# 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:

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