# nt.number theory – From asymptotic Goldbach to an upper bound of \$r_{0}(n)\$

This question is a follow up to my question About Goldbach’s conjecture whose beginning of the content I now copy paste so as to get a self contained question:

“let’s consider a composite natural number $$n$$ greater or equal to $$4$$. Goldbach’s conjecture is equivalent to the following statement: “there is at least one natural number $$r$$ such as $$(n-r)$$ and $$(n+r)$$ are both primes”.
For obvious reasons $$rleq n-3$$. Such a number $$r$$ will be called a “primality radius” of $$n$$.

Now 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 $$x$$.
$$(n+r)$$ is a prime only if for all prime $$p$$ less or equal to $$sqrt{2n-3}$$, $$p$$ doesn’t divide $$(n+r)$$. 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 $$C_{k}(n)$$, as the 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 $$n$$.

A sufficient condition to make $$r$$ be a primality radius of $$n$$ is that for all integer $$i$$ such that $$1leq ileq ord_{C}(n)$$, $$(n-r) mod p_{i}$$ differs from $$0$$ and $$(n+r) mod p_{i}$$ differs from $$0$$. If this statement is true, $$r$$ will be called a “potential typical primality radius” of $$n$$.
Moreover, if $$rleq n-3$$, then $$r$$ will be called a “typical primality radius” of $$n$$.

Now let’s define $$N_{1}(n)$$ as the number of potential typical primality radii of $$n$$ less than $$P_{ord_{C}(n)}$$, where $$P_{ord_{C}(n)}=2times 3times…times p_{ord_{C}(n)}$$, $$N_{2}(n)$$ as the number of typical primality radii of $$n$$, and $$alpha_{n}$$ 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 $$forall varepsilon>0, alpha_{n}=O_{varepsilon}left(n^{frac{1}{2}+varepsilon}right)$$.”

So let’s consider a generalization of the counting function of the primality radii of $$n$$ as follows:

Let $$N_{2}(n;k)$$ denote the number of primality radii of $$n$$ not exceeding $$k$$ and define $$beta_{n,k}$$ writing the following equality:

$$N_{2}(n;k)=dfrac{kN_{1}(n)}{P_{ord_{c}}(n)}left(1+beta_{n,k}right)$$

Of course $$N_{2}(n;n)=N_{2}(n)$$ and $$beta_{n,n}=frac{alpha_{n}}{n}$$.

Denoting by $$r_{0}(n)$$ the smallest primality radius of $$n$$, one has trivially $$N_{2}(n;r_{0}(n))=frac{r_{0}(n)N_{1}(n)}{P_{ord_{c}}(n)}left(1+beta_{n,r_{0}(n)}right)=1$$.

So, can one get bounds on $$beta_{n,k}$$ so as to provide an upperbound for $$r_{0}(n)$$?