Thursday, March 14, 2013

Two Complementary Zeta Functions (2)

As we have seen, all mathematical symbols can be given two complementary interpretations (analytic and holistic) respectively.

This is then reflected in the number system which itself has Type 1 (analytic) and Type 2 (holistic) aspects which are complementary. Then indirectly the Type 1 aspect can be given a Type 2 interpretation while then Type 2 indirectly can be given a Type 1 formulation!

Then with respect to the famed complex zeta function, it too has too complementary aspects.

So corresponding to Type 1 we have the recognised Riemann Zeta Function (which I refer to as Zeta 1).

Then corresponding to Type 2 we have a - largely unrecognised – complementary function (which I refer to as Zeta 2). Strictly, though the Zeta 2 Function is in fact well known; its true holistic significance is not at all appreciated.
Once again the Zeta 1 is defined as the infinite series:

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….

And the non-trivial (pair) solutions of the form s = a + it and a – it (where a according to the Riemann Hypothesis = ½) occur when

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + …. = 0

The Zeta 2 is then in inverse complementary terms (i.e. where the natural numbers now represent the powers and s the base quantities) as the finite series:

1 + s^1 + s^2 + ….+ s^(n – 1).

And the non-trivial solutions of the form s = a + it arise from the solutions of the equation,

1 + s^1 + s^2 + ….+ s^(n – 1) = 0

Now in a qualified sense we can equally define this second function in an infinite manner!

So if we take terms in in a strict cyclical order in accordance with the value of n, then the infinite equation will hold.

Thus for example in the simplest case where n = 2, the non-trivial 2nd root as solution to

1 + s^1 = 0 gives s = – 1.

So when we take terms two at a time the value of the infinite series

1 + s^1 + s^2 + s^3 + ….. = 0.

Now in this blog entry I will try to highlight some of the key complementary relations existing between both series and their implications.

Indeed when one begins to appreciate the very nature of the number system from this dynamic complementary perspective it leads to a completely new type of insight that itself directly fosters appreciation of its holistic nature.

Customarily we tend to view the number system in (real) linear terms with the prime numbers viewed as its independent building blocks! So using the terminology that I customarily employ this reflects an analytic (quantitative) interpretation.

Then parallel with this finding we find that the system of non-trivial zeros relating to the Zeta 1 in turn - assuming the truth of the Riemann Hypothesis – represents a corresponding linear number system (this time however in imaginary terms).

Now properly understood this indicates a direct complementary relationship.

So the numbers on the real number line can represented as separate (independent) points i.e. in analytic terms within this frame of reference.

Likewise the non-trivial zeros on the imaginary number line equally can be represented as separate (independent) points i.e. in analytic terms within this different frame of reference.

However the clear implication here is that when we seek to switch reference frames so as to interpret the imaginary points (from a real perspective) they now assume a holistic – rather than an analytic – identity.

In other words the significance of the non-trivial zeros, from a real number perspective, is that they can be used as an entire group to reconcile the unique individual identity of each prime (which is random with respect to the natural numbers) with an overall collective nature (where they are perfectly synchronised with the natural number system).

In other words, from the perspective of the imaginary number line, each non-trivial zero has an (extreme) analytic identity; however from the perspective of the real number line the non-trivial zeros as an entire group (which is always ultimately indeterminate in finite terms) have a complementary (extreme) holistic identity!

The same in fact applies to the prime numbers.

From the perspective of the real number line, each prime number has an (extreme) analytic identity i.e. as a unique building block of the natural number system!

However from the complementary perspective of the imaginary number line, the prime numbers as an entire group (the ultimate nature of which is necessarily indeterminate in finite terms) play a remarkable holistic role in serving to completely reconcile the random individual identity of each non-trivial zero with their overall collective identity (i.e. through being perfectly synchronised with the imaginary number system).

So we see that when we adopt both perspectives that:

1) Each individual prime number has an extreme analytic identity as separate (in real number terms).

2) That the prime numbers as an entire collective group have a corresponding extreme holistic identity (in imaginary number terms).

3) That each individual non-trivial zero likewise has an extreme analytic identity (in imaginary number terms).

4) That the non-trivial zeros as an entire collective group have a corresponding extreme holistic identity (from the real number perspective).

However to properly appreciate such relationships we must preserve both analytic (Type 1) and holistic (Type 2) perspectives with respect to the number system.

Clearly this cannot be achieved in conventional mathematical terms, as it is based formally on mere Type 1 interpretation.

So each number type contains have both particle and wave identities with an individual (analytic) and collective (holistic) identity respectively.

We can then approach the same number relationship issues in an inverted fashion through the Type 2 aspect of the number system.

Now whereas the Type 1 is directly geared to analytic type appreciation (and indirectly holistic), Type 2 is - by contrast - directly geared to holistic type appreciation (and indirectly analytic).

With Type 2, rather than a linear, we adopt a circular notion of number.

Therefore, from a real perspective instead of representing the natural numbers as equidistant points on the number line, we represent them in alternative fashion as equidistant points on the unit circle (where now they strictly represent an ordinal - rather than cardinal - identity)!

So in this approach the prime number 3 - for example – can be represented by the 3 equidistant points labelled 1, 2 and 3 respectively relating to the three corresponding ordinal members of 3 i.e. 1st, 2nd and 3rd respectively.

Thus once again whereas in the cardinal (Type 1) approach the natural numbers, (as an entire collective group) are all uniquely derived from individual prime constituents, in the complementary ordinal (Type 2) approach each prime number (as an individual group) is uniquely defined by its individual natural number members.

This in turn leads to a fascinating alternative explanation of prime number identity.

From the Type 1 perspective, a prime number has no factors (other than itself and 1).

From the Type 2 perspective a prime number - by definition - will always be a factor of the sum of its individual members.

For example 3 as prime number is necessarily a factor of 1 + 2 + 3 = 6 (and this relationship will always hold where the number is prime). However this is not unique for prime numbers and in fact is shared by all odd numbers.

Put another way, expressing this in modular (clock arithmetic) if the modulus is prime, then the sum of the individual natural number members of this prime = 0.

So with the Type 1 a prime number is defined by having no non-trivial factor in natural number terms (i.e. other than itself and 1).

In Type 2 terms, in inverse complementary fashion, a prime number is defined as always constituting a non-trivial factor of the sum of its natural number members!

This in fact therefore provides an alternative means in practice for testing for primeness (which I have already illustrated in other contexts). See "Interesting Prime Result"

However this circular (Type 2) system of (real) numbers has a counterpart system of (complex) numbers as the non-trivial zeros arising as solutions to the equation,

1 + s^1 + s^2 + ….+ s^(n – 1) = 0, where n is prime.

In contrast to Type 1, appreciation of the nature of these zeros is directly of a holistic nature.

Though 1 is always included as one of the prime roots of 1, in a certain sense it is trivial in that it is thereby not unique!

So the non-trivial strictly refer to the remaining n – 1 roots!

We could perhaps illustrate the true holistic nature of such solutions with respect to the simplest case of the 2 roots of 1.

So once again these are 1 and – 1 respectively.

The holistic interdependence of these roots can be illustrated by the fact that the sum of these roots i.e. 1 – 1 = 0.

Now this interdependence strictly relates to an energy state!

For example at the sub-atomic level, a particle and anti-particle are + 1 and – 1 with respect to each other. Then when combined their separate identities are annihilated resulting in a pure energy state.

So in a very literal sense such a pure energy state thereby characterises the holistic nature of 2.

Indeed if we were to equally consider virtual (imaginary) as well as real particles the combined annihilation with anti-particles at both levels would characterise the true holistic nature of 4!

Though other numbers are indeed more difficult to illustrate, the basic principle is clear in that the holistic nature of number in fact relates directly to a physical energy state!

And as physical and psychological aspects are complementary, this equally implies that associated with each number in holistic terms is a corresponding psychological energy state relating to the (unconscious) intuitive aspect of number experience.

For example, in the dynamics of experience external and internal polarities of understanding are positive and negative with respect to each other.

As the very process of understanding necessarily entails integrating both of these polarities + 1 and – 1 as directions (i.e. dimensions) the holistic understanding of 2 from a psychological perspective necessarily relates to a psycho-spiritual energy state! So intuitive recognition (as a psychological eneregy state) always reflects the holistic aspect of understanding. Unfortunately such intuitive recognition is then reduced to merely rational (analytic) interpretation in formal conventional mathematical terms.

So when one looks at the issue from a number perspective it should come as no surprise that likewise the holistic appreciation of the non-trivial zeros (with respect to Type 1) equally relate to energy states.

Though a degree of recognition of this fact has admittedly been obtained in recent years (with reference to the behaviour of certain quantum chaotic physical systems) Conventional Mathematics still lacks the means to intuitively explain why this is so (due to the complete absence in formal terms of a holistic aspect to its means of interpretation).

In my own case, as I had been specialising for some years in the holistic appreciation of number, I had long reached the conclusion that the Riemann zeros necessarily had quantum mechanical implications.

However even more, I had firmly reached the conclusion that these same zeros necessarily possess extremely important psycho-spiritual implications (which is completely missing as yet from conventional type understanding).

Furthermore the appropriate way to view both these aspects - physical and psychological - is in a dynamic complementary manner!

Moreover, correctly understood, we have in fact two sets of zeros (in accordance with Type 1 and Type 2 appreciation respectively).

Whereas Type 1 directly relates to the understanding of prime numbers as discrete phenomenal entities, by contrast Type 2 directly relates to the same prime numbers as (formless) energy states in both physical and psychological terms.

The deeper implications of all this is that - correctly understood - number itself must be viewed as inherent in all physical and psychological processes as the most fundamental way in which their quantitative and qualitative attributes are encoded.

In this sense, we can truly say that all living phenomenal forms at their most fundamental level represent but the dynamic interaction of number processes with respect to both their quantitative and qualitative aspects!

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