# k-central numbers and bounded gaps between primes

As in the previous articles, let’s assume Goldbach’s conjecture so as to define properly the notion of $k$-central number. The number of $k$-central numbers less than $x$ $\pi_{C,k}(x)$ should verify the following relation:

$\pi_{C,k}(x)=\vert\{n\le x, k_{0}(n)=k\}\vert$

and thus $\pi_{C,k}(x)\le\dfrac{\pi(x+\max_{n\le x}r_{0}(n))}{k}(1+o(1))$.

I now formulate the following conjecture:

Negligible fundamental primality radius conjecture (NFPR conjecture for short):

$\forall\varepsilon>0,\forall x>2, \max_{n\le x}r_{0}(n)=O_{\varepsilon}(x^{\varepsilon})$

One can deduce from this conjecture and the prime number theorem that $\dfrac{\pi(x+\max_{n\le x}r_{0}(n))}{k}\sim\dfrac{\pi(x)}{k}$.

Thus $\pi_{C,k}(x)\le \dfrac{\pi(x)}{k}(1+o(1))$.

Hence, defining $\mathcal{N}_{k}(x)$ as the quantity equal to:

$\displaystyle{\sum_{l=0}^{k}\pi_{C,l}(x)}$,  one gets:

$\mathcal{N}_{k}(x)\le\pi(x)(1+H_{k})(1+o(1))$

where $H_{k}$ is the $k$-th harmonic number.

So that one should have:

$\mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n})\le k(1+H_{k})(1+o(1))$

hence:

$\mathcal{N}_{k}(x)=O(k\log k)$.

Now, from the prime number theorem, one has:

$\mathcal{N}_{k}(x)\sim x$ for $k$ large enough and less than $x$

So, it should be possible to prove rigorously that the conjunction of Goldbach’s conjecture and NFPR conjecture would entail that:

$\displaystyle{\liminf_{n\to\infty} p_{n+k}-p_{n}=O(k\log k)}$

This last relation, as stated in http://arxiv.org/pdf/1306.0948.pdf, follows from Hardy-Littlewood’s prime k-tuples conjecture.

Let’s now define the quantity $\alpha(x,k)$ as follows:

$\alpha(x,k):=\frac{\pi(x)}{k}-\pi_{C,k}(x)$

It seems that there exists $C>0$ (and possibly not much bigger than $1$) such that:

$\forall(x,k)\vert \alpha(x,k)\vert0$

Then there exists a unique $m$ such that $n=\frac{p_{m}+p_{m+k}}{2}$.

One has obviously $p_{m}, hence:

$m\leqslant\pi(n)\leqslant m+k$

and thus $m\geqslant \pi(n)-k$

Moreover $m\leqslant\pi(x)$, so that the total number of $k$-central numbers below $x$ verifies:

$\pi_{C,k}(x)=\delta\vert\{m', \frac{p_{m}+p_{m+k}}{2}\leqslant x\}\vert+h_{k}(x)$

where $\delta$ is the probability for $n'=\frac{p_{m'}+p_{m'+k}}{2}$ to be $k$-central,

and $0\leqslant \vert h_{k}(x)\vert\leqslant 1$

There are $k$ possibilities for the value of $k_{0}(n')$, namely $k_{0}(n')=1, 2, \cdots, k$

Since $n'$ is $k$-central if and only if $k_{0}(n')=k$, one gets:

$\delta=\frac{1}{k}$

Thus $\pi_{C,k}(x)\geqslant\frac{\pi(x)-k}{k}$

Since $\pi_{C,k}(x)=\frac{\pi(x)}{k}-\alpha(x,k)$ one finally gets:

$\vert\alpha(x,k)\vert\leqslant max(\vert h_{k}(x)\vert,1)$

hence $\vert\alpha(x,k)\vert\leqslant 1$.

Obviously the next step consists in showing that:

$\displaystyle{\liminf_{n\to+\infty} p_{n+k}-p_{n}=O(\mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n}))}$

and hopefully:

$\displaystyle{\liminf_{n\to+\infty} p_{n+k}-p_{n}\sim \mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n})}$

One has:

$p_{n+k}-p_{n}=\mathcal{N}_{n+k}(p_{n+k})-\mathcal{N}_{n+k}(p_{n})$

and thus:

$\mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n})=p_{n+k}-p_{n}-(n+k)(H_{n+k}-H_{n})+n(H_{n+k}-H_n)+O(n+k)$

hence:

$\mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n})=p_{n+k}-p_{n}-k(H_{n+k}-H_{k})+O(n+k)$

Thus $\dfrac{p_{n+k}-p_{n}}{\mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n})}=1+O(\dfrac{n+k}{k\log k})+O(\dfrac{\log n}{\log k})$

As Maynard proved that the quantity

$\displaystyle{\liminf_{n\to\infty}p_{n+k}-p_{n}}$

only depends on $k$, one should obtain:

$\displaystyle{\liminf_{n\to\infty}\dfrac{p_{n+k}-p_{n}}{\mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n})}}$

by substracting the divergent part of the error term above.

So that we finally get:

$\displaystyle{\liminf_{n\to\infty}\dfrac{p_{n+k}-p_{n}}{\mathcal{N}_{k}(p_{n+k})-\mathcal{N}_{k}(p_{n})}=1+O(\dfrac{1}{\log k})}$

and thus:

$\displaystyle{\liminf_{n\to\infty}p_{n+k}-p_{n}\sim k\log k}.$

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# 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.

# Upper bound for r0(n) (2)

Writing:

$\mathcal{N}_{n}(x)=k_{n}x(1+\frac{\alpha_{n}(x)}{x})$

and using the same kind of statistical heuristics as in « Primality radius »,

one gets:

$\alpha_{n}(x)=O(\sqrt{x}\log^{2} n)$

hence:

$r_{0}(n)=\dfrac{P_{ord_{C(n)}}}{2N_{1}(n)}+O(\sqrt{r_{0}(n)}\log^{2} n)$

that is:

$r_{0}(n)=O(\log^{2} n)+O(\sqrt{r_{0}(n)}\log^{2} n)$

Writing:

$r_{0}(n)=O(\log^{m} n)$

one gets:

$r_{0}(n)=O(\log^{2} n)+O(\log^{2+m/2} n)$

so that:

$m=4$

and $r_{0}(n)=O(\log^{4} n)$.

# Average value of r0(n)

Let’s denote by $r_{0}(n)$ the smallest potential typical primality radius of a given positive integer $n>1$.

Writing:

$\mathcal{N}_{n}(x)\approx k_{n}x$

where $\mathcal{N}_{n}(x)$ stands for the number of potential typical primality radii of $n$ below $x$ and $k_{n}$ is a quantity depending only on $n$,

one gets:

$\mathcal{N}_{n}(r_{0}(n)+\varepsilon)=1$

and

$\mathcal{N}_{n}(r_{0}(n)-\varepsilon)=0$

where $\varepsilon$ is any positive real number less than $1$.

Hence $\mathcal{N}_{n}(r_{0}(n))$ should be taken as equal to $1/2$,

and one would thus get:

$r_{0}(n)\approx 1/2k_{n}$

Considering the definition of the quantity $\alpha_{n}$ in the article « Primality radius », the only possible value of $k_{n}$ such that:

$\mathcal{N}_{n}(x)\approx k_{n}x$

is:

$k_{n}=\frac{N_{1}(n)}{P_{ord_{C}}(n)}$

Therefore one should expect the following relation to hold:

$r_{0}(n)\approx\frac{P_{ord_{C}(n)}}{2N_{1}(n)}$

As it can be proven that:

$\frac{P_{ord_{C}(n)}}{2N_{1}(n)}=O(\log^{2} n)$

one gets the following average value for $r_{0}(n)$:

$r_{0}(n)\approx C\log^{2}n$ for some $C>0$.

This might help to establish the asymptotic Goldbach’s conjecture (every large enough even integer is the sum of two primes) and Cramer’s conjecture that says:

$p_{n+1}-p_{n}=O(\log^{2}p_{n})$.

Let’s start with the notion of primality radius of a positive integer, which I came to think of while working on Goldbach’s conjecture.

By definition, given a positive integer $n>1$, the non negative integer $u$ is a primality radius of $n$ if and only if both $n-u$ and $n+u$ are prime numbers.

A prime has trivially a primality radius equal to $0$, but the concept becomes really interesting when one considers a composite integer. For example, it is easy to check that $3$ is the smallest primality radius of $14$.

The famous twin prime conjecture is equivalent to the following statement:

« $1$ is a primality radius of infinitely many positive integers »

whereas Goldbach’s conjecture simply becomes:

« every positive integer $n>1$ admits a primality radius ».

Obviously, $u$ can only be a primality radius of $n$ provided $u.

Here comes an almost copy-paste version of a question of mine on MathOverflow:

« Let’s define the number $ord_{C}(n)$, which depends on $n$, in the following way:

$ord_C(n):=\pi(\sqrt{2n-3})$ where $\pi(x)$ is the number of primes less or equal to $x$.

$(n+u)$ is a prime only if for all prime $p$ less or equal to $\sqrt{2n-3}$:

$p$ doesn’t divide $(n+u)$.

There are exactly $ord_{C}(n)$ such primes.

The number $ord_{C}(n)$ will be called the « natural configuration order » of $n$.

Now let’s define the « $k$-order configuration » of an integer $m$, denoted by $C_{k}(n)$, as the following sequence:

$(m \ \ mod \ \ 2, \ \ m \ \ mod \ \ 3,...,m \ \ mod \ \ p_{k})$

For example: $C_{4}(10)=(10\ \ mod \ \ 2,\ \ 10 \ \ mod \ \ 3, \ \ 10 \ \ mod \ \ 5, \ \ 10 \ \ mod \ \ 7)=(0,1,0,3)$

I call $C_{ord_{C}(n)}(n)$ the « natural configuration » of $n$.

To make a non negative integer $r < 2\times 3\times\dots\ p_{ord_{C}(n)}$ be a primality radius of $n$, one can require the following condition:

1) For all integer $i$ such that $1\leq i\leq ord_{C}(n)$:

$(n-r) \ \ mod \ \ p_{i}$ differs from $0$

and

$(n+r) \ \ mod \ \ p_{i}$ differs from $0$.

If this statement 1) is true, then $r$ will be called a « potential typical primality radius » of $n$.
Moreover, if $r\leq n-3$, then $r$ will simply be called a « typical primality radius » of $n$.

Now let’s define:

$N_{1}(n)$ as the number of potential typical primality radii of $n$ (hence less than $P_{ord_{C}(n)}$ but potentially greater than $n$)

where:

$P_{ord_{C}(n)}:=2\times 3\times...\times p_{ord_{C}(n)}$

$N_{2}(n)$ as the number of typical primality radii of $n$

$\alpha_{n}$ by the following equality:

$N_{2}(n)=\dfrac{n.N_{1}(n)}{P_{ord_{C}(n)}}\left(1+\dfrac{\alpha_{n}}{n}\right)$

It is quite easy to give an exact expression of $N_{1}(n)$ and to show that:

$\dfrac{n.N_{1}(n)}{P_{ord_{C}(n)}}>\left(c.\dfrac{n}{\log(n)^{2}}\right)\left(1+o(1)\right)$, where $c$ is a positive constant.

A statistical heuristics* makes me think that: $\forall \varepsilon>0, \ \ \alpha_{n}=O_{\varepsilon}\left(n^{\frac{1}{2}+\varepsilon}\right)$.

I wonder whether this last statement is equivalent to RH (Riemann Hypothesis) or not. If so, it would mean that RH implies that every large enough even number is the sum of two primes. »

*the statistical heuristics I refer to is:

$\vert p-f\vert <\frac{1}{\sqrt{n}}$

with $p$ the « probability » of an integer less than $P_{ord_{C}(n)}$ to be a potential typical primality radius of $n$

hence $p=\dfrac{N_{1}(n)}{P_{ord_{C}(n)}}$

and $f$ the « frequency » of the event « being a typical primality radius of $n$« .

so that $f=\dfrac{N_{2}(n)}{n}$.

This gives:

$\alpha_{n}=O(\sqrt{n}\log^{2}n)$

This is, up to the implied constant, the error term in the explicit formula of $\psi(n)$ under RH.

I was told several years ago by a brilliant youngster that such an upper bound for $\alpha_{n}$ implies GRH (Generalized Riemann Hypothesis, that deals with Dirichlet L-functions). The big deal is thus to prove some kind of converse implication.