Basically this (finite) function provides the (t – 1) roots of 1 (with the default root of 1 excluded).
Then the actual solutions for these roots are estimated through the Euler Identity.
So eiπ = – 1.
Therefore e2iπ = 1 (in what I refer to as the fundamental Euler Identity(.
Then according to De Moivre's Theorem eix = cos x + i sin x.
Therefore when x = 2π (radians),
e2iπ = cos 2π + i sin 2π = cos 3600 + i sin 3600
There is a direct connection as between the fundamental Euler Identity and the Type 2 aspect of the number system.
So e2iπ = 11 and in more general terms,
e2kiπ = 1k = cos k(360)0 + i sin k(360)0
Therefore for the two roots of 1
s2 = 11 and s2 = 12 respectively,
so s = 11/2 and s = 12/2 = 1.
So eliminating the default root 1,
s = 11/2 represents the solution for the general Zeta 2 function,
1 + s1 + s2 + s3 + ... + st – 1 = 0.
Thus, as in this case, t = 2,
1 + s1 = 0.
So s, i.e. s1 = 11/2 = cos (360/2)0 + i sin (360/2)0
= cos 1800 + i sin 1800 = – 1 + i (0) = – 1.
So once again, this represents the first - and in many ways the most important - of the Zeta 2 zeros.
And once again the importance of these zeros is that it enables us to give an indirect quantitative expression to ordinal notions of a strictly qualitative nature.
So 1st (in the context of 2 units) is indirectly expressed as – 1 (which thereby literally negates or excludes the 2nd member).
Now it would not need the Euler Identity (through De Moivre's Theorem) to calculate the non-trivial root in this case.
However it quickly becomes extremely difficult to do so (in the absence of De Moivre's Theorem).
So for example if we were to satisfy the Zeta 2 function for - say - t = 37, we would follow the same general procedure setting k (with respect to the general expression of De Moivre's Theorem) = 1/37, 2/37, 3/37,....., 36/37 whereby we could thereby calculate the 36 zeros (in this case) for the Zeta 2 function.
And these values in turn would then give indirect quantitative expression to the notions of 1st, 2nd, 3rd,......., 36th (in the context of 37 unit members).
Thus the Euler Identity (especially in the form of the fundamental Euler Identity) is extremely important because of its direct connection with the Type 2 aspect of the number system.
In this way, it is directly linked to the qualitative - as opposed to the quantitative - nature of number.
An though respecting the equal importance of the Type 2 aspect (with the Type 1) we can properly clarify many issues of interpretation, that are not properly dealt with in conventional mathematical terms.
For example in conventional mathematical terms,
e2iπ = 1;
Also, e – 2iπ = 1;
Finally, e0 = 1.
This might then seem to suggest that 2iπ = – 2iπ = 0, which is clearly nonsensical from a quantitative point of view!
However, the clue to clarification of this seeming mathematical dilemma is to define the RHS of the equations more precisely in terms of the Type 2 aspect of the number system.
So e2iπ = 11,
e – 2iπ = 1 – 1, and
e0 = e2iπ * e – 2iπ = 11 * 1 – 1 = 10.
So in fact each of the three equations is associated with 3 distinct results when expressed in terms of the Type 2 aspect of the number system!
Finally, it is also customary in conventional mathematical terms to maintain that
e2iπ = e4iπ = e6iπ =.... e2kiπ = 1, where k = 1,2,3,.....
However, properly interpreted, these all relate to distinct numbers with respect to the Type 2 aspect of the number system.
So e2iπ = 11, e4iπ = 12, e6iπ = 13 and e2kiπ = 1k.
Likewise, as we have seen, the various roots of 1 for example, refer to distinct aspects of the Type 2 aspect of the number system.
So, again with respect to the two roots of 1,
+ 1 corresponds to 12/2 = 11, and – 1 corresponds to 11/2 respectively.