We are still looking at the holistic mathematical significance of the Euler Zeta Functions for positive integer values.
We have seen that in qualitative terms, each dimensional value of s is associated with a unique structural manner of rational interpretation.
Once again from this perspective the great limitation of conventional mathematical appreciation is that in formal terms it is solely confined to linear type appreciation (i.e. where s = 1).
And the key defining characteristic of this mode of interpretation is the qualitative aspect of understanding is not recognised and in effect is thereby reduced to the quantitative!
However an infinite set of more refined alternative "higher" interpretations exist where both quantitative and qualitative aspects are given distinctive recognition.
This entails that Mathematics is no longer formally defined in mere conscious rational terms but rather in more subtle terms where both conscious and unconscious aspects interact.
In direct terms the unconscious aspect of such understanding relates to holistic intuition. However indirectly this holistic aspect can be indirectly translated in rational terms as imaginary (based on a circular manner of interpretation).
So just as in quantitative terms the structure of 1 (for 3 or more roots) always entails both real and imaginary parts, likewise in corresponding qualitative terms, the holistic structure of 1 (for 3 or more dimensions) always contains both real (conscious) and imaginary (unconscious) aspects of interpretation.
Though all my essential thinking on these matters had been long formulated before I became interested in the Riemann Hypothesis, this Hypothesis does indeed provide an extremely important application of such notions. And in reverse manner understanding the Hypothesis in this way has helped to considerably clarify my own understanding with respect to the higher structures of psychological understanding.
Using spiritual type language, the higher integral dimensions (positive) relate to the transcendent type structures of understanding that progressively unfold through the process of contemplative type development. So through increasing dynamic interaction of both rational and intuitive aspects of understanding, interpretation becomes ever more refined in rational terms.
Once again the even numbered structures relate to integral type understanding at such higher dimensions (where one attempts to define the appropriate manner through which overall holistic interdependent nature of reality can be rationally encoded).
Then the odd numbered dimensions relate to corresponding differentiated understanding at these higher dimensions (where one now tries to equally define the appropriate manner through the relatively independent nature of specific phenomena of form can be likewise encoded).
Now ultimately both the - relatively - continuous nature of the integral aspect and the discrete nature of the differentiated aspect become so refined as to be indistinguishable from each other which would relate to pure contemplative experience of reality. However this represents a limiting state that can only be approximated imperfectly in human terms.
Incidentally this brings us back to the holistic mathematical nature of e (which plays such crucial role with respect to the behaviour of prime numbers).
Just as in conventional mathematical terms both the integral and differential of e^x are indistinguishable from each other, in holistic mathematical terms it is quite similar. So here we are no longer able to distinguish the discrete (differentiated) aspect of phenomenal form from the corresponding continuous (integral) aspect of holistic emptiness.
And again we can identify such a state as one of pure contemplative awareness (in the proper balanced sense of fully harmonising both intuitive and rational aspects of understanding). So this ideal state - insofar as it can be humanly approximated - combines both profound intuitive depth with incredible rational clarity).
Indeed it is in this state that the mystery of the prime numbers is finally resolved (which is the same state any lingering problem with respect to involuntary primitive instincts) is also resolved. So once again the quantitative nature of prime number behaviour and the qualitative nature of primitive instinctive behaviour entail the relationship of both specific conscious and holistic unconscious aspects of understanding (corresponding to both the linear and circular aspects of understanding respectively).