# 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.

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