We have now looked at the Euler Zeta Function for positive even integers of s (2, 4, 6,...) to find that the resulting value can be always be expressed in terms of Pi.
I have also been at pains to indicate the qualitative significance of this fascinating numerical behaviour.
One might initially think therefore that a similar expresion would exist for corresponding odd integers of s (3, 5, 7,...) but, as is well known, this is not the case. No closed value expressions have yet been found (though several ingenious closely approximating formulae have been derived).
Once again this is where holistic mathematical understanding can prove very illuminating.
In all psychological behaviour we have the two related aspects of differentiation and integration respectively entailing both conscious and unconscious appreciation. Differentiation is (analytically) associated with the linear logical system thus enabling the separation of opposite polarities in experience (as independent).
By contrast integration is (holistically) associated with the circular logical system thus enabling the complementarity - and ultimate identity - of these same polarities (as interdependent).
Now when we look at the higher dimensions of understanding, whereby rational understanding becomes refined in an increasingly intuitive manner, both processes of differentiation and integration are at work.
We have already identified 1-dimensional appreciation as the representative of the linear and 2-dimensional as representative of the pure circular aspect (that exhibits full complementarity) respectively.
So 1-dimensional appreciation is properly geared for the differentiated interpretation of reality (in analytic terms); then 2-dimensional appreciation is properly geared for integrated interpretation (in holistic terms) .
In quantitative terms we see this reflected in the corresponding root structures (of 1).
The 1st root of unity = + 1 is fully independent of its negative and thereby exhibits no complementarity; by contrast the two roots of unity = + 1 and -1 exhibit perfect complementarity (as interdependent).
Now, the higher even numbered dimensions (that are positive) can best be seen as the movement towards ever more refined integral sttructures . So, as we have seen, when we obtain the roots of unity for any positive even integer, we are always able to match each of these roots with a corresponding negative. For example the four roots are + 1, - 1, + i and - i. So here each root (as positive) can be matched with a corresponding root (as negative).
In like manner, the holistic mathematical interpretation of such even dimensions relate to qualitative structures of higher integration (representing ever more refined rational appreciation of interdependence).
However the higher positive odd integral dimensions do not represent such perfect symmetry. In fact with all such root structures we will always have one root that is + 1 separated from the rest (with all other complex roots representing symmetry only with respect to their imaginary parts). Though the sum of the corresponding real parts will indeed sum up to - 1, this represants a form of broken symmetry (with respect to the real part). So - quite literally - in psychological terms, there is an inevitable delay with respect to the unconscious process of negation of phenomenal forms. In this way though experience is indeed of an increasingly refined nature, the very process of differentiating phenomena creates a degree of linear rigidity (by which they are given a degree of independence).
So the holistic mathematical interpretation of such odd dimensions (in qualitative terms) is perfectly reflected in the quantitative structure of its corresponding roots.
The clear implication therefore is that pure circular complementarity (requiring the perfect complementarity of opposites) cannot exist with the positive odd integral dimensions).
Because of a degree of breaking (of integral symmetry) in qualiative terms, such dimensions cannot represent the pure relationship of circular to linear understanding. In like manner the quantitative value of such zeta expressions cannot be conveyed through closed pi expressions.
Now a great deal of debate has continued as to whether such zeta values (for positive odd integer values of s > 1) are rational or irrational.
It has indeed been proved that - at least when s = 3 - the value is irrational.
Now there are good holistic mathematical reasons for believing that not alone is the value of such expressions always irrational (but necessarily of a transcendental nature).
If the value was in fact rational (for s > 3) this would imply a rational value in the prime product formula over the infinite range of terms. This would imply that each term bears a relationship to previous terms that can be expressed in a rational manner. However the very nature of prime numbers is that their individual quantitative nature is uniquely distinct from their overall holistic behaviour with respect to each other. Therefore no such rational formula could exist.
Therefore the convergent sum of terms inevitably entails a relationship involving both linear and circular notions (which directly implies that it is transcendental). And as we have seen what distinguishes the even from the odd dimesnions here, is that with the even we have a pure relationship of circular to linear (whereas with the odd this is not the case).
Once again the key assumption here (regarding the inherent nature of prime numbers) literally transcends conventional mathematical interpretation. Indeed it is the same assumption that ultimately establishes the axiomatic nature of the Riemann Hypothesis!