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The assumption of Langlands functoriality conjecture ensures that the considered class of L-functions is closed under both usual and tensor products. This last property holds unconditionally for the class of L-functions generated by the non negative powers of the Riemann Zeta function, which entails by the same arguments the truth of RH.

]]>We thus obtain some kind of duality, where every statement of the form where is a ternary predicate, is equivalent to .

As a consequence, we get that the reformulation of Goldbach’s conjecture as « every positive integer admits a natural primality radius » is equivalent to this weak form of de Polignac’s conjecture: « every positive integer admits a relative primality radius « , which is obviously equivalent to saying that every even integer is the difference of two (non necessarily consecutive) primes.

The strong de Polignac conjecture is equivalent to saying that « every positive even integer is the difference of two consecutive primes in infinitely many ways » or that « every positive integer is the fundamental (natural) primality radius of infinitely many -central numbers « .

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Let’s define the « gap-variance » of , denoted by , as follows:

where is the total number of typical primality radii of .

From the observation above, one can write

Hence .

As, whenever is -central, one has , one gets for such :

So that

Hence, , so that it’s very likely that the strong form of Cramer’s conjecture, namely , is true.

]]>Let’s now formulate the following Prim conjecture:

There exists a primorial greater than such that both and are primality radii of .

If true, this would entail that there are at least twin primes greater than and not . Hence the infiniteness of twin primes.

I postulate that can be expressed as a function of by solving the following equation:

.

This is just the very beginning of a sketch of proof which obviously needs further investigations.

]]>This and the previous articles of this blog entail that every even integer greater than is the sum of two primes.

]]>Hence (since one has at least different Goldbach gaps, namely of size , or ).

The same reasoning can be applied for even and coprime with by replacing with and with .

]]>is « rather close » to

This follows from the very definition of what a typical primality radius is. Indeed, writing

one gets:

Hence

So

Thus

Finally

Assuming (which remains to be proven), one gets . Maybe some kind of central limit theorem could shed a new light on the true nature of .

]]>the following ratio:

is asymptotically equal to the ratio:

Hence, for large enough, the quantity should be « close » to the expression above times .

This implies , as one can easily figure out replacing by its exact expression given by:

in the following limit:

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

.

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