We return again to the important issue of why in qualitative terms all results in the Riemann Zeta Function for negative odd number integers (- 1, - 3, - 5,...) result in rational values.
I have already referred to St. John of the Cross in the context of explaining the qualitative significance of the negative even integer values of s.
Now the key thing about the negative dimensions is that it psychological qualitative terms they literally imply dynamic negation with respect to phenomenal experience.
As we have seen with negative integer dimensions (that are even), dynamic negation with respect to secondary rational elements of form is required, leading to a purely holistic intuitive awareness (which is - literally - 0 in phenomenal terms).
It is this form of negation (i.e. psychological purgation) that St. John of the Cross addresses when referring to the passive nights of sense and spirit.
However an issue that is not perhaps fully addressed by St. John is the need for ongoing active (as well as passive) purgations at the higher (dimensional) stages of development.
The problem with merely emphasis on intuitive type development is that experience can become unduly passive whereby a person loses the ability to properly engage with the world of form. So properly understood each new higher passive stage (culminating in a purer type of contemplative awareness) should be followed in turn by a more refined active stage. Now whereas the even dimensions refer to more passive intuitive type development with reality, the odd dimensions refer to more active involvement relating to engagement with phenomena of form.
Thus the negative odd integer dimensions relate to the purgation or dynamic negation of such active phenomenal understanding.
Fascinatingly, just as the more passive intuitive purgations take place through the negation of associated secondary rational elements, the more active rational purgations take place - in reverse manner - through the negation of secondary associated intuitive elements.
The ease with which - for example - one can deal with the normal familiar tasks of life, owes a great deal to the customary light or supporting intuition that is associated with such activity. However unfortunately it is likewise this same customary ease that provides an unwitting opportunity for all sorts of unrecognised attachments to develop.
So the manner in which one is thereby purged of such attachments is through the gradual diminishment of the light so that in extremes one is left totally in psychological darkness with no visible support of an unconscious nature.
Put another way, in such circumstances one is led - while carrying out phenomenal activity - to operate purely in faith.
This has the beneficial effect of forcing a person to greatly economise on conscious effort only tending to what is immediate and essential with respect to a task.
So just as pure intuition (which is qualitatively 0 in phenomenal terms) is developed through negation of associated rational support in personality, pure reason in converse manner is developed through negation of associated intuitive support.
And the negative odd integer dimensional values refer precisely to such pure reason.
Not surprisingly just as all the negative even dimensions give rise to 0, the negative odd dimensions give rise to rational values (which once again in this context have a direct qualitative significance).
As before, because of the complementarity of psychological and physical domains, these rational values also necessarily have a qualitative significance for the physical behaviour of matter.
One way perhaps of appreciating such a connection is the realisation that the phenomenal independent manifestation of particles requires in a sense blotting out the dynamic energy interactions with which they are necessarily associated.
Likewise as we have seen in psychological terms when dynamic interaction greatly increases as a result of authentic contemplative development, to keep one's feet on the ground as it were, one must be able to blot out such intuitive interaction to concentrate on specific tasks of a mundane nature.