half plane \Re(s)>\frac{1}{p} and the density of some function space in an L^{p} space: could we prove that \vert\vert E_{k}\vert\vert_{m}=m with m=\inf\{p\in\mathbb{Z}_{>0}\mid \Re(s)>\frac{1}{p}\Longrightarrow\zeta(s)\neq 0\} and that \vert\vert F_{k}\vert\vert_{M}=M=\vert\vert F_{k}\vert\vert_{m} with M=\sup\{p\in\mathbb{Z}_{>0}\mid\Re(s)>\frac{1}{p}\Longrightarrow\zeta(s)\neq 0\}, we would get that RH implies Cramer’s conjecture.