Thursday, December 29, 2016

Zeta 1 Zeros - Key Significance (5)

As we have seen, the Zeta 1 (Riemann) zeros serve as an indirect quantitative means of expressing the holistic qualitative aspect of the primes (through their role as unique factors of the natural numbers).

And again this can only be properly viewed in a dynamic interactive manner.

So from one perspective, we have the quantitative aspect of the primes as the relatively independent "building blocks" of the natural numbers in cardinal terms.

Then from the other complementary perspective, we have the qualitative aspect of the primes - represented by the Zeta 1 zeros - as the relatively interdependent relationship of the unique factors of the natural numbers in ordinal terms.

Thus from one valid perspective, the Zeta 1 zeros represent, in an indirect quantitative manner, the ordinal nature of the primes i.e. in the way that they maintain a collective order with respect to the natural number system, just as the Zeta 2 zeros represent in complementary fashion, the ordinal nature of the natural numbers, with respect to each individual prime number (represented indirectly as the unique prime roots of 1).

However we get a even closer idea of the nature of the Zeta 1 zeros by focusing in on the divisors (i.e. natural number factors) of each successive member of the number system.

And notice once again the complementarity involved here! Firstly, the natural numbers complement the primes and then secondly the factors of each natural number (as considered internally with respect to each individual number) complement the natural numbers (as considered externally with respect to the collective number system).

Therefore once again, if we wish to find the true complement of the primes - as considered externally with respect collectively to the overall number system - then we must look to the natural number factors (as considered internally with respect to each individual number).

And just as in the Zeta 1 case, we ignore 1 (as the non-unique root which must always necessarily arise when we take the prime roots of a number), likewise with respect to the consideration of factors we ignore 1 (which is necessarily a factor of all numbers). However, we do in this case always include n (as a factor of n) as this number is necessarily a factor of all numbers. So for example though 24 is clearly a factor of 24, it is not however a factor of the next natural number i.e. 25 (though 1 is necessarily a non-unique factor of both numbers).

A simple formula then exists to approximate the average number of divisors (i.e. natural number factors) of a given number n.

In fact this internal measurement with respect to the individual factor composition of a number can be simply given as log n; this again shows direct complementarity with a similar approximation for the average gap (or distance) between each prime (as considered externally with respect to the number system as a whole) which is also log n.

The deeper implication of this is that these two features are ultimately of a purely relative nature.

So from one perspective, the internal behaviour of factors (within each individual number) appears to be determined by the external behaviour of the primes (collectively with respect to the number system); however from the equally valid opposite perspective, the external behaviour of the primes appears to be determined by the internal behaviour of factors!

So again like left and right turns at a crossroads, the behaviour in each case is revealed to be of a merely relative nature, which points directly to the truly synchronous identity of primes and natural numbers (in an ultimately ineffable manner).  

So this dynamic interactive manner of looking at the number system immediately opens up the way in which important complementary features characterise the intrinsic behaviour of number that cannot be properly recognised from within the customary analytic approach (of a static absolute nature).

Returning to the average frequency of divisors (i.e. as natural number factors) of the number n, Dirichlet proved in 1838 that this approached log n 1 + 2γ (where γ is the Euler-Mascheroni constant = .5772...).

This would therefore work out at log n + .1544...

 Then, there is the issue as to whether one includes all factors (including 1 and n) in this calculation.

So accepting that the original result is based on the inclusion of all factors as divisors, we would then subtract 1 from this result to equate with the definition of factors that I have adopted.

Therefore the result could now be given as log n 1 + .1544.

 Then to calculate the total number of factors up to n we would multiply by n.

This would give a slightly exaggerated result (i.e. too large) as earlier numbers < n would not contain the same number of  average factors. However given that log n changes very slowly (especially when n is large) and that the average number of factors is slightly greater than log n 1, we can therefore take 
n (log n 1) as a good approximation of the total number of natural number factors (up to n).

When we then look at the corresponding formula for the calculation of Zeta 1 (i.e. Riemann non-trivial) zeros,
i.e. t/2π(log t/2π 1) = t/2π(log t/2π) t/2π, it bears a very close similarity.

In fact it is the same formula where n =  t/2π.

In other words, when one obtains the total of Zeta 1 zeros up to t, it bears a remarkably close similarity with the corresponding total of the combined natural number factors up to n (where n =  t/2π).

In fact, the Zeta 1 zeros can be easily seen to represent the measurement (on a circular scale) of the corresponding natural number factors (on a  real linear scale).

Imagine for example 1 as a single unit on the real number line. If we now draw a circle (using this line as radius) the corresponding length of the circumference that will be traced out = 2π. Therefore to convert such circular units to corresponding linear format we  divide by 2π.

Now it must be remembered that - relative to base numbers (in Type 1 terms), that the factors of numbers relate to the dimensional aspect of number (in a Type 2 manner). Therefore the relationship between both - in dynamic interactive terms - is analytic to holistic or alternatively linear as to circular.

Therefore in this context, the factors of numbers rightly conform to a circular rather than linear frame of reference!

We then further saw in holistic mathematical terms that to represent what is inherently of a circular i.e. holistic nature, indirectly in a linear (i.e. analytic) manner one uses an imaginary rather than real scale.

And this is precisely why the Zeta 1 (non-trivial) zeros lie on an imaginary line, as this represents an indirect analytic mode of representing information that inherently should be understood in a holistic manner!

In other words, the Zeta 1 zeros express - indirectly in a quantittaive manner - the (hidden) qualitative nature of the primes through their collective interdependence (as unique factors) of the natural numbers. And this qualitative aspect of the primes is thereby directly complementary with their corresponding quantitative nature as the independent "building blocks" of each individual natural number. 

And what should be clear now is that this two-way relationship of the primes to the natural numbers (in quantitative and qualitative terms) can only be properly understood in a dynamic interactive manner that combines notions of both relative independence and relative interdependence respectively.    

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