Riemann Hypothesis: New Perspective (3)

We have seen the important basis for the existence of the Riemann zeros in the previous entry.

Once more, this relates to the unrecognised qualitative nature of number, whereby the natural numbers (representing factors)  acquire a unique form of interdependence, through their relationship with other natural numbers.

And the frequency of these Riemann zeros are intimately related to the corresponding accumulated frequency of the (proper) factors of the natural numbers.

So we can now see that the Riemann zeros are directly connected with the attempt to give an indirect quantitative meaning to mathematical; relationships that are - directly - of a qualitative nature.

However in dynamic interactive terms - which is the only appropriate way for viewing such number relationships - it is vital that balance be maintained as between both relative independence and relative interdependence respectively. Alternatively we could express this as the requirement to maintain dynamic balance as between both analytic (quantitative) and holistic (qualitative) aspects of the number system.

Thus once again we start by viewing the primes as the independent building blocks of the natural number system (in quantitative terms). However we then must come to the equal appreciation of the composite natural numbers as representing the unique interdependence of primes (in a qualitative fashion).

Therefore for example, 2 and 3 (as uniquely distinct primes) represent independent building blocks of the quantitative aspect of the number system. However the number 6 (which is composite in nature) now represents, through multiplication, the interdependence of these two primes.

And this interdependence is of a strictly qualitative nature, representing the fact that through multiplication of these two numbers a transformation in their dimensional nature takes place.


So once again - as separate distinct primes in quantitative terms, both 2 and 3 are defined in a 1-dimensional manner i.e. represented by two distinct points on the number line.

However the product of 2 and 3, i.e. 2 * 3, now entails (as in concrete terms with a rectangular table) a transformation in their dimensional nature, which is strictly - in relative terms - qualitative in nature. 

Put another way, in the context of the composite number 6, both 2 and 3 acquire through multiplicative interdependence, a qualitative resonance (reflecting a new shared dimensional nature).

So the question then arises as to how an (indirect) quantitative meaning can be given for all the interdependent relationships that arise through the multiplication of primes.

And just as - in the context of 6 - the qualitative identity of the primes can be expressed through the fact that 2 and 3 now represent unique factors of 6, this equally applies to the factors of all composite numbers.

Also, because natural number divisors (that are not prime) can be factors of a composite number, these must also be included. However as these natural numbers themselves necessarily reflect a unique combination of primes, the factors of the (composite) natural numbers - to which the qualitative identity of the primes relates ultimately relate - reflect factors that are either prime (or represent unique combinations of primes).

So for example, both 2 and 3 represent unique prime factors of 12 (as well as 6) whereby they acquire a distinctive qualitative resonance. However 6 is also a factor of 12 (now representing a composite natural number).

But 6 itself represents a prior unique combination of primes i.e. 2 * 3. So not alone do 2 and 3, as individual factors of 12, acquire a new qualitative resonance, but also 6 (= 2 * 3) which already reflects a prior unique combination of primes.

And in terms of the balanced dynamic appreciation of the number system, both the relative independence of the primes (as quantitative building blocks) and the relative interdependence of the primes (through their qualitative relationship to the composite natural numbers), must be given equal priority.

In other words, they must be viewed ultimately as fully synchronous with each other in an ineffable manner (that defies linear rational explanation).


And this is vital to appreciate because the Riemann zeros in effect represent the perfect balance as between Type 1 (quantitative) and Type 2 (qualitative) notions of the primes i.e. where from one perspective the independence of the primes (in quantitative terms) is maintained and yet from the other the interdependence of the primes (in corresponding qualitative terms) is likewise preserved.