A PNT based variance inequality for Cramer’s conjecture

The Prime Number Theorem (PNT for short) says that the average gap between two consecutive primes of size n is \sim\log n. Defining the quantity \rho_{i}(n) as \frac{2r_{i-1}(n)}{\pi(n+r_{i-1}(n))-\pi(n-r_{i-1}(n))}, where r_{k-1}(n) is the k-th typical primality radius of n, one can expect \rho_{i}(n) to get closer to \log n as i increases for a given n.

Let’s define the « gap-variance » of n, denoted by \sigma^{2}_{\rho}(n), as follows:

\displaystyle{\sigma^{2}_{\rho}(n):=\frac{1}{N_{2}(n)}\sum_{i=1}^{N_{2}(n)}\vert \rho_{i}(n)-\log n\vert^{2}}

where N_{2}(n) is the total number of typical primality radii of n.

From the observation above, one can write \displaystyle{\sigma^{2}_{\rho}(n)\leq \frac{1}{N_{2}(n)}\sum_{i=1}^{N_{2}(n)}\vert\rho_{1}(n)-\log n\vert^{2}}

Hence \sigma^{2}_{\rho}(n)\leq\vert\rho_{1}(n)-\log n\vert^{2}.

As, whenever m is 1-central, one has \rho_{1}(m)=2r_{0}(m), one gets for such m:

\sigma^{2}_{\rho}(m)\leq(2r_{0}(m)-\log m)^{2}\leq 4r_{0}(m)^{2}-4r_{0}(m)\log m+\log^{2} m

So that \sigma_{\rho}^{2}\leq \log^{2}(m)+4r_{0}(m)(r_{0}(m)-\log m)

Hence, r_{0}(m)\leq \log m\Rightarrow\sigma^{2}_{\rho}(m)\leq\log^{2}m, so that it’s very likely that the strong form of Cramer’s conjecture, namely \displaystyle{\limsup_{k\to\infty}\frac{p_{k+1}-p_{k}}{\log^{2}p_{k}}=1}, is true.