So the Type 1 formulation (n/log n) measures the frequency of primes with respect to the natural numbers (considered collectively), whereas the Type 2 formulation (n1/log n1, where n1 = log n) measures the frequency of the prime with respect to natural factors (for each individual number).
And in dynamic interactive terms, both of these formulations are truly complementary.
Therefore from one valid perspective, the (Type 2) distribution of the prime with respect to the natural factors (of each number) depends on the corresponding (Type 1) distribution of the primes with respect to the natural numbers (as a collective group).
However, equally from the other valid perspective, the (Type 1) distribution of the primes with respect to the natural numbers, depends on the corresponding (Type 2) distribution of the prime and natural number factors.
In other words, both distributions mutually depend on each other, which implies that - ultimately - they are synchronistically determined in an ineffable manner.
Here, both the primes and natural numbers mutually reflect each other as identical; equally the quantitative and qualitative aspects of number mutually reflect each other in a similar manner.
However, once we enter the world of dualistic phenomenal distinction, both the primes and natural numbers and quantitative and qualitative aspects of number - and indeed as we have seen the addition and multiplication operations - become separated within distinct polar reference frameworks, that however - from a dynamic interactive perspective - can be seen to be fully paradoxical in terms of each other!
When we look more closely at the issue, we can perhaps appreciate how the Zeta 2 function operates very much in the background of the standard multiplicative approach to number.
Once again we have both the additive and multiplicative approaches to the number system.
So the additive approach operates through attempting to define each number through the successive addition of unit numbers.
Therefore once again from this (Type 1) perspective,
2 = 1 + 1.
However this represents solely the quantitative approach to number (where the individual units are completely homogeneous and absolutely independent of each other, thereby lacking any qualitative distinction).
However for numbers to be related with each other (as interdependent) a corresponding qualitative approach is required.
Using Jungian type terminology (which in fact is very appropriate in this context) a shadow interpretation of number exists that is of a qualitative interdependent nature.
And remarkably - when appropriately understood - this directly relates to the corresponding operation of multiplication
Thus from the Type 2 perspective
2 = 1 * 1 (where the emphasis is now on the dimension or power of the unit involved).
So expressed more fully in Type 1 (quantitative) terms,
21 = 11 + 11 .
Then expressed more fully in Type 2 (qualitative) terms,
12 = 11 * 11
As we have seen In Type 1 terms, 2 has the actual quantitative meaning of two "independent" units.
However in Type 2 terms, 2 has the potential qualitative meaning of "twoness" where each unit can be interchanged with each other.
So implicitly, our very ability to grasp the qualitative notion of "2" (i.e. as "twoness") requires that we recognise that both units can be interchanged as 1st and 2nd respectively depending on context.
And whereas the quantitative notion of 2 relates directly to (analytic) reason of a conscious, the corresponding qualitative notion, by contrast, relates directly to (holistic) intuition of an unconscious nature..
And as I have illustrated on innumerable occasions this is what happens when we recognise that the turns at a crossroads are necessarily left and right (and right and left) with respect to each other.
So the choice as to which turn should be labelled 1st or 2nd is thereby merely arbitrary, depending on relative context. In other words, potentially we recognise that the two turns can interchange as between 1st and 2nd (and 2nd and 1st), though in any actual context, they will be a given a definite ordinal position (depending on context).
However when we look at the standard multiplication approach to number, we equally come to recognise that a shadow qualitative approach (that is unrecognised in conventional mathematical terms) exists.
So in Type 1 terms, the derivation of the (composite) natural numbers is explained as the unique product of (distinct) primes.
Therefore, to give a simple case,
6 = 2 * 3.
So once more, we try to explain the derivation of this (composite) natural number in a merely (reduced) quantitative manner.
61 = (2 * 3)1 .
However we can now equally express this (in Type 2) terms as
16 = 1 (2 * 3) .
Thus, whereas the Type 1 formulation expresses the derivation of the quantitative notion of "6", the Type 2 formulation expresses the corresponding derivation of the qualitative notion of "6" (i.e. as "sixness").
Looked at another way, whereas the (Type 1) formulation expresses "6" as the unique combination of the two primes (2 and 3) in a quantitative manner, the (Type 2) formulation expresses "6" (as "sixness") as the unique combination of the indirect quantitative expressions of the ordinal notions of 1st and 2nd (in the context of 2) and 1st, 2nd and 3rd (in the context of 3).
In other words, to derive the indirect quantitative expression of the ordinal notions of 1st, 2nd, 3rd, 4th, 5th and 6th (in the context of 6 units) we multiply the 3 roots of 1, by the corresponding 2 roots of 1.
The 3 roots of 1 are 1, – .5 + .866i and – .5 – .866i and the two roots of 1 are 1 and – 1.Therefore multiplying both sets we now obtain the 6 roots of 1 which are
1, – .5 + .866i, – .5 – .866i, – 1, .5 – .866i and .5 +.866i .
So these 6 values express in an indirect quantitative manner the ordinal notions of 1st, 2nd, 3rd, 4th, 5th and 6th respectively (in the context of 6 units).
When we then combine both the Type 1 and Type 2 approaches for the additive and multiplicative approaches, we can see how both addition and multiplication are intimately related to each other in both quantitative and qualitative terms.
It also helps to clarify, as we have seen, how the Type 1 and Type 2 distributions, of the primes among the natural numbers and the prime factors among the natural factors respectively, are mutually dependent on each other.
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