Saturday, April 23, 2011

The Trivial Zeros (8)

As is well known two key features characterise prime numbers.

As discrete independent numbers, there is seemingly no discernible pattern connecting them.
However in terms of their overall relationship to the natural numbers an amazingly continuous pattern of interdependent regularity is in evidence.

Though this should immediately suggest both quantitative and qualitative aspects in the behaviour of prime numbers, due to the quantitative bias of Conventional Mathematics this is conveniently overlooked.

Even momentary reflection on the matter should suggest a problem.

From the linear (discrete) view, prime numbers are seen as the independent building blocks of the natural number system.

However from the corresponding holistic (continuous) view, prime numbers are seen as intimately dependent on the natural numbers for their general distribution.

So clearly, both prime and natural numbers are co-determined in a dynamic interactive manner.

To meaningfully refer to prime numbers implicitly requires the use of the natural numbers. The very listing of the prime numbers requires the ordinal use of the natural numbers 1, 2, 3, and 4 etc. And even here we have the relationship as between quantitative and qualitative (in the cardinal identification of prime numbers, together with the ordinal identification of natural numbers).

As is well-known the non-trivial zeros have a vital role to play with respect to the precise distribution of the prime numbers.

The trivial zeros also have an - unappreciated - qualitative significance with respect to their general distribution.

Because of the interdependence of quantitative and qualitative, all quantitative aspects of number have a corresponding qualitative meaning (and all qualitative a corresponding quantitative meaning).

However because of the merely quantitative bias of Conventional Mathematics, both in relation to individual prime numbers and their general distribution, only the quantitative aspect is formally recognised.

In quantitative terms, the simplest recognised expression for the general spread of the primes (i.e. average distance between successive primes) = log n.

Remarkably this in turn can be directly related to all the zeta values for 1, 2, 3,…

The Zeta Function for s = 1 generates the harmonic series. So where n is finite this will have a finite value and already approximates well to log n (for large n).

By successively subtracting and adding zeta values for s = 2, 3, 4,… (again taken over a finite range for n) and dividing by 2, 3, 4,.. we then progressively approximate closer to log n.

Now the Zeta Function for s = – 1, – 2, – 3, – 4,... provides the corresponding qualitative appreciation of this general distribution.
One advantage of such appreciation would lead to the realisation that prime numbers cannot be understood in a merely quantitative manner (but in fact entails a close intimate connection as between both quantitative and qualitative aspects).

The fruit of such understanding is the eventual realisation that the Riemann Hypothesis itself points to the ultimate condition necessary for the full reconciliation of both quantitative and qualitative aspects.

So the quantitative appreciation relates to rational type understanding, whereas the qualitative relates to directly intuitive type awareness (of the interdependence of primes).

Each trivial zero therefore can be seen as a further progression in the pure intuitive realisation of the interdependence of primes. The negative odd numbered integer dimensions (which we will look at more closely) then represent a disturbance of the existing intuitive realisation as not yet sufficient to fully incorporate the inherent mystery of the primes.

Ultimately however for very high numbered dimensions, contemplative type intuitive understanding would become so pure that discrete phenomenal elements would no longer be even distinguishable in experience (as rational understanding itself would attain an incredibly refined nature).

There is another remarkable qualitative connection here. We have already stated that the spread of the primes is quantitatively given as log n. This is based on the natural log (involving e).

Now with e (raised to any power x) both the differential and integral are the same in quantitative terms!

Likewise in dynamic holistic terms, with appropriate qualitative appreciation of the nature of primes, overall awareness becomes so refined that (differentiated) rational become indistinguishable from (integrated) intuitive aspects.

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