k-central numbers and Cramer’s conjecture

Assuming Goldbach’s conjecture (that is, r_{0}(n)<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.

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