I have found the odd numbered dimensions more difficult to properly understand than the even.
As we have seen the even are more properly geared for holistic integral interpretation of reality and as my main concern over the years has been to articulate such an approach in appropriate mathematical terms, it is not surprising therefore that the even dimensions held more resonance for me.
The clue to understanding however of the higher odd dimension (focusing initially on the positive) is the realisation that human development necessarily entails both differentiation and integration which need to be maintained in healthy balance.
So even for one committed to the process of growing in pure contemplative awareness a certain level of activity at each stage must be maintained.
Thus for example when one returns to active involvement (following the development of the more passive even numbered dimension), the next odd numbered dimension will then unfold.
Thus arriving at the first of the higher dimensions i.e. 2 leads to a new integral appreciation of phenomenal reality as being based on the dynamic complementarity of opposite real poles (e.g. internal and external respectively). This new "higher" rational appreciation of the structure of reality is in turn supported by an enhanced quality of spiritual intuition (not available at the linear level).
However such appreciation is of a holistic nature. So to maintain balance one must equally strive for appropriate appreciation at the more active level of dealing with specific phenomena. So this leads to the unfolding of 3-dimensional understanding.
If one looks at the three roots of unity, the first is + 1, which in a sense stands separate from the other roots (which are paired in a more complementary manner). And this feature is shared by all higher odd numbered roots.
The corresponding qualitative significance of this is that likewise associated with the higher odd dimensions is a certain return in qualitative terms to linear understanding (corresponding with + 1). However unlike the earlier 1-dimensional level, this is circumscribed to a considerable extent by other forms of understanding.
It is easiest to illustrate here with respect to 3-dimensional interpretation and I will do so from my own particular experience.
Having gone through an extended "dark night" lasting about 5 years, which in holistic mathematical terms represented the negation of 2-dimensional understanding, I noticed a distinct lull with respect to customary purgation taking place. This gradually enabled me to turn the focus outwards again and resume many of the activities that I had given up for some considerable time. In one way it seemed like a welcome return to normality. As I came to share again the same interests as others, the preceding "dark night" appeared like a lengthy aberration that had now thankfully passed.
However I slowly realised that things were not quite the same as before and that I had become extremely sensitive to projections of all kinds. Now a projection represents an involuntary eruption of the (repressed) unconscious which then becomes embodied - and thereby confused - with conscious phenomena. So I found myself spending considerable time in the attempted negation of the many projections that had come to interfere with my activities.
Now if one looks at the two other roots (of the 3 roots of unity) one can see that there is a real part (which in both cases is -.5 and an imaginary part that is complementary (i.e. one with a plus sign and the other with a minus).
I began to understand that there is a fascinating holistic mathematical significance here (with intimate relevance for psychological development).
The real part represents the attempt to split dynamic negation of phenomena as between the external and internal aspects of phenomena. However because the unconscious is not yet sufficiently developed, this leads to projections becoming associated with both aspects.
So it is only when the unconscious has matured that such involuntary projections cease and the new integral 4-dimensional appreciation unfolds. Now in a suitably refined manner one clearly realises how the unconscious is necessarily involved in the appreciation of all phenomena so that involuntary projections (which represent a failure of such realisation) largely cease.
So in a valid sense the odd numbered dimensions can be seen as transition points between successively higher integral stages (represented by the even numbers).
In this way a new "higher" stage of integration requires a corresponding preceding stage of differentiation before it can successfully unfold.
Indeed this qualitative explanation would also help to explain why in the Riemann Zeta Function, pi values do not conveniently arise where odd numbered dimensions are concerned.
As we have seen integration is associated with pure circular understanding (based on the complementary of opposites). However for odd numbered roots we never obtain matching complementarity. And likewise with odd numbered dimensions, which represent a new form of differentiated experience, likewise we do not obtain full complementarity. Thus in a sense with each new stage of differentiation there is inevitably symmetry breaking with the circle therefore incomplete (until integration with the next even numbered dimension can be restored).
However it is the negative odd dimensions that are perhaps the most perplexing!
Once again if we go directly to the Riemann Zeta Function, we have the unusual finding that the value of the Function always results in a rational number (where the value of the dimensional power i.e. s, is odd and negative). Also this value keeps alternating as between positive and negative with successive "higher" (absolute) values of s.
So when for example s = - 1, the value of the Function = - 1/12 and then when s = - 3 the value = 1/120.
Now once again - though this is not yet recognised - such values (where s is odd < 0) possess a holistic mathematical rather than conventional linear meaning.
Indeed there is a famous account of when Ramanujan first wrote to Hardy, he included the finding 1 + 2 + 3 + ......= - 1/12. Both Hardy and his collaborator Littlewood thought initially that perhaps Ramanujan was mad and then realised that this actually represented the Riemann Zeta Function for s = - 1. So quite independently of Riemann, Ramanujan without any formal training had been able to derive a key result in complex analysis. Not surprisingly their opinion as to Ramanujan's talent quickly changed.
Now from a holistic mathematical perspective, I grappled for years to find why this value is rational. And then eventually it dawned on me, providing in the process much greater clarity regarding the nature of spiritual development.
As we have seen the negative even dimensions are required through a necessary form of passive purgation (or cleansing) so that the spiritual light can shine as pure intuition without any phenomenal attachments remaining. In a scientific context, this would entail the removing of all rigid rational interpretation from such dimensions (whereby the spiritual light to a degree becomes reduced to the phenomenal structures used in representation).
However - though St. John does not specifically refer to it - there is an equal need for a new form of active purgation following each "higher" stage of differentiation. So in a more healthy - and balanced - view of the spiritual journey, passive does not replace active purgation at the more advanced contemplative stages, but rather such advanced stages require both active and passive purgation. Now of course the degree that is appropriate at each stage will largely depend on personality factors and special circumstances.
In this regard I remember being impressed by a lengthy account given by Eveylyn Underhill in her Chapter "Dark Night of the Soul" (from "Mysticism") on the Dominican monk Henry Suso. There is little doubt that he traversed an authentic spiritual journey (entailing a lengthy "dark night") that was very arduous. However many of his key trials - even at the more advanced stages - were decidedly of an active kind (i.e. relating in holistic mathematical terms to negative odd dimensions). So this served in my mind to counteract somewhat the account of St. John in his "Dark Night" where the emphasis is very heavily on passive purgation (corresponding in turn to the negative even dimensions).
We can now provide a fascinating qualitative reason as to why all the negative odd dimensional values of the Riemann Zeta function are associated with rational values.
Again in an intellectual context we explained how in the case of the even numbered dimensions, negation with respect to any (rigid) rational attachment would be required so as to enable purely intuitive understanding (which is literally 0 in phenomenal terms) to arise.
Now an exactly counter experience is required in order to sharpen reason. Reason is cleansed by in turn removing all unrefined intuitive elements.
Normally the use of reason is facilitated by a ready supply of intuition (like oil in an engine) which is taken for granted. However it becomes much more difficult to use the rational faculties when this intuitive light is taken away. What then happens is that one is forced to greatly economise on reason by concentrating on what is truly essential with respect to every task.
Now without such radical purgation, most people never use their minds efficiently constantly wasting intuitive energy on distractions. However when one is required to do tasks in faith (with all supporting consolation of the light taken away) reason is greatly sharpened.
So intuition is refined and purified (through negating rational support); in counter manner, reason is likewise refined and purified (through negating intuitive support).
So in an intellectual context in this sense, pure (linear) reason is developed through the negation of the odd numbered dimensions (in the "dark night" of active involvement).
It has always been my position that reason and intuition are of equal importance (and essential for the proper working of either aspect). However what I did not realise until recently is that the holistic mathematical appreciation of the Riemann Zeta Function has an intimate and precise bearing on the manner through which "higher" spiritual psychological development unfolds.
We can in addition provide a holistic mathematical explanation of some of the other aspects of the negative odd dimensional values.
For example why do the zeta values successively alternate as between positive and negative?
Well the reason is this. Whereas the even numbered integral dimensions relate to the complementarity of both poles, the odd numbered relate to the temporary separation of the positive from the negative pole. Now it is in the very nature of differentiation therefore that either the external or the internal is given temporary prominence. So when negation takes place it tends to be one-sided. So typically at one stage the external aspect will be more dominant before being negated; then at the next stage the internal will now likewise undergo - relatively - more development and be negated etc. Therefore through the continual positing and negating of each pole separately as with an iteration one gradually moves to the position where both are equally refined.
Another fascinating aspect relates to the fact that the absolute value of the Function keeps falling until s = - 5 where it reaches - 1/252, before then continuing to increase, gradually at first and then more and more rapidly.
The qualitative significance here relates to the fact that with contemplative development, rational type use of faculties tends to diminish to be used very sparingly (as the emphasis is primarily on intuitive awareness at this stage). Even before the 8th dimension is reached, a certain rebalancing is required so that the holistic transcendent drive to pure intuitive union with reality does not become too extreme. However by the time that the radial stages unfold, the value has reached 1. This would imply that only then has the emphasis of form been brought back into balance with (intuitive) emptiness. Put in spiritual terms, both the transcendent and immanent aspect would now be in substantial balance. And then the progressive increase in the value would signify an ever greater ability to use the rational faculties, as the radial stages proceed.
Now I can see why I laid so much emphasis on the dimensions up to 8 in my earlier approach (as these are the ones indeed most important in terms of holistic development). I had already worked out that a lengthy transition phase would be necessary to reincorporate the rational faculties (which I had identified with the dimensions from 8 to 16). So it is remarkable that the actual value for the Zeta Function only passes one, for the negative dimensional numbers of s between 15 and 17.
Interestingly where pure intuitive development reaches its zenith, one would expect that the positive and negative polarities of rational experience would be in substantial balance. And this indeed is the case for - 1/252 is followed by 1/240. So the absolute ratio of the two numbers is nearly 1 here (and much closer than the ratios of any other successive zeta values for negative odd dimensions).
All in all therefore, though it is more difficult to precisely "nail" the reasons for the precise magnitude of the vales of the Zeta Function (for negative odd values of s) we can once again show strong associations with spiritual psychological development amply demonstrating that these dimensional values (as with the even) have a precise holistic mathematical significance. In fact the meaning of all the values of the Function for negative dimensional values is - directly - of a holistic mathematical nature (as their values cannot be defined here in standard linear terms).