However we must keep reminding ourselves that there are in fact two complementary sets of these zeros both of which are vitally necessary for true appreciation of their very nature.
So I refer to these two sets as Zeta 1 and Zeta 2.
The Zeta 1 correspond to the customary set of zeros recognised (though not their true nature) in conventional mathematical terms.
Once again I refer to this infinite series as ζ1(s), where
ζ1(s) = 1–s + 2–s + 3–s + 4–s +……..,
1–s + 2–s + 3–s + 4–s +…….. = 0,
Assuming the Riemann Hypothesis is true these solutions occur in pairs and are all of the form
s = a + it and s = a – it respectively.
The Zeta 2 corresponds to an alternative set of zeros corresponding to a finite series where both base (quantitative) and dimensional (qualitative) numbers are inverted with respect to the Zeta 1.
So whereas s represents a dimensional value with respect to the Zeta 1, it represents a base value with respect to the Zeta 2; likewise, whereas the natural numbers 1, 2, 3, 4,.... represent successive base numbers with respect to the Zeta 1, they represent successive dimensional numbers with respect to the Zeta 2.
So the Zeta 2 is (initially) defined as the finite series
ζ2(s) = 1 + s1 + s2 + s3 +….. + st – 1 (with t prime), .
So the (non-trivial) zeros arise in this case as solutions for ζ2(s) = 0,
i.e. 1 + s1 + s2 + s3 +….. + st – 1 = 0.
Now these solutions in fact conform exactly to the t roots of 1 (with the omission of the non-unique root of 1).
In other words as 1 is always one of the t roots of 1, in this sense it represents a trivial root.
So the other roots - which are uniquely defined for all prime number roots - thereby represent the non-trivial roots.
In certain respects, I had already become aware of the crucial starting significance of this Zeta 2 Function from about the age of 10 and it may help to illuminate its significance by reciting it again.
When we square the number 1, this can be written as 12.
Now clearly no quantitative change takes place in the expression which is still 1.
However, a qualitative change has taken place in the nature of units involved. In other words we have moved from linear (1-dimensional) to square (2-dimensional) units.
Now I use "base" here to refer simply to a number (which is then raised to a certain value).
So in this context whenever the base number remains fixed as 1, the corresponding dimensional number takes on a purely qualitative meaning.
Therefore in the expression 12, 2 has a purely qualitative meaning.
The upshot of this is that every number can be given both a quantitative and a qualitative definition (according to what I refer to as the Type 1 and Type 2 aspects of the number systems respectively).
The Type 1 aspect is geared to the standard quantitative definition where every base number (as a variable quantity) is defined with respect to a fixed dimensional value of 1.
So 2 from this quantitative perspective is defined as 21.
The Type 2 aspect, by contrast is geared to the (unrecognised) qualitative definition, where 1 (as fixed base number) is defined with respect to a variable dimensional quality.
So 2 from this complementary qualitative perspective is defined as 12.
Now the really important thing to grasp is that Conventional Mathematics is solely defined in terms of the Type 1 aspect of the number system.
So for example, when we have an expression (where the dimensional value ≠ 1) a merely reduced quantitative result will be given.
Therefore to illustrate from the Type 1 perspective, the result of the numerical expression 22 is given in a reduced quantitative manner as 4 i.e. 41.
Now if we looked on this expression geometrically we would represent it as a square (of side 2) which is qualitatively different from 4 linear (1-dimensional) units.
However because of the grossly reductionist nature of Conventional Mathematics, this crucial qualitative distinction is completely overlooked.
In other words, in a very precise qualitative manner, the fundamental paradigm of Conventional Mathematics is 1-dimensional.
I have gone into the deeper nature of what this means in several places.
Basically what it entails is that in any context that qualitative meaning is reduced to quantitative interpretation. This is turn results from using just one isolated polar reference frame.
In other words, in dynamic terms all experience is necessarily conditioned by the interaction of fundamental poles. These relate (i) to the interaction of external (objective) and internal (subjective) aspects and (ii) quantitative (analytic)) and qualitative (holistic) aspects.
Therefore to avoid such relative interaction, from a 1-dimensional perspective, in every context, interpretation is rigidly frozen in terms of just 1 pole.
So for example numbers are treated as absolute objective entities (thus avoiding dynamic interaction with the internal subjective aspect); Likewise numbers are viewed in a merely quantitative analytic manner thereby avoiding dynamic holistic interaction (of a qualitative kind).
As I say, even as a 10 year old, I could see that there was something fundamentally wrong with mathematical interpretation.
Though of course I had not yet the intellectual capacity to properly articulate the nature of this problem, my conviction was so strong that I never subsequently accepted the highly reduced nature of Mathematics.
So I was soon after determined - even if I never was to receive any support - to reformulate basic mathematical notions such as number, in a properly coherent dynamic interactive manner (as I saw it).
It initially baffled me as to why everyone was not likewise identifying this obvious reductionism as the most significant basic problem intimately affecting all mathematical interpretation!
Unfortunately I eventually came to realise that established mathematical conventions are so strong, that aspiring mathematicians in effect quickly learn to conspire with accepted practice. Then this mode of thinking becomes so habitual that its crucial limitations become quite impossible to see clearly.
We have already come a long way!
So we see that every number - and by extension every mathematical notion - has distinctive quantitative and qualitative aspects (in terms of the Type 1 and Type 2 systems of interpretation).
Secondly, as both of these aspects necessarily interact in experience, we need to move to a dynamic interpretation of number that is relative - rather than absolute - in nature.
Just as we have properly have Type 1 and Type 2 aspects to the number system, this equally implies that we need Zeta 1 and Zeta 2 aspects to properly define the Riemann Zeta Function (and of course its associated Riemann Hypothesis).
And as the Type 2 refers directly to the qualitative aspect of the number system, the Zeta 2 refers directly to the qualitative aspect of interpretation (with respect to the Riemann Zeta Function).
Now the Zeta 2 Function - when appropriately interpreted in a qualitative manner (i.e. where a qualitative interpretation is given to the notion of number as dimension) leads to the startling realisation that potentially an unlimited number of possible mathematical interpretations (all with a limited relative validity) exist.
So once again Conventional Mathematics is defined in a fixed absolute type manner (even when dealing with dynamic interactions).
This reflects that fact that is defined solely in a 1-dimensional fashion (always using isolated independent poles of reference).
This then leads to the inevitable reduction (in any context) of qualitative to quantitative meaning.
Now I am not for a moment denying the great value of this type of Mathematics! However it is vital to realise that it represents just one special limiting case.
In all other mathematical interpretations (corresponding to dimensional numbers ≠ 1), a dynamic relative interpretation is taken of mathematical symbols, where quantitative and qualitative aspects (while preserving a certain relative independence) are seen to interact with each other (in a relatively interdependent manner).
A truly startling deficiency regarding Conventional Mathematics is that by its very means of interpretation, it is totally lacking any genuine notion of interdependence.
Just reflect on this simple observation for a moment! If numbers can be absolutely defined as independent entities, then how can we then establish their interdependence with other numbers?
Alternatively, in an equivalent manner, how can we derive the ordinal nature of number from its cardinal identity?
Well! the answer to both questions is simply that we can't (without gross reductionism).
If you can appreciate this, then you have already taken the first step towards realising the crucial role of the zeta zeros in the number system.