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.