The True Nature of the Zeta Zeros (3)

In the last two blog entries, I have outlined how the the two sets of zeros enable conversion of both cardinal and ordinal notions (as analytically understood) in a complementary holistic manner.

So again with the Zeta 1 (Riemann) zeros, we start with the quantitative notion of primes (i.e. 2, 3, 5, ...), which serve as the building blocks of the natural number system, except 1, (again in a quantitative manner).

However through the process of multiplication of primes, a qualitative dimension is also crucially involved, whereby the primes (and by extension other natural numbers) now serve uniquely as factors of other composite numbers.

Therefore the absolute independent identity of primes is thereby lost, when they are combined as factors of subsequent natural numbers!

So in truth, a new unique qualitative identity (of a merely relative nature) is thereby established.

For example if we take the composite number 12 to illustrate, we perhaps readily appreciate that it is composed from two initial prime building blocks i.e. 2 and 3.

However this means that in the context of 12, both 2 and 3 as constituent factors obtain a new qualitative identity. In other words through their relationship to 12 - and the qualitative aspect inherently relates to number relationships - 2 and 3 are thereby uniquely reflected in a new light.
However other natural numbers (already attained from prime building blocks) are now also uniquely reflected in a new light i.e. 4, 6 and 12.

Thus, from this perspective associated with 12, are 5 natural number factors (including primes and other composite natural numbers derived from primes).

The factors - where each is uniquely reflected in the light of the composite natural number concerned - thereby express the qualitative nature of the natural number system.


And there is a direct link as between the accumulated frequency of such factors and the corresponding frequency of the famed Zeta 1 (Riemann) zeros.

So a stated before the frequency of these non-trivial zeros to t (on the imaginary line) matches very closely the corresponding (accumulated) frequency of factors to n (on the real line) where n = t/2π.

For example, the frequency of zeros to t = 628 (on the imaginary line) = 91.
The corresponding (accumulated) frequency of factors to n = 100, i.e. n = t/2π (on the real line) = 98, which already in relative terms is fairly close.

Indeed there is equally a close relationship as between the accumulated sums of factors and corresponding sum of zeros.

(In attaining the sums of factors we multiply each composite number - the primes are excluded - by the number of its factors. So for 12 (as illustrated), we would add 60 i.e. 12 * 5.

Thus in this manner, the accumulated sum of factors to 100 (on the real line) = 20367 (according to my estimate).
The corresponding sum of zeros to 628 (on the imaginary line) with the result then divided by 2π = 20133. So the relative accuracy here is already close to 99%!

This thereby reveals the very nature of the Zeta 1 (Riemann) zeros as an imaginary (i.e. holistic) expression of the qualitative nature of the primes. And because of  complementarity as between analytic and holistic notions, the primes and zeros thereby both necessarily lie on lines (real and imaginary respectively) which mutually mirror each each other in a perfect manner!

However once more, this can only be properly understood from a dynamic interactive perspective, where the zeros, as representative of holistic qualitative appreciation, are clearly recognised as directly complementary with the standard analytic appreciation of the primes in quantitative terms.

So we start with the (conscious) analytic appreciation of the primes as absolute in a merely quantitative manner.

Then we are ultimately led, through the zeros, to this (hidden) shadow recognition of the corresponding (unconscious) holistic recognition of the primes, as purely relative in a qualitative manner.

Thus the true integration in understanding of both aspects (i.e. primes and zeros) ultimately requires the full incorporation of both analytic and holistic modes of mathematical appreciation.

This in turn requires from a psycho spiritual perspective the consequent full integration of both conscious and unconscious aspects of all mathematical understanding.

Thus the poverty of present conventional understanding is starkly revealed, where no formal recognition whatsoever of the holistic aspect yet exists!

And quite simply the true role of the zeros cannot be grasped in the absence of this holistic aspect.


So the Zeta 1 (Riemann) zeros express, in a holistic numerical manner, the hidden qualitative aspect of the cardinal primes (through their collective relationship to the natural number system).

Then the Zeta 2 zeros likewise express, in a complementary holistic mathematical manner, the hidden quantitative aspect of the ordinal natural numbers (in their relationship to individual primes representing groups).

So once again the ordinal notions of 1st, 2nd, 3rd,...are inherently of a qualitative nature (expressing in any given finite context, a relationship between a group of numbers).
Then, through obtaining the successive prime roots of 1, these ordinal members, except 1, can be uniquely expressed in an (indirect) circular quantitative manner.
And because of the fundamental importance of primes as building blocks (now as unique "circles of interdependence"), the ordinal nature of all numbers can likewise be expressed in a quantitative manner (through the corresponding roots of that number).

Now, the Zeta 2 zeros  represent all these roots, except 1, thereby expressing the qualitative nature of ordinal numbers in an (indirect) quantitative manner.
And once again the interpretation of such roots is properly of a holistic - rather than analytic - nature.

Thus, again for example, the 3 roots of 1, i.e. 1, .– 5 + .866i, and  – .5 – .866i express in an indirect quantitative manner the ordinal notions of 1st, 2nd and 3rd (in the context of 3 members) .

However the interpretation is properly of a circular (holistic) nature, where the relative independence of each individual root as quantitative (representing each distinct ordinal identity) can only be properly understood in the context of the collective interdependent identity (through addition) of all 3 members. 

And in every case, this collective sum = 0, representing the corresponding qualitative nature of these members. In this way, through appropriate holistic understanding, both the (individual) quantitative and (collective) qualitative nature of the roots (expressing their ordinal identity) can be fully balanced in a dynamic relative manner!


And again it is similar in a complementary manner with respect to the Zeta 1 (Riemann) zeros.

Here, through multiplication, each individual zero has a qualitative identity. This expresses, at each such point on the imaginary scale, a location where opposites (from an analytic perspective) are reconciled in a dynamic interactive manner. So randomness and order are opposites from an analytic perspective! However these two notions are dynamically reconciled for each point representing a non-trivial zero. Likewise the notions of prime and (composite) natural numbers are opposite in analytic terms. However once again these two opposing notions are reconciled in a holistic manner with respect to each zero!

However each individual zero can only be properly understood in the context of the collective nature of all zeros! And it is from this latter collective perspective that the quantitative nature of these zeros can be appreciated, in their ability to smooth out discrepancies as between the general behaviour of primes (as quantities) and their unique unpredictable behaviour in local regions of the number system.