By LFT, , for , ie. . When , then . From there, we have and , which leads to for a fixed , where is -th square of a chosen integer, thus . From there, we have . Haven’t gone further with this yet, as more in-depth analysis of the very sumations is required.
Another interesting thing, for of the LFT, is that from and for , where it is trivial to show that , with substitutions and , we have determinant where can’t must never be integer for contradiction to hold, assuming no leaks in analysis. As for the substitutions and for some , we must make sure so that generality is not affected. Wanted to use this transformation as some sort (long-vision) of equivalence between the elementary NT and ring homomorphism (and its numerical criterion) from Wiles .
When it comes to -function, it’s trivial to show that for either nor are not the borderlines for , so the questions that arise are: is limited by either the amount of primes equal to the argument, or – for instance – the smallest prime appearing in the sum equal to the very argument. I will update you on this shortly as soon as I have more information.
Yet another update on learning.