Zeta 3 Zeros - Key Significance (3)

In discussing the Zeta 2 zeros, I was at pains to clarify the distinction as between the analytic and holistic interpretation respectively of the ordinal notion of number.

In the context of a given number t - which initially is defined as prime - the corresponding t roots of 1 provide an indirect quantitative means of expressing the natural number ordinal members of that prime (considered as a group of related members).

So once again, for example, in the context of 3 (as prime) the 3 roots of 1 express - in an indirect quantitative manner - the qualitative notions of 1st 2nd and 3rd respectively which in terms of the Type 2 aspect of the number system are represented as 1^1/3, 1^2/3 and 1^3/3 respectively.

However the last root here representing the notion of the 3rd of 3 (with the two other positions fixed) = 1, is quite distinct from the other roots. And likewise this is always true of the last ordinal position (with the other positions fixed).

So this last root in fact represents the conventional analytic interpretation of the ordinal notion, whereby it is directly identified with the corresponding quantitative definition of the number in cardinal terms.

Thus in this analytic context, 1st is identified as the last unit of 1 (which of course = 1). Then with this position fixed, the 2nd is identified with the last unit of 2 (= 1). And finally in the case of our example of 3, the 3rd is identified with the last unit of 3 (= 1).

So the cardinal definition of 3 = 1 + 1 + 1 then happily coincides with the corresponding ordinal definition of 3 = 1st + 2nd + 3rd.

However when we strip out this "trivial" case where the root = 1, we are then left with the remaining roots that directly relate to the Zeta 2 zeros.

So again in the case of 3, we obtain the solution of

1 + x¹ + x² = 0, which yields the other 2 roots (of the 3 roots) of 1.

And these roots representing the notions of the 1st of 3 and 2nd of 3 (where positions are not fixed) - again in an indirect quantitative manner -  express the true holistic meaning of these ordinal notions.


However, once again we are left with a paradox.

Though we start with a prime (as  dimensional number in the Type 2 aspect) the ordinal notion relating to this prime, remains undefined by the Zeta 2 zeros.

So again though we have obtained, in the case of 3, an indirect quantitative expression of the holistic notions notions of 1st and 2nd (with respect to 3), this does not apply to 3rd (which reduces down to its conventional analytic meaning).

And this applies in turn to each of the primes which cannot holistically be given an ordinal interpretation (in relation to the corresponding group of prime members).

So we can give a holistic meaning to 3rd, for example in the context of 5 members; however we cannot do this in relation to 3 members!

So the Zeta 2 zeros relate to a holistic notion of internal order (i.e. with respect to the individual members - excepting the last - of a prime group of members).

However we also have, in complementary dynamic fashion, a corresponding holistic notion of external order (i.e. with respect to the collective relationship of the primes with the natural numbers).


Now it is puzzling in a way that the term "ordinal" in a mathematical sense is solely identified with the Peano based additive approach to the number system i.e. where each natural number is obtained through the addition of 1 to the previous natural number. So the notion of order (1st, 2nd, 3rd,...) is here identified successively with each of the natural numbers in a strictly linear fashion.  

As we know, there is a corresponding way of deriving the natural numbers based on the prime numbers and the operation of multiplication. Thus from this alternative perspective, each natural number represents a unique product of prime factors. 

However, surprisingly though we readily associate corresponding ordinal notions with the first additive approach to the natural number system, this is greatly lacking with respect to the second multiplicative approach.

Now clearly we are talking about a different kind of order here in relation to the primes, which is not of a neat successive nature. However it is an extremely important kind of order nonetheless!

Thus the primes while maintaining their distinct individual identity in quantitative terms, can be collectively combined with each other in an apparent seamlessly integrated fashion.

So when we talk about the "ordinal nature of the primes", we are referring to this qualitative relationship of factor interdependence, which uniquely enables the consistent generation of the natural number system.

And just as the Zeta 2 zeros provide an indirect quantitative means of expressing the internal ordinal  nature of the individual natural number members of a prime number group (excepting the prime itself), likewise in complementary fashion, the Zeta 1 (Riemann) zeros provide an indirect quantitative means of expressing the external ordinal nature of  the unique collective relationship of prime factors with respect to the natural number system.

And again, as in the previous case, we have a paradox in that on this occasion the individual primes are themselves defined in an analytic manner (directly with respect to their quantitative value).

However, when these primes are then combined with each other as the unique factors of composite natural numbers the qualitative aspect of the primes then operates in a holistic manner.

Thus the Zeta 1 zeros strictly relate to this holistic qualitative nature of the collective order of the primes. In this way they are the mirror opposite of the primes.

So the individual primes have a recognised quantitative identity (as the" building blocks" of the natural numbers). Then the collective relationship of the primes through the unique products of prime factors (that consistently enable the generation of the natural numbers), expresses the (unrecognised) qualitative holistic identity of the primes. In fact the proper intuitive recognition of this - literally - represents the collective energy of the primes.

Then it operates in reverse in relation to the Zeta 1 (Riemann) zeros. Each individual zero represents a unique holistic point of qualitative identity (approaching a pure energy state). Like the two turns at a crossroads, this arises from the paradoxical recognition of the two-way relationship of primes and natural numbers (indirectly expressed on an imaginary scale), which coincides with these zeros.  

Then the collection of zeros represents their quantitative identity, whereby they can be used to correct the errors arising from the general estimate of prime frequency to a given number so as to eventually zone in on the correct absolute value.