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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.
The Prime Number Theorem (PNT for short) says that the average gap between two consecutive primes of size is . Defining the quantity as , where is the -th typical primality radius of , one can expect to get closer to as increases for a given .
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
As, whenever is -central, one has , one gets for such :
Hence, , so that it’s very likely that the strong form of Cramer’s conjecture, namely , is true.
As mentioned in the previous article, it appears that the upper bound would follow from the following reasonable assumption:
is « rather close » to
This follows from the very definition of what a typical primality radius is. Indeed, writing
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 definition of a « typical » primality radius of leads to the following assumption:
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: