# A conjectural expression for N2(n)

The definition of a « typical » primality radius $r$ of $n$ leads to the following assumption:

the following ratio:

$\dfrac{N_{2}(n)}{N_{1}(n)}$

is asymptotically equal to the ratio:

$\dfrac{n-\sqrt{2n-3}}{P_{ord_{c}(n)}-\sqrt{2n-3}}$

Hence, for $n$ large enough, the quantity $N_{2}(n)$ should be « close » to the expression above times $N_{1}(n)$.

This implies $\alpha_{n}=o(n)$, as one can easily figure out replacing $N_{2}(n)$ by its exact expression given by:

$\frac{n.N_{1}(n)}{P_{ord_{C}(n)}}(1+\dfrac{\alpha_{n}}{n})$

in the following limit:

$\frac{N_{1}(n)}{N_{2}(n)}\left(\frac{n-\sqrt{2n-3}}{P_{ord_{C}}(n)-\sqrt{2n-3}}-\frac{N_{2}(n)}{N_{1}(n)}\right)\to 0$

We will show in the next article that one might even get:

$\alpha_{n}=O_{\varepsilon}(n^{1/2+\varepsilon})$.

Publicité