I mentioned in a previous entry how a very simple formula could be used to calculate to a very high degree of accuracy the deviation from 2/π (= i/ ln i) of both the (reduced) average cos and sin values of the n roots of 1.
I then illustrated this with respect to the prime number n = 53!
So the formula again for the cos part deviation = π /12(n ^ 2) and the sin part
= π /6(n ^ 2).
Having written this piece I then searched for an underlying explanation of this phenomenon and now can provide a clue to what is involved.
As we know for non-trivial zeros with respect to the Zeta 1 Function that the Riemann Hypothesis maintains that the real part = 1/2.
Interestingly when we obtain the 6th root of 1, the real part of the non-trivial zero for the Zeta 2 (i.e. cos 60) likewise = 1/2.
Then when we obtain the 12th root of 1, the imaginary part of the non-trivial zero (i.e. sin 30) = 1/2!
So it does seem that this very fact (which is unique in the case of both the 6th and 12th roots of 1 respectively) indeed represents the underlying explanation with a mirror significance to that of the Zeta 1 Function.
Some time ago I explained how the Zeta 2 distribution can be used to explain values of the Zeta 1 distribution, where real non-intuitive values for the Function arise with the negative odd integers where s = – 1, – 3, – 5, – 7, – 9 etc.
For example when. ζ (– 1) = – 1/12 the denominator is divisible by 12 (and of course 6).
When ζ (– 3) = 1/120 the denominator is again divisible by 12 (and 6).
When ζ (– 5) = – 1/252 the denominator is again divisible by 12 (and 6).
When ζ (– 7) = 1/240 the denominator is divisible by 12 (and 6).
And when ζ (– 9) = – 1/132 with once again the denominator divisible by 12 (and 6).
Indeed this is a feature that seems to universally hold for all denominator values (where s is a negative odd integer).
This would not only strongly support my contention that the Zeta 2 Function is embedded as it were in the Zeta 1 (for values of s < 0). It would also strongly suggest that the underlying explanation lies in the unique nature for the 6th and 12th root of 1 (where the real and the imaginary parts = 1/2 respectively).
For some time now I have also been aware of the fact that apart from the first two (3 and 5) the sum of twin primes always appear to be divisible by 12 (and thereby of course 6).
For example 5 + 7 = 12 (divisible by 12 and 6)
11 + 13 = 24 (divisible by 12 and 6)
17 + 19 = 36 (divisible by 12 and 6)
29 + 31 = 60 (divisible by 12 and 6)
41 + 43 = 84 (divisible by 12 and 6).
As I say this pattern continues and appears to be universal.
This would again suggest that perhaps the same underlying reason applies (ultimately tied to the fact that the real and imaginary parts of the 6th and 12th roots of 1 = 1/2 respectively).
This would also strongly indicate (as I have long suspected) that the Twin Prime Hypothesis is intimately tied to the Riemann Hypothesis.
Therefore as the Riemann Hypothesis can have no solution (in conventional mathematical terms) this would imply that likewise the Twin Prime Hypothesis can have no solution!