Thursday, July 28, 2016

Riemann Hypothesis: New Perspective (5)

Once again, I keep returning to the distinction as between number independence and number interdependence respectively.

It is important to bear in mind that the quantitative notion of number is related directly to the independence of units.

Therefore when one uses for example the number 3 in its accepted quantitative sense, this assumes the identity 3 = 1 + 1 + 1 (where each of the units is considered as  independent relationship of each other).

However one when uses the notion of number in a dimensional sense to represent factors, the very nature of number thereby subtly changes.

So for example if we now say that a number contains 3 factors, this implies that each of the "unit" factors now bears a relationship of (multiplicative) interdependence with the number (of which they are factors).

Thus confining ourselves to proper factors,  1, 2 and 3 are factors of 6.

However this clearly assumes a relationship of interdependence as between 6 and each of its member factors.

Now again we could indeed validly say that in this context that 6 has 3 factors.

However if we wish to express 3 in terms of its unit members 3 = 1 + 1 + 1, this  now implies that the individual members share a relationship of interdependence with each other (through each being a factor of 6).

So in fact - though this observation is missed entirely in conventional mathematical interpretation - we have switched, using the language of quantum mechanics, from the particle aspect of number (implying independence) to its corresponding wave aspect (implying interdependence).

Expressing this more fundamentally, we have switched from the quantitative aspect of number (again implying independence) to its corresponding qualitative aspect (implying interdependence).

Once again, this is vital to understanding the true nature of the Riemann zeros.

As I have stated before a very close relationship connects the Riemann zeros to the factors (i.e. divisors) of the natural numbers.

So once again if we count the (accumulated) factors of the natural numbers to n and then measure the frequency of the Riemann zeros to t (where n = t/2π), the ratio of factors to Riemann zeros → 1, when n and t are sufficiently large.

In fact, the very function of the Riemann zeros is to smooth out the unevenness in occurrence of factors with respect to the primes and composite natural numbers.

We start initially with the primes as "building blocks" of the composite natural numbers (in quantitative terms). 

However, as we have seen, through their relationship as factors of these composite natural numbers, the primes and other natural numbers (representing unique combination of primes) acquire a new relationship of qualitative interdependence with the (composite) natural numbers.  

So in terms of proper factors, the quantitative independence of the primes (as building blocks of the natural numbers) is expressed as 1 (i.e. each prime contains one proper factor = 1).

By contrast the qualitative interdependence of the (composite) natural numbers, as the relationship of primes (or other natural numbers already expressing a unique combination of primes) is expressed through the number containing 2 or more composite factors.

Therefore, a very uneven pattern is in evidence with respect to the occurrence of factors of the natural numbers.

Thus looking at numbers up to 10, starting with 2, as prime this represents 1 factor; 3 also as prime represents 1 factor; 4 then as the first composite number represent 2 factors; 5 as prime then represents again 1 factor; 6 as composite now represents 3 factors,; 7 as prime represents 1; 8 as composite represents 3; 9 as composite represents 2 and 10 as composite represents 3.

The Riemann zeros represent the attempt to harmonise both the quantitative and qualitative aspects of the number system i.e. with respect to the unitary nature of factors associated with the primes and the multiple nature of factors associated with the composite natural numbers.   

Now just like the recognition that left and right turns can interchange with each other in paradoxical fashion at a crossroads (depending on the reference frame N or S from which the crossroads is approached), likewise the recognition of the true paradoxical nature of the Riemann zeros (in holistic terms) is derived from the ability to simultaneously view number with respect to both its quantitative and qualitative aspects (i.e. in Type 1 and Type 2 terms). 

Thus aspects clearly separated with respect to number in an analytic manner (as distinct) are yet seen from a holistic perspective to approach mutual identity (in a dynamic interactive manner).

Therefore in proper analytic terms, the aspect of number as representing the base is clearly separated from the corresponding aspect representing the dimensional aspect of number.  Because of the gross reductionism of conventional mathematical interpretation, this however is not attempted (with both aspects confused with each other). However I have demonstrated proper separation is achieved through defining both the Type 1 and Type 2 aspects of the number system! 

However in holistic terms, both of these aspects are then seen as identical with each other - or more accurately as approaching perfect identity with each other - in a dynamic interactive manner.

Thus from this perspective, each Riemann zero represents a point (on the imaginary number line) where the Type 1 and Type 2 aspects of the number system approach identity.

Again. in analytic terms, the notion of primes and natural numbers are clearly separated.

So in conventional (Type 1) terms the (composite) natural numbers clearly depend for their quantitative identity on the primes (as fundamental "building blocks"). However in unrecognised (Type 2) terms, the primes (and unique combinations of primes) obtain their qualitative identity through their relationship with the (composite) natural numbers as factors.

So from this perspective, the Riemann zeros holistically represent the points on the imaginary line where both their quantitative and qualitative aspects approach dynamic identity.

Also in analytic terms, addition and multiplication are clearly separated through the Type 1 and Type 2 aspects of the number system (where multiplication with respect to the Type 1 represents addition with respect to the Type 2 aspect and multiplication with respect to the Type 2, addition with respect to the Type 1 aspect  respectively.  

From this perspective, the Riemann zeros holistically represent the points on the imaginary number line, where both addition and multiplication approach a common identity.

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