We now will bring the various elements together to show how the relationship as between the primes and the natural numbers (and natural numbers and primes) is one of true interdependence (thereby revealing itself in dynamic terms through a precise form of two-way complementarity).
Once again from a Type 1 (linear) perspective, the primes are viewed as the basic building blocks for the natural number system in a merely quantitative manner.
Thus from this perspective, each natural number (in cardinal terms) represents a unique combination of prime factors.
However we have indicated many times the key problem with this approach whereby uniqueness in - solely - quantitative terms, strictly rules out any distinctions of a qualitative (ordinal) nature.
Again, a cardinal prime is defined by its collective whole nature (in quantitative terms).
So the prime number 3 is thereby defined uniquely in terms of unit parts that are completely homogeneous in nature (i.e. lacking qualitative distinction). So 3 = 1 + 1 + 1.
Therefore the cardinal approach to this key relationship - of the primes to the natural numbers - leaves us with no means of making ordinal distinctions of a qualitative nature. And without making such distinctions we cannot relate numbers and thereby achieve order with respect to the number system!
Thus in Type 2 terms, the natural numbers are viewed - in reverse fashion - as the building blocks of each prime number.
Therefore from this counter perspective, each prime number (in ordinal terms) represents a unique combination of natural number members.
So the prime number 3 is now viewed uniquely in terms of its natural number members i.e. 1st, 2nd and 3rd in ordinal terms.
Indirectly, each of these members can be expressed in quantitative terms through the corresponding 3 roots of 1.
Then their true qualitative significance (as interdependent) is revealed through combining (in dynamic terms) these - relatively - separate members = 0. So the significance of 0 in this context reveals the true qualitative meaning of the number 3 (which - literally - is nothing in quantitative terms).
From the Type 2 (circular) perspective, all prime numbers are defined (except 1) by a unique set of natural number members in ordinal terms (indirectly expressed in a circular quantitative manner by the roots of the prime number).
And the corresponding sum of these unique set of roots, representing the true holistic meaning of the prime number (as the dynamic interdependent notion of the number in question) = 0.
We now come full circle!
We started in Type 1 terms by defining the natural numbers as representing unique combinations of prime number factors (in quantitative terms).
Fro example the natural number 30 = 2 * 3 * 5 (from this perspective).
Now strictly we should express this Type 1 representation as 30 ^ 1 = (2 * 3 * 5) ^ 1
However we can equally express 30 as representing a unique combination of prime number factors (in a qualitative manner)
So in Type 2 terms, 2 * 3 * 5 = 30 is represented as 1^(2 * 3 * 5).
This means in effect that 30 is equally defined in a qualitative manner with respect to its 30 corresponding roots!
And as we know when we obtain the n roots of 1 (where n is any natural number ≠ 1) the sum of roots = 0.
Therefore when we interpret this in Type 2 (circular) terms, the prime numbers equally represent the qualitative building blocks of the number system (where each prime number is uniquely defined by its natural number members).
So once again, a matching qualitative (i.e. ordinal) interpretation exists both (internally) within each prime and and (externally) for the number system as a whole for the recognised quantitative interpretation.
Therefore let us now summarise the full picture.
1. In linear Type 1 (quantitative) terms, every natural number is defined (externally) in terms of a unique combination of prime number factors.
2. Again in linear Type 1 (quantitative) terms, each prime number is defined (internally) in terms of a unique combination of homogeneous units i.e. 1 + 1 + 1 +... Unique in this quantitative sense simply means that each unit = 1.
3. In circular Type 2 (qualitative) terms, each prime number is defined (internally) in terms of a unique combination of ordinal number members i.e. 1st, 2nd, 3rd,... (indirectly represented in quantitative terms as the corresponding roots of 1). Since 1 is common to all roots, strictly uniqueness in this context implies all roots ≠ 1st. So even here, we have in the definition of qualitative uniqueness, complementarity with the quantitative (internal) definition!
4. Again in circular Type 2 (qualitative) terms, every natural number is defined (externally) in terms of a unique combination of prime number factors (based on the corresponding ordinal number of roots).
So - when properly understood - the ultimate relationship of the primes to the natural numbers (and the natural numbers to the primes) is one of perfect complementarity.
In other words the primes and natural numbers (and natural numbers and primes) are fully interdependent with each other externally and internally (in both quantitative and qualitative terms) totally mirroring each other in an ultimate identity that is ineffable in nature. In other words in this ultimate mutual embrace, their objective (external) nature cannot be divorced from their corresponding (internal) interpretation! likewise quantitative cannot be divorced from corresponding qualitative identity!
Thus the mystery of the primes in relation to the natural numbers (and the natural numbers in relation to the primes) essentially relates to the manner in which both the quantitative (cardinal) and qualitative (ordinal) aspects of the number system interact.
Put another way the very manner in which the quantitative (analytic) and qualitative (holistic) aspects of the number system are mediated is through this key relationship of the primes to the natural numbers (and natural numbers to the primes).
So in dynamic terms, the qualitative represents the - initially unrecognised - shadow of the corresponding quantitative aspect; the quantitative likewise represents the - initially unrecognised - shadow of the corresponding qualitative aspect.
The zeta non-trivial zeros (both Zeta 1 and Zeta 2) represent these shadow number systems, which mediate the ultimate perfect relationship of quantitative and qualitative aspects.
In the case of the Zeta 2 zeros, each natural number is given - through the unique interdependence of its prime individual members - a corresponding qualitative dynamic meaning. So the perfect relationship here as between quantitative and qualitative aspects is established internally ultimately with respect to each natural number.
In the case of the (recognised) Zeta 1, the zeros represent the direct shadow correspondent of the prime numbers which in their totality - which can only be approximated in finite terms - perfectly mediate the dynamic interaction of quantitative and qualitative aspects for the natural number system as a whole.
Clearly once again, it is futile trying to appreciate the ultimate nature of the number system in merely quantitative external terms as in Conventional Mathematics. In dynamic terms the number system has both external and internal interpretations (with both quantitative and qualitative aspects) as I have sought to demonstrate in this blog entry.
The significance of the Riemann Zeta Function cannot be properly appreciated in this limited conventional manner. Likewise the Riemann Hypothesis, which relates to this central relationship as between quantitative and qualitative aspects (externally and internally), cannot of course be proved within its limited axiomatic system (which gives no formal recognition whatsoever to these key polar distinctions).