We now move closer towards interpretation of the true nature of the non-trivial zeros.
The Riemann Zeta Function (with which these zeros are directly associated) extends the well known Euler Function,
1/(1^s) + 1/(2^s) + 1/(3^s) + (1/4^s) +….. to the complex plane (where s - as dimensional power - now contains real and imaginary parts) .
More precisely it extends it to the complex plane (where symbols are interpreted in a merely quantitative manner)!
However just as real and imaginary notions can be defined in quantitative terms, equally they can be defined in a qualitative manner.
So the qualitative implication of defining the Zeta Function in complex terms is that interpretation is now of a dynamically interactive nature. Therefore it takes place in both an analytic and holistic manner, containing cardinal and ordinal aspects (that are not reduced in terms of each other).
Also when appropriately understood, the Riemann Functional Equation serves as the means of connecting both the analytic and holistic type interpretation of symbols.
It is clear that when s > 1, the value of the Function corresponds to the standard linear interpretation with a quantitative value (in accordance with accepted intuition).
So we can readily see for example that when s = 2, if we keep adding terms that the value of the Function will approximate ever more closely to (π^2)/6.
However for values of s < 0, from a standard linear perspective, the Function necessarily diverges to infinity.
For example when s = – 1,
ζ(– 1) = 1 + 2 + 3 + 4 +…. (which from a linear perspective → ∞).
Yet the value given by the Riemann Function = – 1/12!
The root reason for this non-intuitive result (from the standard linear perspective) is that we have now switched to a holistic (circular) mode of interpretation, directly in keeping with a (qualitative) ordinal rather than a (quantitative) cardinal type result.
Indeed there is a clear clue to its holistic nature given by the fact that the value – 1/12 is only meaningful in the context of all the values of the Function (taken as whole in an infinite manner).
By contrast with the analytic results for s > 1, we can break up the terms of the Function in finite terms thereby approximating ever closer to the true result!
So here is a vital point, which springs directly from dynamic interpretation of ζ(s) in both (quantitative) analytic and (qualitative) holistic terms!
The Riemann Functional Equation can now be readily seen as a means of relating all results (interpreted in an analytic quantitative manner) for s > 1 with the corresponding results (interpreted in a holistic qualitative fashion) for s < 0.
Quite simply, because the standard approach to interpretation of the Riemann Zeta Function is in terms of the mere (quantitative) analytic aspect, it thereby generates a whole series of results for s < 0 which are rendered non-intuitive from a linear perspective!
However we are still left with the need to interpret results within the critical strip where 0 < s < 1.
Now it is clear for example that if we set s = .9 that from a standard linear perspective, the result will diverge. Then through the Functional Equation this result will be matched in a complementary manner with the corresponding value of the Function for s = .1 (which also diverges).
However in the context of the Riemann Zeta Function, both of these values are of a finite nature.
We interpret this situation by recognising that within the critical strip, all values of s contain a relationship of both analytic and holistic aspects. So values therefore to the right of .5 bear a complementary relationship with values to the left where the Function for s on the RHS is matched with the corresponding Function for 1 – s on the LHS of .5. Therefore the analytic and holistic relationship expressing the value of the Function for s = .6 is matched with the complementary holistic and analytic relationship that expresses its value for s = .4.
This dynamic complementary type interpretation with respect to the Functional Equation then raises the question as to a possible value for s where both holistic and analytic aspects coincide in identical fashion!
And this situation of course occurs where s = .5, where by definition ζ(s) is identical with ζ(1 – s).
So both analytic and holistic interpretations coincide at this value.
However when we set the Zeta Function = 0, we do not obtain a solution for s = .5
Such a solution requires that s be defined in complex terms where s = a + it.
However the requirement of ensuring that both analytic and holistic interpretation coincides for
1/(1^s) + 1/(2^s) + 1/(3^s) + (1/4^s) +….. = 0,
requires that ζ(s) = ζ(1 – s)
This clearly implies for the real part of s, a = 1 – a.
Therefore a = .5
The zeta zeros, or more correctly the non-trivial zeros therefore relate to solutions of
ζ(s) = 0 where s is a complex number.
All of these therefore have the same real part = .5, and an imaginary part which can vary.
As is now well known, potentially an infinite number of solutions to the equation exist.
The Riemann Hypothesis relates to the requirement that all of these lie on the line with real part = .5 and in the 150 year or so since Riemann’s initial discovery, repeated attempts have been made (without success) to prove the Hypothesis.
However by now widening the context of interpretation of ζ(s) in both a complex quantitative and qualitative fashion, we can now see that the Riemann Hypothesis stands as the simple condition required to ensure the coincidence of both cardinal (quantitative) and ordinal (qualitative) interpretation with respect to the two-way relationship as between the primes and natural numbers.
We saw that in the Type 1 approach to the number system that the relationship of natural numbers to primes appears one-way and unambiguous with all natural numbers appearing as the unique product of cardinal primes.
Then in the Type 2 ordinal approach, again the relationship seems one way and unambiguous with all primes defined by a unique set of natural numbers in ordinal terms.
However in terms of each other the two results are paradoxical.
So solving the relationship of the primes to the natural numbers (and the natural numbers to the primes) requires establishing the requirement for the mutual identity of both the quantitative (cardinal) and qualitative (ordinal) interpretation of number (both of which relate to differing systems of interpretation).
And as we have shown the Riemann Hypothesis serves as the key condition for establishing this identity.
As it relates to the critical relationship as between the quantitative and qualitative nature of number, it cannot of course be either proved or disproved within the standard mathematical approach (which in formal terms merely recognises the quantitative aspect).
In this sense the very question of proof with relation to the Riemann Hypothesis is seen to be a big non-issue as the very truth to which the Hypothesis points, is prior to the axioms employed in Conventional Mathematics (and therefore already inherent in their very use).
So the real question now shifts to the significance of the zeta zeros which is truly greater than anyone could possibly imagine!