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 .

Thus .

Hence, defining as the quantity equal to:

, one gets:

where is the -th harmonic number.

So that one should have:

hence:

.

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:

and thus

Moreover , so that the total number of -central numbers below verifies:

where is the probability for to be -central,

and

There are possibilities for the value of , namely

Since is -central if and only if , one gets:

Thus

Since one finally gets:

hence .

Obviously the next step consists in showing that:

and hopefully:

One has:

and thus:

hence:

Thus

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:

and thus: