As in the previous articles, let’s assume Goldbach’s conjecture so as to define properly the notion of -central number. The number of -central numbers less than should verify the following relation:
and thus .
I now formulate the following conjecture:
Negligible fundamental primality radius conjecture (NFPR conjecture for short):
One can deduce from this conjecture and the prime number theorem that .
Hence, defining as the quantity equal to:
, one gets:
where is the -th harmonic number.
So that one should have:
Now, from the prime number theorem, one has:
for large enough and less than
So, it should be possible to prove rigorously that the conjunction of Goldbach’s conjecture and NFPR conjecture would entail that:
This last relation, as stated in http://arxiv.org/pdf/1306.0948.pdf, follows from Hardy-Littlewood’s prime k-tuples conjecture.
Let’s now define the quantity as follows:
It seems that there exists (and possibly not much bigger than ) such that:
Then there exists a unique such that .
One has obviously , hence:
Moreover , so that the total number of -central numbers below verifies:
where is the probability for to be -central,
There are possibilities for the value of , namely
Since is -central if and only if , one gets:
Since one finally gets:
Obviously the next step consists in showing that:
As Maynard proved that the quantity
only depends on , one should obtain:
by substracting the divergent part of the error term above.
So that we finally get: