Zeta 1 Zeros - Key Significance (6)

We have seen that - relative to the quantitative base notion of number (in Type 1 terms) - that the Zeta 1 (Riemann zeros) relate to the qualitative measurement of the prime factor interdependence of the natural numbers, which is indirectly expressed in Type 2 terms (where the linear measurement of 1 unit now represents the radius of the unit circle).

Now the imaginary line simply represents the indirect attempt to express this circular measurement - suited to the holistic interpretation of factor interdependence - in a quantitative linear manner.

And as the circumference of the unit circle, stretched out as a line = 2π, Therefore to convert to standard linear units we divide by 2π.

In this way we are able to link up the imaginary part of accumulated total of Zeta 1 zeros to t with the corresponding total of divisors i.e. all factors (except 1) of the number factors to n, where n = t/2π. Again it is important to note that we are including composite natural numbers (as well as primes as factors here). So to illustrate, from this standpoint, the factors of 12 (except 1) are 2, 3, 4, 6 and 12! 

The total sum of Zeta 1 zeros to t = 100 is 29. So n = 100/2π = 15.915.

Now the total sum of factors (except 1) to 15 = 29 (and to 16 = 33). So we already can see a close relationship as between the frequency of the Zeta 1 zeros and the corresponding accumulated frequency of the factors of numbers. 


However, we will attempt here to probe in more detail the precise significance of the Zeta 1 zeros.

It is important to bear in mind - in this dynamic interactive context - that we must keep balancing  two aspects with respect  to the understanding of the primes that are quantitative (analytic) and qualitative (holistic) with respect to each other.

So again we start in analytic terms with the appreciation of the primes as unique independent "building blocks" of the natural number system in a quantitative manner. Though in conventional mathematical terms, the primes are interpreted individually in a reduced absolute manner, strictly in more correct dynamic terms, understanding of this aspect of the primes is of a relatively independent nature that can approach but never fully attain an absolute meaning.

However, then in complementary holistic terms, we have the corresponding appreciation of the primes as comprising unique factor relationships of collective interdependence with the natural numbers in a truly qualitative manner.

The Zeta 1 (Riemann) zeros then express this qualitative relationship - pertaining to the natural numbers as  expressive of unique factor combinations of the primes - indirectly in a quantitative manner.

However the correct appreciation of the nature of these zeros lies at the complementary extreme to that of the primes (as quantitative "building blocks").

In fact, true appreciation of the Zeta 1 zeros, in dynamic interactive terms, approaches a state - though necessarily  never quite fully - of pure relativity (which in a complete manner would represent total ineffability).

Expressed in an equivalent fashion, whereas the analytic aspect approaches the (conscious) rational extreme with respect to understanding, the corresponding holistic aspect represents the (unconscious) intuitive extreme.

Yet again, whereas the analytic aspect (with respect to the individual primes) approaches an extreme where numbers seemingly represent absolute forms, the holistic aspect approaches the opposite extreme (with respect to the Zeta 1 zeros) where numbers now seemingly represent pure energy states!


Therefore, true appropriate understanding of the relationship as between the (independent) primes and the (interdependent) Zeta 1 zeros, from a psychological developmental perspective, requires the full integration of both conscious (rational) and unconscious (intuitive) processes of mathematical understanding.

And let us once more remind ourselves at this point, that because Conventional Mathematics solely recognises the (conscious) analytic aspect in a reduced manner, that it thereby blots out the completely the (unconscious) holistic aspect - at least in formal terms - thereby providing a fundamentally distorted interpretation of all its relationships (especially with respect to number)!

It may help towards the realisation of the true holistic nature of the Zeta 1 zeros to bear in mind the crossroads example, that I have referred to on so many previous occasions.

In this context left and right turns have a clear unambiguous meaning when we use a one-directional (i.e. one-dimensional) frame of reference. So if one approaches the crossroads moving N or S (as single separate directions) left and right turns can be given an unambiguous identity.

However when one simultaneously attempts to combine both directions (N and S), what is a left or a right turn is rendered paradoxical, for what is left from one direction is right from the other, and vice versa.

So we move from a (linear) either/or logic which befits unambiguous analytic, to a (circular) both/and logic that now befits paradoxical holistic appreciation.

It is precisely similar with respect to the relationship as between the primes and natural numbers, the understanding of which can be approached in both external and internal terms.

Thus in conventional mathematical terms, one concentrates on the external relationship as between the primes and natural numbers where the primes are viewed as the independent "building blocks" of the natural numbers in a unique quantitative manner.

However one can also concentrate on the internal relationship as between the interdependent prime factors of each natural number also in a quantitative manner.

So for example one can approximate the external frequency of primes up to n by the simple relationship n/log n.

One can also approximate the internal ratio of (distinct) prime to natural number factors of a number by the simple relationship n₁/log n₁, where n₁ = log n.

So we have two distinct reference frames here (i.e. externally in a collective manner and internally in an individual manner respectively) for viewing the relationship as between the primes and natural numbers.

In each case, we appear to be able to define the nature of this relationship in an unambiguous quantitative manner.

However just like N and S served as complementary opposite directions in our crossroads example, both external and internal serve as complementary opposite directions with respect to the relationship as between the primes and natural numbers.

Therefore, when we simultaneously combine both external and internal directions, the relationship as between the primes and natural numbers (and natural numbers and primes) is rendered paradoxical in a circular manner.

And the Zeta 1 zeros directly represent such paradoxical understanding, which is directly of a qualitative holistic nature (though indirectly represented in a quantitative manner).

There are indeed several equivalent ways in which we can express this paradox.

So from one perspective, each point (denoting a Zeta 1 zero) represents a state where prime and natural number notions become identical (and thereby distinguishable from each other).

So just as at the crossroads - when simultaneously approached from two directions N and S - left and right become identical with each other (for what is left from one direction is right from the other), likewise at each point (denoting a Zeta 1 zero) prime and natural number notions are identical (for what is caused by the primes from one direction e.g. external, is caused by the natural numbers from the other and vice versa).

Thus the Zeta 1 zeros serve as two-way reflecting mirrors where the primes and natural numbers become identical with each other. Now this cannot be understood directly in a dualistic analytic manner (through reason) but rather in a nondual holistic manner (through intuition).

Then when one indirectly tries to convey its meaning in a rational analytic fashion, its truth is revealed as circular (i.e. paradoxical).

We could also refer to each Zeta 1 zero as a point where notions of randomness and order with respect to the number system are identical. In fact from a dynamic interactive perspective, we can only define notions of number randomness against an implicit background of number order and vice versa. 

Thus the apparent randomness of primes, as viewed in an independent individual manner from the external quantitative perspective, appears as remarkable order when viewed internally from an interdependent qualitative perspective (as prime factors) and vice versa.

Thus randomness and order from a dynamic interactive perspective are revealed to be clearly two sides of the same coin.

We can also refer to each Zeta 1 zero as a point where the qualitative notion of interdependence becomes inseparable from the quantitative notion of independence.

This entails that true appreciation of the Zeta 1 zeros requires the intimate balancing of both highly refined reason and highly refined intuition respectively.

Finally, we can say that each Zeta 1 zero represents a point where the operation of addition is indistinguishable from that of multiplication (which follows from the recognition that they are - relatively - quantitative and qualitative with respect to each other)