6. Zoning in on Riemann

6. Zoning in on Riemann

Radial Band

We are now approaching the three levels which comprise the Radial Band where both rational (linear) and intuitive (circular) understanding can be combined in a mature fashion.

Once again it might be helpful here to trace the major events that take place when the full spectrum of possible development successfully unfolds.

The first Band (i.e. the Lower) is concerned with the gradual differentiation of experience as the confused starting entanglement of both conscious and unconscious aspects is gradually left behind.

The second Band (i.e. the Lower Middle) is then associated with the specialised development of differentiated rational understanding as exemplified by (conventional) science and mathematics. This is based on (either/or) linear logic leading to unambiguous dualistic type distinctions as between polar opposites in experience.

However - where further substantial spiritual development is destined to unfold - the later stages of this Band lead to existential conflict. This is due to the resurfacing of intuitive unconscious energies that increasingly expose the limitations of the rational dualistic worldview.

The third Band (i.e. the Higher) leads to growing refinement with respect to spiritual intuitive awareness. In rational terms, this causes paradox with respect to accepted dualistic notions. So the growth of (nondual) intuition can thereby be associated with considerable negation of (former) linear understanding.

The fourth Band (i.e. the Upper Middle) is then associated with the specialised development of such integral intuitive awareness i.e. the purely contemplative spiritual worldview. However, over specialisation of form through rational understanding is unbalanced, likewise this is true with respect to emptiness and spiritual intuition. So once again - where substantial further growth is set to take place - this Band in its later stages reveals the limitations of the extreme contemplative position (requiring significant withdrawal from phenomenal reality).

The fifth Band (i.e. the Radial) is thereby concerned with the balanced and mature interplay of both linear and circular understanding (where the refined reason and pure intuition can combine in both a creative and productive manner).

Put another way the Radial Band is associated with the balanced interplay of both the differentiated quantitative aspects of varied phenomena (revealed through rational understanding) and the integral qualitative aspects of overall reality (revealed through holistic intuition). This in turn intellectually entails the growing reconciliation of both linear and circular interpretation.

Finally the sixth Band (the Advanced Radial) is associated with the specialised interaction of both reason and intuition (used in close conjunction with each other).

Clearly here, only an approximation - which at this time in evolution will inevitably fall well below what can potentially be attained - is possible.

Especially with respect to appropriate scientific and mathematical understanding, development at this Band is scarcely yet in evidence.

If we briefly trace the same Bands with respect to mathematical understanding, the following pattern emerges.

The first Band is concerned with the gradual growth of analytic ability with respect to the quantitative interpretation of mathematical symbols.

Then the second Band relates to specialised development of such ability in what - I term - Conventional Mathematics. As we have seen however such Mathematics, from a formal perspective, is almost entirely conducted within a fixed qualitative method of interpretation that is literally one-dimensional. In other words such Mathematics is based on the (merely) rational interpretation of symbols using linear (either/or) logic. This is associated with the considerable reduction (in any relevant context) of qualitative to quantitative meaning.

The third Band is then associated with the emerging development of pure intuitive understanding combined with the additional refined circular (paradoxical) use of mathematical symbols. This leads in turn to the growth of Holistic Mathematics which is directly concerned - not with quantitative meaning - but rather with the potentially infinite range of qualitative (i.e. dimensional) interpretations that can be given to mathematical symbols. This approach is best encapsulated with reference to an alternative definition of the natural number system, where against the background of a (default) quantity of 1, each of the natural numbers refers to successive dimensions (or powers) to which this default quantity can be raised.

With Holistic Mathematics, the true qualitative - as opposed to reduced quantitative - meaning of number as dimension is gradually revealed.

The fourth Band relates in turn to the specialised development of this qualitative interpretation of number which - as we have shown in the previous Chapter - is intimately associated with the holistic interpretation of the (fundamental) Euler Identity.

So with the fifth Band we are now at the stage where both the quantitative and qualitative interpretations of mathematical symbols can be fruitfully combined with each other in - what I refer to as - Radial Mathematics. Though these have already been developed separately (in relative isolation from each other) through the understanding that is Conventional and Holistic Mathematics respectively, it is only now that they can be combined together in an immensely creative and potentially highly productive manner. In truth however I only have the ability to briefly sketch out here - specifically with respect to the Riemann Zeta Function - some of the most basic insights with respect to this approach.

Central to this understanding however is the key point that appropriate understanding (constituting “radial proof”) requires that a satisfactory quantitative and qualitative interpretation be given with respect to any proposition.

For example this would require with respect to the Riemann Hypothesis, not alone that an appropriate “quantitative proof” be supplied (as the goal of Conventional Mathematics), but equally that an appropriate “qualitative proof” be likewise made available (as the goal of Holistic Mathematics). Thus whereas the quantitative explanation would provide an appropriate rational explanation of how the Riemann Hypothesis is true, the corresponding qualitative explanation would provide the deeper intuitive philosophical explanation of why the Hypothesis is true. And it is this important qualitative dimension that is greatly missing from current mathematical understanding.

It would undoubtedly be a great achievement if the long sought after “quantitative proof” of the Riemann Hypothesis could be provided. This would still hold if such a “proof” required a lengthy and difficult exposition which only a few especially gifted mathematicians could verify! However even if this were to happen, it would add little to our deeper philosophical understanding of why the Hypothesis is true!

Indeed I would go considerably further. Some propositions by their very nature can only be solved in radial terms. In other words with these propositions the quantitative and qualitative aspects of understanding are so closely related as to be ultimately inseparable. Furthermore this is likely to be especially true of those propositions - such as the Riemann Hypothesis - that that are most fundamental in terms of mathematical understanding.

Indeed this could provide a clue as to why the Riemann Hypothesis has proven incredibly elusive to prove (in merely quantitative terms). In other words if the Hypothesis is so fundamental - as I am suggesting - that the (qualitative) intuitive are ultimately inseparable from the corresponding (quantitative) rational aspects of understanding, then a merely rational proof (in the currently accepted manner of Conventional Mathematics) may not even be possible.

However by switching to a new basis of understanding i.e. radial (where quantitative and qualitative understanding is inextricably mixed) the Riemann Hypothesis, because of its truly fundamental nature, could then perhaps have an incredibly simple “proof”.  So from the radial point of understanding, the truth of what underlies the Riemann Hypothesis would then be seen as directly related to a very basic axiom in a more comprehensive understanding of mathematical reality.

Once we generally admit the three respective domains of Conventional Mathematics, Holistic Mathematics and Radial Mathematics, then all sorts of new possibilities emerge where some propositions - in terms of existing understanding - may be  provided with “quantitative proofs” (but lacking qualitative verification); others may be given “qualitative proofs” (while lacking quantitative verification).  Yet others may be provided with both quantitative and qualitative type explanations.

Indeed in this context it is worthwhile referring back again to - what I call - the Pythagorean Dilemma. Clearly it was long known that √2 is irrational (and a quantitative proof certainly existed in ancient times). However what caused such great concern for the Pythagoreans was the lack of a deeper qualitative explanation of why √2 is irrational!

So in discussing the Riemann Zeta Function (and its related Riemann Hypothesis) I will be attempting to adopt in barest outline a radial approach.

Indeed in this context I would distinguish variations of the radial approach to mathematics.

In spiritual mystical terms we can have three types of development with respect to the Radial Bands.

The first type i.e. active - though well developed with respect to spiritual intuitive awareness - remains especially grounded in the phenomenal world of form.

This would support a radial mathematical approach which is more closely related to analytic rather than holistic concerns i.e. where quantitative rational work of a linear nature is implicitly fuelled by a plentiful supply of creative intuition. So I will call this the Type 1 radial approach

The second type of spiritual person at the Radial Bands, though attempting to achieve an appropriate level of phenomenal involvement, is much more passive and remains closer to contemplation rather than worldly action. This is turn supports a radial mathematical approach that is now more holistic than analytic i.e. where quantitative type relationships essentially serve as a background for intuitive philosophical understanding (of a qualitative nature). This is the Type 2 radial approach which with respect to the Riemann Zeta Function (and Riemann Hypothesis) will be represented in the continuing discussion (in a very preliminary manner).

The third - and most gifted - mixed type of spiritual person is able to combine in an immensely creative and productive fashion worldly action with refined contemplative awareness at the Radial Bands.

This potentially supports a radial mathematical approach where both the quantitative (analytic) and qualitative (holistic) aspects of understanding can be maintained in equal balance, greatly enhancing ability for substantial developments with respect to both aspects.

This is the Type 3 radial approach which serves as the ideal to which the others can only dimly attain.

Indeed the sixth and final Band of development - which I refer to as the Advanced Radial (AR) Band - especially, relates to further ongoing development with respect to the most talented of these mixed types. Here the possibility for the most extraordinary developments with respect to mathematical understanding can take place where both an appropriate rational (quantitative) and intuitive (qualitative) explanation can be given for any mathematical relationship (irrespective of how abstract it might seem).

However - even for those who are destined for substantial development at the radial levels - very few will ever experience the advanced Radial Band in any real measure. 

Radial 1 (R1) – Riemann Zeta Function 1

Basically, in the discussion that follows, I will be adopting - in terms of my definitions - a Type 2 radial approach (of a necessarily very preliminary nature). This will attempt to unveil - while using quantitative formulations – some of the immense qualitative mathematical riches that are embedded in the Riemann Function (ultimately suggesting indeed a fundamental “radial proof” of the Riemann Hypothesis).

Once again as consistent with my approach so far, mathematical understanding is closely linked with the main stages (i.e. levels) which potentially unfold through appropriate psychological development.

In this sense there is a (Type 2) radial approach that is consistent with the understanding that unfolds during the three main levels of this Band i.e. Radial 1 (R1), Radial 2 (R2) and Radial 3 (R3) respectively which can gradually uncover more and more of the qualitative mysteries inherent in the Function.

We dealt at the previous levels of the upper middle level Band with development that would most typify the (passive) contemplative mystic type. Here remaining rigid attachment to phenomenal form is dissolved enabling the pure experience of spiritual emptiness.

In corresponding mathematical terms this is associated with the specialisation of the qualitative interpretation of mathematical symbols (i.e. Holistic Mathematics). In particular as we have seen, this results in the unreduced appreciation of number as (holistic) dimension, with the corresponding realisation that each such number (as dimension) serves as a unique means of logical interpretation of mathematical reality.

I must stress once again that the implications of this finding are truly enormous as it leads to the clear recognition that conventional mathematics is conducted - at least in formal terms - within the default dimensional interpretation corresponding to the number 1. So quite literally the established rational mode of interpretation for mathematical relationships is linear (i.e. one-dimensional).

Looked at from another perspective we can view mathematical - as indeed all -understanding of necessarily involving a dynamic interaction of both rational (linear) and intuitive (indirectly circular) modes. So one-dimensional (i.e. linear) interpretation here relates to an extreme position where interaction of both aspects (rational and intuitive) is frozen. Corresponding interpretation is then conducted explicitly in merely rational terms (though implicitly intuition is still required to fuel rational linkages).

However when we explicitly allow for interpretation to be based on the dynamic interaction of both modes the other (potentially infinite) set of number dimensions (as qualitative dimensions) then come into play.

What is especially remarkable here is that the precise logical structure corresponding to each dimensional interpretation is (indirectly) provided by the corresponding root of unity with respect to the dimensional number in question.

So as we have already briefly illustrated the important (qualitative) dimensional interpretation corresponding to “2” is given though obtaining the two roots of unity i.e. + 1 and – 1.

What this then entails is that two dimensional interpretation is based on the circular paradoxical interplay of “real” (i.e. conscious) opposites in experience where both (independent) linear and (interdependent) circular aspects are both necessarily involved.

Linear interpretation in any context requires the fixing of reference frames (which ultimately are arbitrary). So to unambiguously denote right and left turns on a road we must initially fix the direction of movement along that road (either in an up or down direction). However this entails that there are always two equally valid reference frames possible for such dualistic interpretation.

So if we designate the direction along the road as “up” a right turn can be unambiguously designated + 1 and a left by – 1. Then in terms of the “down” direction, the right is now – 1 and the left + 1. Within both reference frames (taken separately) designation of left and right is unambiguous.

However problems arise when we try to simultaneously integrate both reference frames for what is + in terms of one frame is – 1 in terms of the other and what is

– 1 in terms of the alternative is + 1 in terms of the first frame.

We are dealing here with a central issue affecting all understanding. Thus we need a means of successfully differentiating opposites in experience while equally requiring a means of achieving holistic integration of those same opposites.

And two dimensional understanding is the simplest way in which we can attempt to explicitly formulate such dynamic interaction.

Now it is important to realise that this problem is equally central to all mathematical understanding, where in the process of interpreting relationships, elements of both (analytic) differentiation and (holistic) integration are involved.

However whereas two-dimensional interpretation is very important, more intricate forms of interaction are associated with “higher” dimensions.

Ultimately as appreciation of “higher” dimensions unfolds, experience becomes more refined ultimately culminating in the pure spiritual contemplative experience of reality.

In my own work on Holistic Mathematics, I have concentrated especially on the “higher” understanding corresponding to 2, 4 and 8 dimensions respectively. Indeed I have consistently maintained that 2-dimensional understanding corresponds to the structure of H1 (psychic/subtle), 4-dimensional understanding to H2 (causal) and 8-dimensional to H3 (nondual) levels respectively.

Now there are good reasons for maintaining that these dimensions are especially important. 2-dimensional understanding integrates both positive and negative poles; 4-dimensional in addition integrates both real and imaginary aspects of reality with respect to these poles and then 8-dimensional reality represents a special form of complex reality associated with null lines (both psychologically and physically, representing emptiness).

Further “higher” dimensions remain similar in that they continue to relate to varying mixes of complex structures that undergo refinement during the specialised development of contemplative understanding (which I equate with the Upper Middle Band). However because such understanding is concerned with the balancing of complementary complex opposites - representing varying combinations of real (conscious) and imaginary (unconscious) elements - we thereby are confining ourselves at this Band to the qualitative interpretation of the even dimensions i.e. 2, 4, 6, 8, 10,… (the corresponding roots of which are always balanced in complementary terms).   

However, just as (mere) rational appreciation is ultimately unbalanced, likewise this is true equally of (mere) intuitive experience (especially where extreme contemplative specialisation takes place). So ultimately what is required is the ability to combine both reason and intuition (without either being reduced to the other).

This is what I term radial experience.  And we are now dealing with the first of the radial level (proper).

So again in our developmental journey so far we have now reached through pure contemplation the integral state where form can be successfully equated with emptiness.

Thus in holistic terms, 1 (as the symbol of form) is now inseparable from 0 (as the symbol of emptiness).

The task now at R1 is gradually in mature fashion to separate both of these poles while maintaining the ability to successfully integrate them in experience.

This equates with the analytic ability to cognitively express the universal structures of all development, while remaining absorbed in constant contemplative awareness of reality. (And quite deliberately in my own mathematical approach these structures are encoded in a holistic binary fashion).

So in standing back at this level as it were to look at the inherently dynamic nature of al reality one can see clearly - that just all information processes - can potentially be encoded through the analytic use of the binary digits (1 and 0), likewise all transformation processes can be potentially encoded through the corresponding holistic use of the same digits (1 and 0).

Therefore though remaining at a deep level of contemplative absorption at R1, externally, sufficient light is now provided to the mind to satisfactorily outline in cognitive fashion the fundamental dynamic structures which govern all development (human and physical).

Again when viewed from a slightly different perspective the very universality of such a vision facilitates an intimate experience of the radial bodyself i.e. where one’s body is now seen in cosmic terms as ultimately inseparable from everything in creation.

Once more the experience of R1 can, for convenience, be subdivided into three sub-levels, though in truth they are very closely intertwined with each other at this stage.

SL1 is associated with the impressionistic ability to obtain meaning from the most general of sense perceptions (as mediators of the universal light experienced through formless contemplation)

SL2 is then associated with the corresponding ability to see meaning through the organisation of the most general concepts. This represents the refinement of that special ability to see only what is most essential in terms of the basic dynamic structures of reality (which reveal a holistic mathematical identity).

SL3 - while still operating at a very general level - facilitates the growing interaction of refined universal perceptions and concepts of form which serves to gradual facilitate the more detailed phenomenal involvement which typifies the next level.

Now one might well ask what all this has to do with the mystery of prime numbers and in particular the Riemann Hypothesis! Well the answer is that it has a very great deal to do with these issues. However to appreciate the intimate connections involved we will need to recast experience in an appropriate manner.

The journey to pure contemplative awareness requires significant detachment from phenomena of form. In the most obvious manner these phenomena are apparent in the natural phenomena that we consciously observe. So initially, detachment in the spiritual life is with respect to such natural phenomena.

However as progress is made, the attention gradually switches from these outer phenomena to more primitive instincts that are unconsciously projected into experience in an imaginary manner.

As these prime instincts comprise the basic building blocks from which natural phenomena are organised, the root of all disordered self interested attachment lies ultimately at a deeper unconscious level.

Then when - after a lengthy process of specialisation in contemplative awareness - one starts to emerge back into the radial light with the restoration of phenomenal involvement, the integral relationship as between primitive instincts and their consequent organisation into stable natural phenomena remains central to experience.

And when we think of it, the Riemann Hypothesis in a very special manner is concerned with the relationship of prime numbers to the natural number system.

Thus the connection that I am making is that the radial levels provide the dynamic experiential counterpart through which a fuller radial understanding of the Riemann Hypothesis can take place.

And each of the radial levels has a special contribution to make.

I have made the distinction before as between the original prime numbers (1 and 0) and the secondary primes (2, 3, 5, 7…) which are all literally rooted in 2 (and ultimately 1 and 0).

Now when correctly seen, R1 is directly concerned with further refinement with respect to original prime behaviour.

I have explained before the unique nature of the transcendental number e, which combines the notions of ever more refined differentiation (with respect to time periods) with continuous integration with respect to those same time periods.

The holistic counterpart of this nature, is the organically dynamic notion of e where the discrete differentiation of phenomena in human experience becomes so refined as to be indistinguishable from the holistic overall integration of those same phenomena. And this is what is precisely meant by a state of pure contemplation, where phenomena of form are now so short-lived that they no longer even appear to even arise in experience.

Clearly the attainment of such a pure state requires a satisfactory means of interpreting the nature of primitive instincts and their relationship to overall natural experience of phenomena. So the spiritual adept can intuitively predict the degree of “primitive” disturbance that is likely to arise from any given level of involvement with natural phenomena and thereby learns to adjust accordingly.

However many contemplatives may only successfully manage involvement through limiting experience to a narrow and predictable range of possible phenomena. Indeed this is exemplified by the traditional monastic approach where the external features of natural living tended to be organised on very predictable lines thereby providing safe boundaries within which contemplative monks could strive to maintain contemplative equilibrium.

The remarkable parallel in quantitative terms is with the prime number theorem which is intimately tied up with the behaviour of e. So Gauss discovered that the general probability of a number being prime is related to the inverse of its natural log. For example the probability that 100 is prime is 1/ln_e 100 (which is a little more than .2) and the predicted probability improves as the size of the number increases.

The simplest expression for the prime number theorem is

p(x) → x /ln_e x as x → ∞

In other words the true number of primes i.e. p(x) as the number of primes occurring in the first x natural numbers, approaches the predicted level

x/ln_e x as x → ∞.

However though this in itself is a remarkable finding, it does not enable us to precisely locate the primes. We can make a good estimate of how many primes to find within any given (natural) number range but not exactly where they will occur.

In experiential terms, the qualitative equivalent to the prime number theorem unfolds through the specialised contemplative development of the Upper Middle Band.

Because of the continual exercise in the refined spiritual experience, whereby the differentiation of discrete phenomena becomes inseparable from their corresponding integration, one thereby discovers the practical means for satisfactory general regulation of behaviour (whereby the extent of primitive instinctive occurrences with respect to natural phenomena can be generally predicted and thereby safely managed). In this way contemplative peace is not easily threatened through the onset of such primitive disturbances.  

However this experience of itself does not enable one to predict in precisely what circumstances primitive instincts are likely to surface thus creating a bias to “play safe” by confining experience within customary - and thereby - predictable limits.

This parallels the ease of extracting primes when we confine our attention to the customary limited range of the earliest and most frequently used natural numbers!  

However combined with the prime number theorem, the Riemann Zeta Function (together with a very important assumption known as the Riemann Hypothesis) does provide an ingenious means of zoning in on the precise frequency of the primes.

However in discussing the Riemann Zeta Function we must first start with the function discovered earlier by Euler which connects in a remarkable manner the natural and prime numbers.

Euler’s Zeta Function is defined for any real number s > 1 by the infinite sum

ζ (s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ……….,

where it has a finite answer.1/1² + 1/2² + 1/3² + 1/4² + … = 1/{(1 - 1/2²)} X 1/{(1 - 1/3²)} X 1/{(1 - 1/5²)} X

1/{(1 - 1/7²)} X … ,

Euler then showed that this simple function based on the natural numbers has a profound relationship with the prime numbers for Euler also showed that

ζ (s) = 1/{1 – (1/2^s)} X 1/{1 – (1/3^s)} X 1/{1 – (1/5^s)} X 1/{1 – (1/7^s)} X …

where the product is taken over all the primes.

So for example when s = 2,

Then

So 1 + 1/4 + 1/9 + 1/16 + …  =  4/3 X 9/8 X 25/24 X 49/48 X ….  = p²/6

What is extraordinary here is how such a direct relationship can be shown as between a simple summation series on the one hand involving the natural numbers and a on the other hand a product series involving the primes.

What is also somewhat remarkable here is the nature of the result which directly involves p.

Indeed this is where we can suggest briefly our first bit of holistic mathematical understanding in explaining the nature of this result.

As we have seen in conventional mathematics, quantitative calculations are based on the default linear interpretation (i.e. where numbers are reduced in one-dimensional terms.

So for example the value of 1² = 1 (i.e. 1¹).

However we have already demonstrated how the true qualitative dimensional interpretation of 2 intimately involves the interaction of both linear and circular notions.

So when we geometrically express the two roots of 1 they will lie as equidistant points on the line diameter on the unit circle (of radius 1). So we can readily see therefore that the two points representing the roots can be connected by both a line (i.e. as diameter) and circle (as circumference).

Now when we consider the value of p, we can see that its value (which is constant) is intimately based on the relationship between the circumference (of any circle) and its line diameter.

So though explicitly the quantitative calculations involved in the Euler Zeta Function for s =2 can be carried out in standard linear terms, implicitly the holistic qualitative meaning of what is two two-dimensional is actually contained in the form of the result (i.e. based on p).

Put another way though standard quantitative meaning is linear (i.e. one-dimensional) in conventional mathematics based on the unambiguous separation of polar opposites, the unreduced qualitative meaning of two dimensions is properly circular (based on the complementarity of polar opposites).

Furthermore Euler was able to prove that for all even natural number dimensions i.e. where s = 2, 4, 6, 8,… in the zeta summation series and prime product formulae that corresponding powers of p would be directly involved in the resulting quantitative result.

And the clear holistic explanation of this is that the even numbered roots of any natural number always result in a complementary balancing (where half of the root expressions are exactly balanced by the other half).

x² + bx + c = 0

So we have even here - before consideration of the extended Riemann Zeta function - the first indicators of a hidden qualitative (i.e. holistic) meaning underlying the whole process.

Before however moving on to direct consideration, I will briefly deal with some findings that I made in another connection (which however has a distinct bearing on the holistic interpretation of both the Euler and Riemann Functions).

At one stage, I was very interested in the Fibonacci sequence i.e. that series of numbers

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,…….

Now this sequence of numbers is intimately related to the simple polynomial expression

x² – x – 1 = 0

More generally it is related to the expression

 (where b = - 1 and c = - 1)

Now if we start with 0 and 1 and add – b times the 2nd digit (i.e. 1) with – c times the 1^st digit we get 1 + 0 = 1 (which is the next term in the series).

Then by continuing in the same fashion always concentrating on the most recently acquired numbers), we generate the Fibonacci sequence.

So for example the next term is (1 X 1) + (1 X 1) = 2.

Then working with the most recently acquired digits (1 and 2) we get 2 + 1 = 3, which is the next term and so on.

What is also fascinating about this series is that we can approximate the value of its leading root through generating the ratio of successive terms which steadily improves as the number of terms increases.

The leading root of x² – x – 1 = 0 is 1.618033… (i.e. phi).

So for example if we approximate its value through taking the last two values listed above we get

987/610 = 1.618032… (which is already correct to 5 decimal places).

However what is remarkable is that in principle the same process used to generate the Fibonacci sequence can be used with respect to any polynomial expression thereby generating a unique series of terms.

The interesting question then arises as to whether the natural number series can be generated in this manner!

Well the answer is yes as it is related to x² – 2x + 1 = 0!

So in this case b = -2 and c = + 1.

Therefore starting again with 0 and 1 we proceed to generate further terms by adding twice the 2^nd (most recent term) to – 1 times the preceding term

So therefore the next term is 2 – 0 = 2; then the next is 4 – 1 = 3; the next is 6 – 2 = 4 and so on.

We thus have the natural number series 0, 1, 2, 3, 4, 5, 6, 7,……

What is also fascinating is that the equation to which it is related is (x – 1)(x – 1) = 0.

So from a holistic mathematical perspective, this directly implies the linear approach (where polar opposites in experience are both treated in a positive direction).

To briefly illustrate this let us consider the interpretation of the movement of time which is considered to move forward (in a positive direction) in experience.

Actual human experience is made up of the interaction of both external and internal poles. So in linear terms with respect to the external frame (considered separately) time moves in a positive direction; likewise with respect to the internal frame (considered separately) time also moves in a positive direction. However if we were to treat these poles as complementary opposites, then time would be seen to be paradoxical moving relatively in both a positive and negative direction. 

So we can see here in the nature of the natural number series a profoundly interesting connection with the holistic nature of linear (one-dimensional) understanding. And of course in quantitative terms the natural numbers are generally represented as lying on a straight line!

We can also see from the natural number series that we do indeed approximate the roots (= 1) by taking the ratio of successive terms and that this steadily improves as we move to higher terms.

Even more fascinating holistic connections can be made however through looking at other series.

For example consider the simple equation x² = 1 which qualitatively is the basis - as we have seen - of two-dimensional understanding.

Alternatively written this is x² – 1 = 0 where in terms of the general equation mentioned b = 0 and c = - 1.

Therefore again starting with 0 and 1 we generate its unique number series by adding 0 times the most recent term with 1 times the previous term.

So in this way the next term will be 0 then followed by 1 and then 0 in an alternating fashion.

So the series is 0, 1, 0, 1, 0, 1, 0, 1,…..

So it is most interesting how we have produced a series containing just 1 and 0.

Now the roots of this equation are + 1 and – 1.

However when in standard terms we try to extract an approximation for the principle root using successive terms, the process seems to be meaningless for we get 0/1 = 0 or alternatively 1/0 = ∞.

However when we now interpret these results - not from the linear perspective - but rather from the correct two-dimensional perspective (of complementary opposites) it makes great sense.

For here the ratio of 1/0 (or alternatively 0/1) in holistic terms intimately suggests the interaction of both linear and circular (or alternatively circular and linear) understanding respectively (which as we have demonstrated already exactly characterises the nature of two-dimensional understanding).

Also, rather than approximating (unambiguously) the same ratio through taking successive ratios (as with linear understanding) we get complementary opposites (i.e. 0 and ∞) which is what we would expect (with circular understanding).

So in this context, a simple quantitative procedure itself throws up relationships which require for meaning a qualitative holistic (rather than quantitative analytic) interpretation. 

Now the reason why the earlier polynomial expression generating the natural numbers (also of degree 2) generates a result which can be interpreted in linear terms, is due to the fact that it also contains a linear term in x (enabling interpretation to default as it were to the linear mode)!   

The Riemann Zeta Function has its starting base in the Euler Function. However whereas the Euler Function is confined to real values of s (with a finite value for s > 1), The Riemann Function is defined for complex values of s (i.e. where s = a + it).

Furthermore through some ingenious mathematics, Riemann was able to extend the domain of definition of his Function to cover all values of s (except for the one point where s = 1).

I am not even going to attempt to convey the analytic means by which Riemann achieved this complex mapping of his Function. Rather I will concentrate on some of the holistic implications of this mapping (which are neglected in conventional discussion).

I have already commented on the extremely important fact that - rather than just one possible explanation of mathematical reality based on default linear (one-dimensional) qualitative interpretation - a potentially infinite set of possible explanations exist (using the logical systems based on the corresponding real dimensions involved).

Furthermore I have shown that when the dimension is imaginary rather than real that this enables an infinite set of quantitative results for the same mathematical relationship.

Thus when we employ complex dimensions in mathematical expressions (combining both real and imaginary aspects) we open up the possibility of obtaining a variety of differing qualitative interpretations of the same (quantitative) result or alternatively a variety of differing quantitative outcomes with respect to the same qualitative procedure.

Because the very notion of an imaginary number embodies (in a hidden manner) the alternative (both/and) logical system of a circular kind, when we embody complex numbers (as dimensions) we must use a radial method of interpretation that combines both quantitative and qualitative appreciation.

So we cannot hope to properly understand the nature of the complex mapping of the Riemann Zeta Function without maintaining a close connection as between (analytic) quantitative results and (holistic) qualitative type appreciation.

Let me illustrate this issue with respect to one specific problem of interpretation.

When we adopt the standard linear method of interpretation the when s = - 1 in the Zeta Function, that the resulting series diverges to infinity. Thus from the Euler perspective the Function would not be defined for this value.

For in the Euler formulation, when s = - 1

ζ (s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ……….,

Thus ζ (- 1) =  1/1^-1 + 1/2^-1 + 1/3^-1 + 1/4^-1 + ……….,

= 1 + 2 + 3 + 4 +…….

So from the linear perspective, the sum of this series does indeed diverge to infinity.

However, by an ingenious process called analytic continuation, the domain of definition for the Riemann Function can be extended to all values of s ≠ 1.

For example a very important special relationship - called the Riemann functional equation - can be used to calculate the Riemann Function for all negative values of s.

And using this equation it can be shown that

ζ (- 1)  = - 1/12.

So clearly there is a problem here to explain as to how we get two very different results for the same quantitative expression.

Now the remarkable answer to this question is that - hidden in the very form of Riemann’s functional equation - is the means by which the standard linear interpretation that works for s > 1 is transformed into a circular interpretation where negative values of s are involved.

Quite simply, the very (qualitative) method of interpretation that must be used to meaningfully interpret quantitative results in the Riemann Zeta Function, for negative values, must be changed from linear to circular.

So now instead of interpretation through the standard default dimension of 1, we interpret expressions with respect to the actual qualitative dimensions (used in the expression).

This is easiest to explain for the even number dimensions – 2, - 4, -6, - 8, etc. which result in the so-called trivial zeros for the corresponding zeta expressions.

So for example using the Riemann functional equation,

ζ (- 2) = 0;

Now once again this clashes with the standard linear interpretation.

Here  ζ (- 2)  =  1/1^-2 + 1/2^-2 + 1/3^-2 + 1/4^-2 + ……….,

= 1² + 2² + 3² + 4² +…….  which again clearly diverges to infinity.

However the key to realisation of what is happening in the Riemann formulation is - not the standard linear interpretation based on 1 as dimension but rather - the qualitative circular interpretation with respect to – 2 as dimension.

Now when we used the positive dimension of 2 we saw that the result is intimately related to p (i.e. p²/6).

So the form of the result here entailed the ratio of circle and line i.e. circular circumference to line diameter.

In holistic terms, this entails the relationship of circular to linear understanding (conveyed in a reduced linear fashion). Once again the very reason why such rational understanding seems so paradoxical is because we are trying to convey a logic that is properly appropriate to a (formless) intuitive comprehension through a linear form of rational expression!

In psychological terms the very notion of negation - as exemplified by “the dark night of the soul” - is the means by which one moves from phenomenal (rational) understanding at a conscious level to formless (intuitive) awareness in unconscious terms.

So by switching the dimension from + 2 to – 2 appropriate understanding now becomes directly intuitive (which is empty in quantitative terms).

One way of interpreting the quantitative result for ζ (- 2) where the quantitative value = 0 is in terms of combining complementary opposites for each term.

So 1² would be interpreted in perfect complementary terms (where opposites exactly cancel out) as + 1 – 1, 2² as 2 – 2, etc.

We have already seen how for each of the positive even values for s, that the resulting quantitative value is intimately related to p. By the corresponding means of switching from linear (rational) to true circular (intuitive) recognition, the zeta expression (using these negative even values for s) = 0.

The significance of the trivial zeros i.e. for s = - 2, - 4, - 6, - 8 etc. is directly related – not to the secondary primes – but rather to the original primes 1 and 0.

As I have already stated in psychological terms, qualitative understanding with respect to the even natural number dimensions takes place during the specialised contemplative development of the Upper Middle Band.

Here both positive and negative dimensions are very closely related in experience. The positive dimension is revealed through the phenomenal intellectual ability to structure reality in an increasingly refined manner (based on the corresponding form of the roots associated with these numbers). The negative dimensions then entail the erosion of these structures (strictly the erosion of lingering phenomenal attachment to such structures) resulting in purely spiritual intuitive awareness (which is empty = 0 in phenomenal terms). 

So the direct psychological value of such development is that it further enhances that basic state where differentiation of discrete phenomena and overall holistic integration are so closely related. And one of the considerable benefits of such a state is that it provides a general means of healthily predicting and managing the primitive instinctive desire (which as the basic requirement for all phenomenal experience to take place can - and should - never be eradicated).

So though “temptation” never fully ceases it can thereby be more successfully controlled and integrated with Spirit so where it no longer threatens a continuous state of contemplative awareness.

However I have already pointed to the limitations of the over specialised contemplative state which may require considerable withdrawal from worldly involvement to be sustained. In other words where the emphasis is on continual erosion of attachment to the forms generated through higher level states (i.e. dimensions) then it can prove very difficult to get a foothold in active affairs.

So R1 - which we are dealing with - is still very closely related to (specialised) contemplative awareness.

The corresponding significance of the trivial zeros for the prediction of the primes is that it can make some (limited) contribution to the overall process of predicting the number of primes within a given range of natural numbers.

Before leaving this section we will make a number of other holistic observations.

I have already mentioned Riemann’s functional equation which in effect enables true circular interpretation (with respect to negative dimensions) from knowledge of corresponding positive dimensions according to linear interpretation. 

It might help in what follows to provide one accessible version of this formula (taken from the P. 147 of the excellent book by Derbyshire “Prime Obsession” on the Riemann Hypothesis)

This is

ζ(1 - s) = 2^{1 – s} p^{– s} sin {(1 – s)p/2} (s – 1)! ζ(s) where the angle for which sin is calculated is measured in radians.

So starting from a positive value of s for the zeta function on the R.H.S. (where s > 1) we can obtain a corresponding value for a negative value of s on the L.H.S.

For example to calculate the first of our trivial zeros i.e. for s = -2 = (1 – 3), we must start with the value of ζ(3) on the R.H.S.

Thus     

ζ(1 - 3)  = 2^{1 – 3} p ^{– 3} sin {(1 – 3)p/2} (3 – 1)! ζ(3)

Thus ζ(- 2) = 2 ^{– 2} p ^{– 3} sin {(– 2)p/2} (2)! ζ(3)

= 1/4p ³ sin {– p} 2 ζ(3)

And because sin {– p}  = - sin p = 0, the value of ζ(- 2) = 0.

So even in quantitative terms we can see how the negative value here for s is based on a circular transformation (though obtaining the relevant sine value) of the original zeta expression (with a positive value of s).

Many other interesting observations can be made.

When we confine ourselves to real values of s, a direct transformation takes place where the value of the zeta expression for the positive value of s on the R.H.S. is subject to linear interpretation (in terms of the default dimension of 1) and the value on the L.H.S. for the newly derived zeta expression subject to circular interpretation (in terms of the qualitative meaning of its actual dimension)

When s > 1 for the zeta expression on the R.H.S., the value of s on the L.H.S will thereby be negative. However when 0 < s < 1 on the right, likewise 0 < s < 1 on the left.

Of particular interest - which is directly tied up with the Riemann Hypothesis - is the zeta expression is the case for which  s = ½ on the R.H.S.

This implies that the derived zeta expression on the left will also be ζ(½).

The extremely important implication of this is that s = ½ is the only real value which can ensure that there is a direct correspondence as between the linear interpretation for ζ(½) on the R.H.S and the corresponding circular interpretation appropriate for ζ(½) on the L.H.S.

And as we have already seen in our earlier discussions this correspondence of linear and circular interpretations is fundamentally inherent in the very notion of a prime number.

I have already dealt with the psychological interpretation of the dimension ½, when dealing with the Pythagoras Dilemma. What it simply entails is that for successful reconciliation of rational (dualistic) and intuitive (nondual) understanding, opposite (real) polarities must be maintained in equal balance with each other.  

For example if we take these polarities as the external (objective) and internal (subjective) aspects of experience, avoiding rigid attachment to either pole - which would be inconsistent with pure nondual (intuitive) awareness - requires that both be kept in perfect balance.

Thus if we consider these polarities in linear terms - as representing one unit - the midpoint will be at .5.

In geometrical terms therefore starting at this midpoint we can circumscribe a circle through both extreme points of the line (thereby reconciling line and circle).

Though we will have a lot more to say about the Riemann Hypothesis, this in essence is the basic reason why the Hypothesis must hold. In psychological terms, the full reconciliation of primitive instinctive behaviour - projected into consciousness in an imaginary fashion - with naturally ordered experience requires the perfect balance of opposite real polarities (thus avoiding undue phenomenal attachment to either pole). Only in this way can phenomenal involvement remain fully consistent with the (empty) contemplative awareness of reality.

In like manner the full reconciliation of prime numbers with the natural number system requires that all non-trivial zeros lie on the real line = .5.    

Another fascinating observation is that the only point (argument) on the complex plane, where the Riemann Zeta Function remains undefined, is for s = 1.

Again a very simple holistic explanation can be given for this fact.

Clearly when using standard linear interpretation, the Zeta Function diverges for s = 1. In accepted mathematical language therefore a simple pole exists at this point.

ζ (1) = 1/1¹ + 1/2¹ + 1/3¹ + 1/4¹ + ……….,

This results therefore in the well known harmonic series

1/1 + 1/2 + 1/3 + 1/4 + ………., which diverges.

Now, in all other cases where the Euler Function is undefined, the domain of definition can be extended by essentially switching to a qualitative interpretation in terms of the actual dimensional value of s.

However clearly when we attempt to do this for s = 1 we are left with the same qualitative interpretation (i.e. linear).

So the Zeta Function (where s = 1) is – by definition - the only case that resists an alternative qualitative dimensional interpretation.

We will now look at the interesting case where the Riemann Zeta function is defined for odd natural number values (firstly where s ≥1).

Now, unlike the even number values, the zeta expressions for s =1, 3, 5, 7, 9,.. do not result directly in closed form expressions (based on p).

The holistic explanation for this relates to the fact that full complementarity is not possible for odd valued dimensions. This can be readily seen through extracting the appropriate roots of these numbers.

For example if we take the 3 roots of unity we get 1, (- 1 + i√3)/2 and (- 1 - i√3)/2) which cannot be broken down into two exactly matching equal sets.

However we can see a fascinating picture emerge when we look at the (extended) Zeta Function for the corresponding negative values for s.

For example though ζ(1) is undefined, ζ(-1) based on a distinct dimensional value ≠ 1, has an alternative interpretation = -1/12.

Whereas the values of ζ(1) for odd s > 1 are invariably irrational (based on linear interpretation), the corresponding values of ζ(s) for natural s ≤ - 1, are invariably rational (based on circular interpretation).

So what is illustrated here is that a circular (qualitative) interpretation of a mathematical expression can result in a distinctive quantitative result!

Furthermore another distinctive pattern is in evidence with the negative valued expressions that represents a form of broken complementarity.

This is evidenced by the fact that the sign keeps switching as we proceed through the negative odd values for s.

So ζ(- 1) = - 1/12;  ζ(- 3) = 1/120; ζ(- 5) = -1/252; ζ(- 7) = 1/240 etc.           

There are other interesting values for ζ(s) that we could explore.

For example the important value of ζ(.5) can be calculated and is generally given as – 1.4603545…

Indeed it might be instructive finally to illustrate how this can be achieved by a sort of “mathematical trick” that again has important qualitative ramifications.

Now clearly in conventional linear terms

ζ(.5) is undefined as

ζ (.5) = 1/1^1/2 + 1/2^1/2 + 1/3^1/2 + 1/4^1/2 + ……….,

which diverges to infinity.

However another function called the eta function can be defined – based on the zeta – through organisation of terms in an alternating complementary + and – fashion.

So the zeta function is

ζ(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ……….,

And the corresponding eta function is

η(s) = 1/1^s - 1/2^s + 1/3^s - 1/4^s + ……….,

Now η(s) can be expressed as ζ(s) – 2 X (1/2^s + 1/4^s + 1/6^s + 1/8^s + …)

By dividing (1/2^s + 1/4^s + 1/6^s + 1/8^s + …) by 2^s

We can express this as ζ(s)/2^s

Therefore

η(s) = ζ(s) - 2ζ(s)/2^s = ζ(s) - ζ(s)/2^s-1

= ζ(s){1 – 1/2^s-1}

Alternatively we could express this as

ζ(s) = η(s)/{1 – 1/2^s-1}

Therefore by calculating the value of the alternating series for values in the range of 0 < s < 1 for the convergent series η(s), we can thereby find a corresponding value for ζ(s).

Though η(.5) converges very slowly, it has the value .60489864…

Therefore through our simple derived relationship we get

ζ(.5) = - 1.40603545..

So one again we have somehow managed to derive a coherent finite answer (through a simple bit of series juggling) that otherwise provided an infinite result!

However what has really happened is that we have switched from a linear to a circular interpretation.

The clue to this is that the original series - which diverged to infinity - was replaced by an alternating series (based on the complementarity of opposite i.e. odd and even terms). In this way we have switched from a linear logic (which is inherently defined solely for the finite domain) to a circular logic (defined for the infinite).

Indeed we can perhaps more easily demonstrate the pitfalls exposed by this juggling of an infinitely divergent series (in linear terms) by concentrating on the simple case where s = 1.

Here the eta function = 1 – 1/2 + 1/3 – 1/4 + 1/5 -…. = ln 2.

And this can be obtained from the zeta function by subtracting 2{ζ(1)/2)} from ζ(1) = 0.

Thus, by this logic ln 2 = 0.

However what has really happened here is that we have switched to the circular notion of 2 (where opposite positive and negative real polarities are combined). So from this “higher” perspective they do indeed form a unified dimension (which is revealed when we once again extract the two roots to get + 1 and – 1 in linear terms).

So ln 2 through this transformation is ln 1 (interpreted through a different logic) with the answer having a strictly qualitative (rather than quantitative interpretation).

And of course ln 1 = 0!

In this way we can give a coherent meaning to what otherwise appears a nonsensical result (from a linear perspective).

Also it is worth pointing out that whereas linear type series (where all terms are positive) diverge to infinity, that the finite values relating to corresponding alternate series that do converge will intimately depend on the precise arrangement of terms in the alternate series.

So strictly the complementary series

1 – 1/2 + 1/3 – 1/4 -……  can be given an infinite set of possible values (depending on the precise complementary ordering of terms).

So for example, whereas

1 – 1/2 + 1/3 – 1/4 -……    = ln 2

1 - (1/2 + 1/4) + 1/3 – (1/6 + 1/8) + 1/5 -….   = ln2/2.

and

1 + (1/3 + 1/5) – 1/2 + (1/7 + 1/9) – 1/4 +…  = 3ln2/2

Intriguingly, this same procedure can be used to calculate ζ(0).  

In fact it can be shown that ζ(0) = - .5 (thus bearing comparison as a sort of reverse complementary example of the Riemann Hypothesis where all non-trivial zeros lie on the real line = ½).

In this case (where s = 0)

η(0) = 1/1⁰ - 1/2⁰ + 1/3⁰ - 1/4⁰ +…..

= 1 – 1 + 1 – 1 +………

Now η(0) = 0 (when we confine ourselves to an even number of terms).

However η(0) = 1 (when we confine ourselves to an odd number of terms).

The first result corresponds to a circular logic (of complementary opposites) and the second to a linear logic (of separate terms). 

So when we attempt to get an average (as simple mean) of these two results

 η(0) = (0 + 1)/2 = .5

As we have seen,

ζ(s) = η(s)/{1 – 1/2^s-1}

Therefore

ζ(0) = .5/{1 – 1/2^-1} = .5/{1 – 2¹} = - .5

So implicit in this very result - as with the truth underlying the Riemann Hypothesis - is a reconciliation of linear (1) and circular (0) understanding.

Indeed using Riemann’s functional equation we can throw fascinating light again on the reason why ζ(1) is undefined.

Its value is ζ(1 - s) is obtained from ζ(s) where s = 0.

Thus

ζ(1) = 2^{1 – s} p^{– s} sin {(1 – s)p/2} (s – 1)! ζ(0)

However – as we have seen ζ(0) represents a hybrid mix of both linear and circular interpretation.

Therefore the attempted derived value on the L.H.S. for ζ(1) therefore entails likewise a hybrid mix of both circular and linear interpretation (though complementary transformation).

However - by definition - ζ(1) represents a purely linear form (based on the default dimension of 1).

Therefore it cannot be defined in terms of a transformation (which entails both circular and linear aspects).