Zeta Zeros and the Changing Nature of Number (6)

We are now ready to look at the significance of the Riemann zeros, which I refer to as the Zeta 1 zeros.

An important complementary relationship exists as between these (recognised) Zeta 1 zeros and the Zeta 2 zeros (the significance of which are not yet properly understood).

As we have seen with respect to the Zeta 2 zeros we started with the cardinal notion of a prime number.

Now again from this perspective, if we were to attempt to "crack open" such a prime - say again 5 - we would find it composed of independent homogeneous units (completely lacking in qualitative distinction).

This is akin to splitting open an atom on the physical level and expecting it to be composed of the same uniform atomic "stuff".

However we know now, that properly understood, within the atom is a highly dynamic world made up of interacting sub-atomic particles (that are not composed of uniform "stuff").


Likewise properly understood, the outer identity of the "independent" prime building block likewise conceals an inner world  of interacting natural number elements in ordinal terms (which possess a unique qualitative identity).

Using terminology from Jungian psychology - which is indeed fully appropriate in this context - we can say that each prime has a shadow identity. Thus the shadow to the accepted analytic notion of the cardinal prime (as independent) in conscious terms, is the corresponding holistic notion of  the prime (as the unique  interdependent expression of its ordinal natural number members) in an unconscious manner.    

So the Type 2 (holistic) appreciation of number, properly represents the (unrecognised) unconscious shadow of conventional Type 1 (analytic) appreciation.

However in Jungian terms, the recognised community of practitioners, remains completely blind to this important shadow side of Mathematics.

Therefore instead of recognising that all mathematical relations properly entail the dynamic interaction of conscious and unconscious aspects (entailing both reason and intuition), the mathematical community still blindly insists on the merely reduced interpretation of all concepts in a formal rational manner!

We then went on to show that the Zeta 2 zeros represent this shadow holistic appreciation of the ordinal nature of the number system. Put more simply they  represent therefore the unconscious aspect of number appreciation.

We then went on to show that the Zeta 2 zeros play a vital role with respect to the consistent two-way interaction of the Type 1 and Type 2 aspects of number. In psychological terms, they thereby enable the consistent interaction of both conscious and unconscious with respect to all number understanding!


However Zeta 2 understanding is limited somewhat to the internal relationship as between each individual prime and its ordinal natural number members.

So in a complementary manner, the Zeta 1 understanding concentrates on the external relationship as between the collective nature of the primes and its cardinal natural number members.

Again in the recognised Type 1 manner, each (cardinal) natural number represents the unique combination of prime number factors.

Once again 6 as a natural number is the unique expression of combining (just once) the primes 2 and 3.

So 6  = 2 * 3.

This then leads to the restricted view of the individual primes in analytic terms as the independent  building blocks of the natural number system in a quantitative manner.

However once again, this must be balanced by the holistic shadow interpretation of the collective nature of the primes as fully interdependent with the natural number system in a qualitative manner.

One might then seek to find out how this collective nature of the primes is expressed!

It is here that I find the complementary relationship with the Zeta 2 zeros so helpful.

So once more in this latter context we start from the position of viewing 1st, 2nd, 3rd,... and so on as fixed notions (whereby they can be directly reduced to cardinal interpretation).

However we then saw how in relative terms all these ordinal notions can be given an unlimited number of alternative expressions.

So the default fixed notion of 2nd is given as the last unit of 2 (2nd in the context of 2).

However we can also - sticking initially to prime groups - relative notions of 2nd (in the context of 3), 2nd (in the context of 5), 2nd (in the context of 7) and so on without limit.

And we found that the Zeta 2 zeros uniquely express all these relative notions of the ordinal numbers!

Thus using the non-trivial prime roots of 1 we are able to coherently express the true relative meaning of these ordinal notions in a holistic manner.

So for example, as we saw in the case of 5 the 4 non-trivial roots express (in a Type 1 quantitative manner) the notions of 1st (in the context of 5), 2nd (in the context of 5), 3rd (in the context of 5) and 4th (in the context of 5) respectively. Then the final root gives the (default) notion of 5th (in the context of 5).

Thus while each of these ordinal notions (indirectly expressed in a circular quantitative manner) enjoys a relative independence from each other, when combined together their relative interdependence is indicated by the fact that the sum = 0 (i.e. has no quantitative value).

So this provides a striking demonstration of the inherent qualitative nature of the notion of interdependence!  


Now the Riemann (i.e. Zeta 1 zeros) arise in a similar manner, that is now directly focussed on the cardinal nature of number.

So we start from the (default) position of each of the primes i.e. 2, 3, 5, 7,.... as having a fixed quantitative identity.

However once we start combining primes in new combinations, they thereby attain, for every combination, a unique qualitative type identity (expressing their interdependence  with other primes).

For example we have already seen how 6 represents the unique combination of 2 and 3 i.e. 2 * 3.

Therefore with respect to 6, both 2 and 3 acquire a new qualitative identity through being factors of 6.

So in combining primes to form composite natural numbers, the primes thereby lose their exclusive individual identity. So in this respect it a little bit like combining individual ingredients in a cake recipe, whereby each ingredient is qualitatively changed through interaction with the other ingredients!

So quite simply, we measure these new qualitative interactions of the primes through obtaining the natural number factors of the composite number involved.

So in the case of 6 we have 2 and 3 as factors (now qualitatively changed through interaction) and also 6 (as the natural number combination of both). Just as 1 is not directly considered with respect to the Zeta 2 zeros, likewise it is not considered as a non-trivial factor!


Now, to put it simply, the frequency of the  Riemann zeros bear a remarkably close relationship to the corresponding frequency of natural number factors.

So the cumulative frequency of  natural number factors on a linear scale up to n, bears an extremely close relationship with the corresponding frequency of Riemann (Zeta 1) zeros on a circular scale to t, where n = t//2π.

For example I calculated manually the cumulative frequency of all the natural number factors in the manner described up to n = 100 and obtained 357.

This should then equate well with the frequency of Riemann (Zeta 1) zeros up to t = 628.32 (approx).

And the number of zeros by my estimate = 361. So we can already see this close relationship between the two measurements.

Now just like the primes, the factors of the composite numbers - we do not consider the primes as containing factors in this respect - occur in a discontinuous fashion.

Thus we keep moving along the number line from the primes (as numbers with no factors) to the composite natural numbers (which will contain a varying number of factors).

In fact the Riemann (Zeta 1) zeros can best be understood as the attempt to smooth out in a continuous fashion  these discrepancies with respect to the occurrence of factors.

In this sense each zero represents a harmonisation of the primes with the natural numbers.


Again from an analytic (Type 1) perspective we look at the primes and (composite) natural numbers as distinctive entities with the composites determined by the primes.

However from the complementary (Type 2) perspective, this relationship is reversed with the primes now "determined" by the natural numbers.

Thus the position is very much here like the interpretation of turns at a crossroads.

If we approach the crossroads from just one direction - say heading N - left and right turns will have an unambiguous fixed meaning.

Then when we approach the same crossroads - heading S - left and right turns will again have an unambiguous meaning.

However when we simultaneously consider both N and S directions, our notions of left and right are rendered paradoxical. What is left is also right and what is right is also left (depending on context).

The logic is very similar here.

When we consider the relationship of the primes to the natural numbers, the position appears unambiguous in Type 1 terms (i.e. the natural numbers are derived from the primes).

Then when we consider the relationship of the primes to the natural numbers from a Type 2 perspective, it again appears unambiguous (i.e. the primes are derived from the natural numbers)  

So again in Type 1 terms, the natural numbers are uniquely determined as the product of individual primes i a quantitative manner.

Then in Type 2 terms it is the unique combination of natural number factors expressing the collective interdependence of the natural number system in a qualitative manner, that uniquely determines the location of the primes.

Now when we bring both Type 1 and Type 2 perspectives together - like at the crossroads - paradox results so that we can no longer distinguish the primes from the natural numbers.

And the Riemann (Zeta 1) zeros express this paradoxical nature of primes and natural numbers.

Their nature can only be properly grasped through the dynamic two-way interplay of both Type 1 and Type 2 understanding (entailing both analytic and holistic understanding) which I refer to as Type 3.

When we try to fix their meaning (through adopting just one reference frame) their true meaning will elude us.

Our conventional understanding of number conforms to rigid notions of form in an absolute fixed manner.

However the Riemann (Zeta 1) zeros lie at the other extreme of understanding, approaching pure relative notions (that are rendered paradoxical in terms of fixed reference frames).

Thus these zeros are best understood as the other extreme to form in representing fleeting  energy states as the final partition to the pure ineffable nature of ultimate reality.

They cannot be grasped through reason alone but rather the most refined circular form of understanding that is plentifully infused with pure intuitive insight.