We have looked at the first of the trivial zeros in the Riemann Zeta Function for s = – 2 and explained its qualitative meaning (which is directly implied by the Function).
It now remains to explain why trivial zeros exist at all negative even values for s, i.e. – 2, – 4, – 6,…….
As we have seen the qualitative dimensional structure of a number is related to the corresponding root structure (when given a circular rather than linear interpretation).
So if we take the number 4 (as dimension) to illustrate, the corresponding root structure (i.e. the 4 roots) are + 1, – 1, + i, – i respectively. Now in quantitative terms these are interpreted in linear either/or fashion.
Therefore the corresponding (qualitative) dimensional structure entails the simultaneous appreciation of these as complementary opposites.
Now however as well as the two real opposites + 1 and – 1 that we have already encountered, we have in addition two imaginary opposites + i and – i.
Now the significance of these imaginary opposites in qualitative terms is that they relate to the manner in which the holistic unconscious projects its meaning indirectly in a conscious manner. Thus – as I have outlined in detail in my writings on the stages of development - such imaginary understanding becomes the very means through which circular meaning can be indirectly incorporated within the standard linear framework of Conventional Science.
However, once again though the use of the imaginary is now well established in both Mathematics and Science with respect to its (reduced) quantitative interpretation, little or no appreciation yet exists as to the enormous qualitative significance of the imaginary. For properly appreciated the imaginary points to the hidden holistic dimension of Mathematics (that receives no formal recognition in conventional interpretation).
Long before developing an interest in the Riemann Hypothesis, I had gone into considerable detail on the precise qualitative nature of 2, 4 and 8 dimensional understanding. Once again the positive sign of the dimension points to refined rational appreciation (of a circular paradoxical nature); in turn the negative sign of the dimension points to the erosion of any remaining linear rational elements of understanding in obtaining a purely intuitive appreciation (that is nothing in phenomenal terms).
Now in all cases, where even dimensional numbers are involved, a complementary structure as between the roots can be obtained.
Therefore we can look at the larger even dimensional numbers as representing ever more refined rational appreciation of the complementarity of opposites.
The corresponding negative sign of these dimensions in turn leads to ever purer intuitive recognition (of an empty kind).
So once again all of the trivial zeros relate to increasingly refined intuitive recognition states (which are nothing in phenomenal terms).
Therefore - quite literally - when one intuits the meaning of the Riemann Zeta Function for negative even dimensional values, it is empty of phenomenal rational interpretation.
So once again the 0 that is generated by the Function in such cases relates to a directly qualitative meaning (of a holistic kind).
Quite obviously this cannot be appreciated in conventional mathematical terms as it offers no formal recognition of such qualitative interpretation!
However once it is grasped that the values of the Riemann Zeta Function for s < 0 directly correspond to qualitative - rather than quantitative - meaning, then perhaps we can begin to appreciate the significant limitations of the standard approach.