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}).