We have explained in qualitative terms why the results of the Riemann Zeta Function for negative odd integer dimensional values of s are always of a rational nature.
This qualitative explanation also enables one to appreciate the complementary nature of (negative) even and odd results for s respectively.
Once again for even values degrees of pure intuitive awareness (to which the even values relate) always requires negation of associated rational elements.
In converse manner pure rational awareness (to which the odd values relate) always requires corresponding negation of associated intuitive elements.
The next step is to explain - again in qualitative terms - why these rational results (for negative odd integer values of s) - keep alternating as between negative and positive signs.
Once again the explanation is very revealing as regards the nature of how higher level contemplative development takes place.
Phenomenal recognitions takes place in relation to both external (physical) and internal (psychological) aspects. Typically for example an extrovert will be more engaged with the external and the introvert with the - relatively - internal aspects respectively.
So when dynamic negation with respect to phenomenal experience takes place it entails both external and internal aspects (which are - relatively - positive and negative with respect to each other).
However it is in the nature of experience that this does not take place in a balanced fashion. So typically more attention will initially be given for example to the external structures and then to the internal structures and so on. So for example in St. John.s approach the initial emphasis is on purgation i.e. dynamic negation of the senses (which correspond to the external structures). Then with the more deep-rooted purgation of spirit, the emphasis switches to the internal structures.
And in development until full stable equilibrium is obtained - which in human development can only ever be approximated - a relative switch is emphasis will keep taking place at each higher differentiated stage of development as between external and internal.
So if we denote rational understanding associated with the external physical environment as positive, then in corresponding fashion rational understanding associated with internal psychological development will be - relatively - negative.
Thus dynamic negation of the rational structures takes place through the erosion of all secondary intuitive support (which provides that customary light that greatly facilitates the use of reason).
And this erosion takes place both with respect for intuitive support for the external and internal structures respectively (which are positive and negative with respect to each other).
Indeed St. John in his writings deals well with the problem of scrupulosity which at certain times can become a major problem on the contemplative journey.
Now this problem actually relates to the dynamic negation of the customary intuitive supports that normally facilitates the taking of internal moral decisions.
So as one is left - literally - more and more in the dark as regards the correct decision - in any relevant context to take it becomes increasingly difficult to decide on what is appropriate.
What happens in fact is that reason is used in an ever more refined manner in balancing pros and cons as one waits for a very faint intuitive signal providing inner confirmation that one is making the right choice. However in extremes, such supporting intuitive light is taken away altogether. So without ant supporting intuition, even refined reason can no longer operate and one is left to operate purely by faith (which in this context actually represents the purest degree of reason).
So in experiential terms both reason and intuition are necessarily involved in all phenomenal understanding. Thus the two extremes are:
1) the pure intuitive light (when all secondary rational understanding of a phenomenal nature is removed.
2) the pure intuitive darkness i.e. pure faith (when all secondary intuitive understanding associated with phenomenal understanding is removed.
And such pure faith actually likewise represents the purest form of reason i.e. that is so refined that any remaining phenomenal aspect is impossible to detect in an explicit manner.
Once again a complementary explanation of these alternating rational results (for the Riemann Zeta Function) can be given with respect to physical reality.
Though we - wrongly accustomed to looking at physical reality in a merely external objective fashion, in truth all physical processes entail the dynamic interaction of complementary internal and external aspects that are - relatively - positive and negative with respect to each other.
One way of viewing this would be in terms of intimate sub-atomic reactions where matter and anti-matter aspects of particles are involved.
So in this context the existence of - relatively - independent matter particles requires the blotting out of associated energy reactions with respect to both matter and anti-matter manifestations. And such manifestations can only occur in an alternating fashion (where for example matter excludes its anti-matter equivalent).