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.


\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:




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



Therefore one should expect the following relation to hold:


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:



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