# k-central numbers and Cramer’s conjecture

Assuming Goldbach’s conjecture (that is, $r_{0}(n) whenever $n>1$), let’s write $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$.

I call $k_{0}(n)$ the order of centrality of $n$ and say that $n$ is a $k$-central number if and only if $k_{0}(n)=k$.

Now let’s consider $\rho(n):=\dfrac{2r_{0}(n)}{k_{0}(n)}$ and $\sigma(n):=\dfrac{\log(\rho(n))}{\log\log n}$. Let $\sigma_{+}:=\lim\sup \sigma(n)$ and $\sigma_{-}:=\lim \inf\sigma(n)$ and $\sigma_{m}:=\dfrac{\sigma_{+}+\sigma_{-}}{2}$. Zhang’s theorem implies $\sigma_{-}=0$, while Cramer’s conjecture implies $\sigma_{+}=2$. The Prime Number Theorem asserts that, on average, $\sigma(n)=1$, so that $\sigma_{m}$ is very likely to be equal to $1$.

Let’s now define the following density: $\delta_{\varepsilon,x}(\sigma):=\dfrac{\vert\{n\leq x, \sigma(n)\in(\sigma-\varepsilon,\sigma+\varepsilon)\}\vert}{x}$.

I formulate the following conjectures:

Symmetric Density Conjecture: $\displaystyle{\forall \sigma\in(\sigma_{-},\sigma_{+}), \lim_{\varepsilon\to 0}\lim_{x\to \infty}\dfrac{\delta_{\varepsilon,x}(\sigma)}{\delta_{\varepsilon,x}(2\sigma_{m}-\sigma)}=1}$

Increasing Density conjecture: $\displaystyle{\sigma_{-}\leq \sigma_{1}\leq \sigma_{2}\leq \sigma_{m}\Longrightarrow\lim_{\varepsilon\to 0}\lim_{x\to \infty}\dfrac{\delta_{\varepsilon,x}(\sigma_{1})}{\delta_{\varepsilon,x}(\sigma_{2})}\leq 1}$

It may be worth noticing that these two conjectures are a consequence of a rather natural hypothesis of rotational invariance.

Would these last two conjectures imply Cramer’s conjecture?
By the way, I think that the fact that whenever $r$ is a potential typical primality radius of $n$, then so is $P_{ord_{C}(n)}-r$, implies symmetric density conjecture.

Moreover, the same kind of reasoning can be applied mutatis mutandis replacing the quantities $\rho(n)=\dfrac{2r_{0}(n)}{k_{0}(n)}$ by $r_{0}(n)$, $\sigma_{-}$ by $\varsigma_{-}:=\lim\inf\dfrac{\log r_{0}(n)}{\log\log n}$ and $\sigma_{+}$ by $\varsigma_{+}:=\lim\sup\dfrac{\log r_{0}(n)}{\log\log n}$ to get $\varsigma_{-}=0$ (as a consequence of Maynard’s work on bounded gaps between primes), $\varsigma_{+}=4$ and $\varsigma_{m}:=\dfrac{\varsigma_{-}+\varsigma_{+}}{2}=2$ (as a consequence of both the fact that on average $r_{0}(n)=\dfrac{P_{ord_{C(n)}}}{2N_{1}(n)}$ and that this last quantity is $O(\log^{2}(n)$), hence giving further evidence for the equality $r_{0}(n)=O(\log^{4} n)$ derived under a statistical heuristics in the previous article.