7. Riemann Continued

7. Riemann Continued

Radial Band - R2 Level: Introduction

As I have indicated previously, my approach to the Riemann Hypothesis is a (preliminary) radial interpretation which depends on exposure to the corresponding radial levels of experience.

Appropriate (unreduced) interpretation of prime numbers depends on the balanced interaction of two logical modes of interpretation which are linear and circular with respect to each other.

The linear is based on the clear distinction of polar opposites and is properly suited in any context for the analytic differentiation of phenomena; the circular, by contrast is based on the complementarity (and ultimate identity) of these same opposites and is directly suited for the overall holistic integration of meaning.

Prime numbers in a very special manner intimately embody the interaction of these two logical systems. However, because in conventional mathematics only the linear system is recognised, it gives a somewhat reduced interpretation. Not surprisingly the fundamental secrets of such numbers remain highly elusive when subjected to this partial - and limited - lens of understanding.

In experiential terms a considerable amount of psychological development is required before the mature interpenetration of both the linear and circular systems can take place. 

As we have seen at the Middle Band, of the overall spectrum of potential human development, specialisation of linear understanding takes place. So in a unique manner, rational mathematical understanding is based on the specialised understanding of the Middle Band.

However with the unfolding of the Higher Band – where it occurs – the development of refined intuitive understanding occurs and this has its own specialised development at the Advanced Higher Band. Such understanding lends itself to a distinctive form of qualitative understanding of symbols, which I term Holistic Mathematics.

Basically the seed of holistic mathematical understanding lies in the clear recognition that whenever multiplication (as opposed to the mere addition) of numbers takes place that a distinct qualitative – as well as quantitative change – is involved.

The fundamental limitation, as we have seen of conventional mathematical understanding, is that it simply reduces this qualitative transformation in quantitative terms.

So the value of 2 X 2 (which is properly 2² and thereby two-dimensional in nature) is given in a merely reduced quantitative manner as 4 (i.e. 4¹). So it reduces interpretation to the default dimension of 1 (where interpretation takes place in a merely quantitative manner).

Therefore - quite literally - such interpretation is linear (i.e. one-dimensional).

The purpose of Holistic Mathematics is to give a new qualitative meaning to all numbers as dimensions (which entails ever more refined intuitive type understanding).

However the most comprehensive type of mathematical understanding is radial where both quantitative and qualitative meaning is given to mathematical symbols. And this type of understanding is especially suited for the interpretation of prime numbers, which embodies both quantitative and qualitative aspects of meaning in a unique manner.

We have already seen that the linear (quantitative) and circular (qualitative) aspects of understanding can only be fully reconciled where neither separately exists.

However phenomenal understanding with respect to form requires that such separation necessarily takes place, where objects that can be quantitatively identified become to a degree separated from the (qualitative) dimensional background necessary to give them meaning.

However this separation entails a certain distortion with respect to both quantitative and qualitative aspects.

In spiritual experiential terms, this gives rise to momentary attachment to phenomenal events entering consciousness. These events originate in the unconscious, where dimensions and objects ultimately coincide. However in natural conscious terms, both aspects (objects and dimensions) gradually separate thus enabling phenomenal stability to be attained.

The problem then arises that recognition of both objects (through perceptions) and dimensions (through concepts), thereby inevitably leads to a degree of attachment due to volitional assent to the independence of those phenomena thereby posited. 

So the removal of such attachment requires that its roots be removed at source, through quickly obtaining a new empty state of spiritual contemplative equilibrium (whereby the original interdependence of objects and dimensions is once again restored). 

This entails that the opposite polarities of experience (internal and external) must be maintained in equal balance. In this way, any temporary attachment to the real manifestation of phenomena is thereby minimised.

Where perfect balance is maintained - though this always remains an approximate goal in dynamic experiential terms - no attachment resides. Attachment always requires allowing the internal or external aspect attain a relative degree of independence!

The imaginary aspect then relates to the projection from the unconscious of the holistic dimension (in indirect phenomenal terms).

Where balance is to be obtained this always happens in a complementary vertical manner (where positive and negative polarities balance out).

This can be given a simple psychological explanation.

When the holistic unconscious projects itself in a “higher” level fashion, some attachment is inevitable (due to the need for this holistic experience to be given a conscious form).

So a degree of indirect attachment thereby arises i.e. to the conscious symbols used to convey the - primarily - unconscious experience.

This then leads to a compensatory tendency whereby the “lower-level” instinctive unconscious likewise projects itself in a conscious manner.

Just as earlier a degree of indirect conscious attachment arises in relation to the holistic (spiritual) meaning arising, now again a degree of indirect conscious attachment arises in relation to the specific (physical) meaning conveyed in an instinctive fashion.

Indeed the very means of minimising such attachment (both “higher” and “lower”) is in the clear recognition of the necessarily complementary nature of both experiences.

Then when minimal (indirect conscious) attachment is involved to either form of vertical experience “higher” and “lower”, one can thereby more easily maintain that perfect balance at a “real” conscious level (where external and internal aspects of experience are harmonised without undue attachment).

In this way spiritual equilibrium can be continually restored in experience following slight temporary imbalance relating to the initial prime disturbances (both “high” and “low”).

Thus primitive disturbance is always necessary in allowing the basics of form to enter experience i.e. in providing both the prime holistic dimensions and building blocks of experience.

Paradoxically once the true vertical complementarity of (prime) dimensions and objects is experienced, then they can be increasingly separated in experience without undue attachment arising.

In this way one can increasingly seek to attain pure contemplative peace (i.e. spiritual zero or equilibrium) while remaining actively engaged in the world of phenomena (both “real” and “imaginary”).

Thus the “imaginary” (i.e. unconscious) nature of phenomena is clearly experienced through the recognition of their vertical complementary nature. The “real” i.e. conscious nature is experienced through recognition of their separate nature i.e. as wholes and parts.

So with due mastery of unconscious attachment (equality of wholes and parts) one can equally maintain mastery of conscious attachment (internal and external).      

    

From a radial perspective therefore the Riemann Hypothesis can be read in two ways

From the (conventional) quantitative perspective, it says something extremely important about the nature of prime numbers, requiring two distinct methods of logical interpretation. So the Riemann Hypothesis entails that the secret to understanding the (hidden) distribution of prime numbers lies precisely at the point where these two methods of interpretation are harmonised.

The problem is that when we look at actual prime numbers (as building blocks) this general distribution remains hidden.

Likewise when we look at the actual general distribution (without recourse to specific prime numbers) their actual specific existence also remains hidden.

From the equally important – though still largely unrecognised - qualitative

perspective, it tells us something extremely important about the nature of

prime i.e. primitive phenomena in experience and in complementary physical terms prime particles at the quantum levels of reality.

Now in the past the radial life was best exemplified by great contemplative activists who operated as superconductors of Spirit on behalf of their fellow beings.

Any indirect attachment to “imaginary” (unconscious) phenomena was thereby greatly minimised as a prerequisite for maintaining continual intense involvement with respect to “real” conscious activities.

This in turn required skilled recognition of the complementary nature of all

“primitive” disturbances “higher” and “lower”, thereby greatly facilitating the balanced interplay of the “real” polarities (external and internal).

Thus from a qualitative perspective, the generalised distribution of prime  

numbers requires contemplative mastery of experience (where phenomenal involvement of a primitive kind greatly ceases). This corresponds to what I refer to as the “Higher” and “Specialised Higher” Bands respectively.

However such contemplative mastery in the midst of active involvement i.e. the Radial Bands, requires the gradual separation of “higher” dimensions and corresponding associated “lower” objects in experience (which initially causes a temporary disturbance). Thus the non-trivial zeros here entail the ability to keep returning to spiritual equilibrium in the midst of growing active involvement with phenomena.

Thus contemplative involvement, in isolation from phenomenal activity, is largely based on the circular mode of interpretation entailing complementary opposites.

However both contemplative and active involvement requires that attention is duly given to both circular and linear understanding.

So to sum up, from a radial perspective the Riemann Hypothesis contains both a quantitative and qualitative interpretation (entailing two logical systems).

Because these two systems are inherent to the very nature of prime numbers the understanding of one aspect e.g. quantitative) cannot be properly undertaken in the absence of the other.

So once again true understanding of the Riemann Hypothesis can only be properly undertaken in a radial context.       

A Complementary Riemann Hypothesis

When we looked earlier at the number system we saw that an alternative complementary system exists (which forms the basis for holistic mathematical interpretation).

So to recap, the conventional number system - as the basis for quantitative interpretation – is defined with respect to the default first dimension leading literally to a linear (i.e. one-dimensional) interpretation.

So this number system (relating to the natural numbers) can be written

1¹, 2¹, 3¹, 4¹, …..

However a complementary alternative number system – as the direct basis for qualitative interpretation – can be defined with respect to a default number quantity (1) with varying number dimensions (as qualitative interpretations).

So this alternative natural number system can be written as

1¹, 1², 1³, 1⁴, …..

Once again the qualitative holistic dimension is obtained with reference to its corresponding (partial) root.

So for example the dimension 2 is obtained with reference to the square root of 1 = - 1.

So in holistic terms, this entails - with respect to qualitative interpretation - the negation of rationally posited conscious form.

This indeed is the basis for direct intuitive - rather than rational - understanding.

Thus when we combine the 1^st and the 2^nd dimensions in qualitative interpretation we use both linear (one-dimensional) and circular (two-dimensional) interpretation.

And whereas linear interpretation is based on an either/or sequential logic, the circular interpretation is based on both/and simultaneous logic (which seem paradoxical in terms of the former).

We have illustrated the nature of this logic with illustration to right and left turns on a straight road. If we define the sense of direction e.g. through heading “up” the road, then left and right have a clear unambiguous (either/or) meaning.

Likewise if we now alter the sense of direction i.e. by now heading “down” the road, then left and right again have a clear unambiguous (either/or) meaning.

However if we now consider both reference frames simultaneously - as befits holistic intuitive understanding - then left and right have a paradoxical (both/and) meaning so what is left is also right and what is right is also left.

Now the relevance of all this is that actual understanding of all mathematical symbols is necessarily based on the interaction of polar opposites (such as external and internal) in experience.

So when we define mathematical meaning in linear terms (where polar opposites are treated as relatively independent), actual symbols appear to have an unambiguous (either/or) meaning. This corresponds to standard analytic rational interpretation.

However when we define mathematical meaning in corresponding circular terms (where polar opposites are treated as relatively interdependent) symbols now appear to have a paradoxical (both/and) meaning. This corresponds to alternative intuitive interpretation.

Though in dynamic experiential terms both of these logical systems are necessarily involved in all mathematical understanding, conventional mathematics attempts to give a merely reduced interpretation based solely on the linear mode.

Of course more intricate forms of interpretation apply to even “higher” dimensions though in essence the same point holds i.e. that in dynamic terms a comprehensive interpretation of mathematical symbols entails varying interactions of two logical systems (linear and circular) that are incompatible in terms of each other.

Now the (standard) Riemann Zeta Function is defined for s = ½ + it (where t is a real number) and given by the infinite sum

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

The question then arises as to whether - as with the number system - an alternative complementary form exists (with direct relevance for holistic mathematical interpretation)!

Remarkably this can be shown to be the case through use of the simple expression that defines the various roots of unity

So xⁿ = 1; thus for example when n = 2 we derive the two roots x = 1 and x = -1 which forms – as we have seen - the basis for the simplest type of holistic understanding that combines both linear (one-dimensional) and circular (two-dimensional) logic.

xⁿ = 1 can of course be written as xⁿ - 1 = 0.

Dividing by x – 1 we get x^{n – 1} + x^{n – 2} + x^{n – 3} +…….1 = 0

Then multiplying by x we get  xⁿ  + x^{n – 1} + x^{n – 2} +…….x = 0.

Alternatively we could write this (through reversing terms)

x¹ + x² + x³ + x⁴ +…. + xⁿ = 0 

Going back to the Riemann Zeta Function, it can be written

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

Now if we let x = -s, then this becomes

ζ(s) = 1^x + 2^x + 3^x + 4^x + ……….,

just as

1^x + 2^x + 3^x + 4^x + ……….n ^x, = 0 for selected values of x = a + it (where a = -1/2) and n → ∞

likewise

x¹ + x² + x³ + x⁴ +…. + xⁿ = 0  for selected values of x (i.e. successive roots of 1) where x = a + it (i.e. successive roots of 1).

The other value for which the expression = 0 is for x = 0 (which in holistic terms implies pure intuitive awareness).

Now it might be instructive here to explain a little regarding the holistic interpretation of such quantitative root values.

For example we will look at the very important case of four dimensions (as conventional reality is understood in terms of four dimensions).

Now the first root - which we have eliminated from consideration - is where x = 1, corresponding holistically to linear (one-dimensional) understanding.

Now the other roots represent progressively more refined interpretations of reality (that cannot be accommodated in a merely rational interpretation).

Then - 1 as we have seen corresponds to the negation of form leading to pure intuitive awareness.

The third root i then corresponds to imaginary - as opposed to real - rational interpretation which amounts to an indirect rational means of conveying the (unconscious) holistic nature of understanding.

Finally – i represents the negation of this indirectly projected nature of form leading to an even more refined holistic intuitive awareness.

Now one may ask what on earth has this to do with prime numbers.

Well in holistic terms the psychological equivalent of prime numbers relates to primitive instincts (where “real” conscious and “imaginary” unconscious are combined). In this way the underlying nature of what is primitive can be seen to be literally complex (in qualitative terms) just as the Riemann Zeta Function points to the corresponding complex nature of prime numbers (in quantitative terms).

Now coming to terms in a general manner with primitive behaviour requires that firstly that the “real” conscious and “imaginary” unconscious be successfully differentiated from each other and then negated (thus enabling their successful integration through elimination of any residual attachment).

This precisely is what progression through these four dimensions entails.

So the first root 1 represents the successful positing (i.e. differentiation) of rational form.

However, integral intuitive awareness of a truly holistic nature, requires the corresponding negation of such form (- 1).

Then imaginary rational understanding i.e. where true holistic awareness can be indirectly accommodated to rational form, requires the positing (i.e. differentiation) of such form (i.e. the third root i).

Finally corresponding true integral awareness of such imaginary form requires its corresponding negation (i.e. the 4^th root – i).

Thus from a contemplative perspective, primitive instinctive behaviour becomes properly unravelled through this process of both successfully differentiating and then integrating both conscious and unconscious.

In fact these stages properly correspond with the first of the so-called “trivial” zeros. For example the 2^nd ζ(-2) corresponds to the pure negation of “real” form , the 4h to the negation of “imaginary” form (both of which imply pure intuitive emptiness i.e. 0 in qualitative terms).

Thus the true significance of the “trivial” zeros is that they correspond directly to a qualitative - rather than a strict quantitative - interpretation.

Though the “non-trivial” zeros do indeed have a more established quantitative interpretation, they likewise combine a - yet - unrecognised qualitative meaning!

Some time ago I used these four roots of unity as a basis for generating 24 personality types (16 of which are basically identical to those identified in the well-known Myers-Briggs Typology).

Quite simply the four roots can be permutated in 24 different ways. Now if we identify personality as a composite mix of these four dimensions, each personality can be seen as representing a unique configuration (representing various strengths) of these four dimensions.

Thus the contemplative process of both differentiating and integrating the four dimensions would lead to a much more balanced and refined personality where varying aspects of all 24 types would be present in varying degrees.

Furthermore because physical and psychological aspects of reality are complementary, this also had deep applications as to the fundamental nature of physical reality.

In other words we likewise have 24 “impersonality” or material types with respect to the fundamental nature of matter.

Indeed this configuration of 24 has already appeared has very important both in Group Theory and in one of the approaches to String Theory. So at the very least this provides a way of reconciling conventional notions of 4 physical dimensions with the much larger number appearing in String Theory.

Thus a dimension at the level of string reality - here really implies a configuration of the - still - entangled four dimensions (that become clearly differentiated at the macro level of reality).

However we can go much further in identifying the significance of this complementary formulation of the Riemann Hypothesis.

Recently, I considered that a considerable value would pertain to obtaining the mean absolute value of the sum of roots for various prime values of n.

It quickly became clear to me that some remarkable patterns were unfolding.

The varying roots can be broken down into their (real) cos and (imaginary) sin values.

For example when n = 3 the three cos values are 

1, -1/2 and  -1/2 respectively.

The mean absolute result of the sum of these 3 values = 2/3 = .6667

Likewise for n = 3 we have three sin values 0, .866 and -.866

Again the mean absolute result of these values = .5773.

So the mean cos is greater here than the corresponding sin value.

Further exploration for higher prime values of n confirmed this pattern.

However what becomes quickly apparent is that both mean values converge (from opposite directions) on a central value which is 2/p (.636619772).

For example when n = 127 (the highest prime value for n that I have computed),

the mean absolute result (for the 127 cos values) = .636636004 (correct to 4 decimal places).

The mean absolute result (for the 127 sin values) = .636587311 (again when rounded up correct to 4 decimal places).

So we could sum empirical evidence up in the following manner

p

∑ |Cos{2kp/p}|/p   ~  2/p  (as p increases)

k=1

In the limit as p → ∞,

p

∑ |Cos{2kp/p}|/p   =  2/p 

k=1

Likewise

 p

∑ |Sin{2kp/p}|/p   ~  2/p  (as p increases)

k=1

In the limit as p → ∞,

p

∑ |Sin{2kp/p}|/p   =  2/p  (as p → ∞)

k=1

   

As we have seen the simplest version of the prime number theorem (for the general distribution of primes) can be expressed as

n/log n as n → ∞

Remarkably it can be shown that 2/p  is the imaginary counterpart of this

For e^2pi  = 1

Therefore by raising both expressions to the power of i,

e^-2p  = 1ⁱ

and e^-p/2   = iⁱ

Taking logs

- p/2 = i log i

p/2 = - i log i = log i/i

Therefore

2/p  = i/log i

So we have here the imaginary counterpart to the prime number theorem

where i/log i is reached in the limit where n → ∞.

However the fascinating patterns do not end here.

As we have seen the mean absolute result for the real cos values - relating to the p roots of unity - always exceeds that of 2/p (with the approximation steadily improving as p increases).

Likewise the mean absolute result for the imaginary sin values is always less than that of 2/p (with again the approximation steadily increasing as p increases).

However when we compare the absolute ratio of the difference of cos and sin results respectively (from 2/p) then a remarkably coherent pattern emerges.

Quite simply, as the value of p increases this absolute ratio converges ever closer to .5.

Indeed in the limit, when p → ∞, this ratio = .5.

Expressing this more formally

P                                      p

[∑ |Cos{2kp/p}|/p - 2/p]/[∑ |Sin{2kp/p}|/p - 2/p] → .5 as p → ∞

k=1                                  k=1

For example when (as in our earlier illustration) p = 3,

P                                      p

[∑ |Cos{2kp/p}|/p - 2/p]/[∑ |Sin{2kp/p}|/p - 2/p] 

k=1                                  k=1

= |.6667 - .636619772|/|.5773 - .636619772|

= .03008/05932 = .50708

So here, even with the value of p so low the ratio already approximates well to .5.

However when we now calculate this value for a much greater value e.g. p = 31, the approximation to .5 significantly improves.

Here

P                                      p

[∑ |Cos{2kp/p}|/p - 2/p]/[∑ |Sin{2kp/p}|/p - 2/p] 

k=1                                  k=1

=  |.636892277 - .636619772|/|.66074831 - .636619772|

= .0002725046/.00005449414 = .500030805

So it seems that .5 plays a complementary role in this formulation to that in the standard Riemann Hypothesis.

Indeed a further (more speculative) inference can be made regarding the rate at which the value of .5 is approximated (as the value of p increases).

To do this we concentrate on the residual value of the result (by subtracting .5).

It would then same that the corresponding residual for a larger value p₁ approximates the square of the ratio of the smaller p to the higher p₁

So we can approximate the resulting residual from .5 by multiplying the result for p by (p/p₁)².

An even better illustration of this relationship can be seen from looking at the initial deviations of the absolute mean cos (and sin) values from 2/p.

If we start with an initial prime value p, and look at the deviation (of absolute mean cos value for p roots) we can thereby closely approximate the corresponding deviation for any higher prime number (i.e. p₁) .030046893

For example starting with the simplest case (p = 2 is a special case) of p = 3, the absolute mean average for the 3 cos values (1, .5 and .5) = .66666666 and the deviation of this from 2/p (.636619772)  = .030046893.

Now if we take another prime number p₁ - say - 127, we can calculate the corresponding deviation (of the absolute mean cos value of the 127 roots of unity) from 2/p by multiplying the earlier deviation by (p/p₁)²

= .030046893 * (3/127)²   = .00001676619983

This compares with the true value of .000016231648.

To improve this approximation we next divide by 1 + earlier deviation

= 1 + .0300488693 = 1.030048693.

Dividing in this manner gives a better approximation of.00001627712286.

We can also use a variant of this approach to likewise calculate deviations on the sin values.

Here the first part is the same i.e. multiplying by (p/p₁)².

However to improve the approximation, we now divide by 1 + (original deviation/4).

So here our first approximation = .059269503 * (3/127)² = .003307244882.

The next approximation = .03307244882/(1 + .014817376) = .000032589557.

This compares with correct value of .000032464368. 

Though the approximation here does not perhaps seem that accurate, it has to be borne in mind that it greatly improves (i) as the size of p increases and (ii) the greater is the distance of p₁ from p.  

So for example when p = 61 (and p₁ = 127), the prediction of the absolute mean deviation (for the 127 roots) is accurate to 6 significant figures.

So we have in effect a parallel approach to the Riemann Hypothesis. Here - on the assumption of its validity - we can move from the general approximate prediction of the number of primes (within the corresponding range of natural nos.) to an ever more precise prediction.

Here in corresponding fashion we move from an approximate general prediction of the absolute mean value of prime roots (to an ever more precise prediction 2/p = i/log i) based on the assumption that in the limit the absolute ratio of mean cos to sin deviations = .5.

Thus in principle we can calculate for any finite value the absolute sum of prime roots (with respect both to mean and imaginary parts) without having to actually measure these roots.

Non-Trivial Zeros - Significance of Complementary Imaginary Dimensions

In the last Chapter, I explained how the functional equation provided an ingenious means of converting from the linear to the circular method of interpretation with respect to the numerical values arising in the Riemann Zeta Function.

So for example if we start with the well known value for ζ(s) of s = 2, we obtain the      

value of p²/6. This corresponds to the linear method of interpretation (with an intuitively acceptable numerical result in conventional terms).

However from the functional equation we can also calculate ζ(1 - s), which in this case is ζ(-1). Now the numerical answer here i.e. -1/12, does not have an explanation in conventional (linear) terms, as the sum of the series diverges to infinity. However it does have an alternative qualitative circular explanation. Here, the result in a sense corresponds philosophically to the degree of rationality involved through the refined intuitive type of understanding required to holistically interpret such a result.

And as we have seen, wherever an expression is defined with respect to a power (i.e. dimension other than 1), an alternative circular numerical explanation is given through the functional equation.

Thus the only point where the Riemann Zeta Function is undefined is for s = 1, for here the circular (dimensional) identity - by definition - does not exist and as the sum of the series diverges in conventional (one-dimensional) terms, it thereby has no alternative circular explanation.

So we can quickly move on from this to easily demonstrate that the only value of s, for ζ(s), which ensures that both the circular and linear results are identical is where s = .5.

And this fundamentally provides the holistic explanation of why all imaginary values for s must fall on the real line = .5, i.e. because only here is the inherent identity of both (analytic) linear and (holistic) circular interpretations maintained. And remember that it is this very identity which inherently defines the fundamental nature of a prime number!

However having said this, the value for s, defined only for its real part (=.5) does not generate a zero of the zeta function. 

So it is only when this real part (a = .5) is also combined with an imaginary part (it) for values of s that non-trivial zero values for the function can arise.

So this means that - literally - a more complex version of the transformation from linear to circular identity is given (in terms of the functional equation).

So for non-trivial zeros the zeta function is of the form ζ(.5 + it).

Thus if we start by giving a linear identity to the real part (.5) on the L.H.S. of the equation, this is thereby balanced on the R.H.S. by an identical circular identity.

Now - fascinatingly - if the linear identity of the imaginary part on the L.H.S. is (+ it), then the corresponding circular identity on the R.H.S. is (– it).

So the task therefore is to explain in holistic terms why linear and circular aspects of imaginary understanding are positive and negative with respect to each other!

It is here that the dynamics of contemplative type attainment are so   

valuable, as a marked complementary pattern as between “high-level” and “low-level” experience comes sharply into evidence.

If for example the “high-level” is experienced in transcendent holistic fashion (i.e. in refined dimensional terms empty of specific content), the corresponding “low-level” experience – often referred to as temptation - is of instinctive physical content (that is so intimate and specific that it seems empty of any wider dimensional framework).

So we move here clearly from one extreme to another with respect to the experience of holistic dimensions (on the one hand) and specific objects (on the other).

Though the intuitively inspired holistic dimensional experience e.g. of refined spiritual archetypes, is largely of the paradoxical circular variety based on the ceaseless interaction of complementary opposites, the very recognition of this in intellectual terms necessarily requires a detached cognitive aspect (based on linear recognition).

So where residual attachment relates to this “higher-level” understanding in cognitive terms, this creates an imbalance with respect to the unconscious that creates a complementary “lower-level” disturbance where residual attachment will now apply to specific content. In other words if one gives “superior” spiritual recognition to the holistic dimensional aspect, this creates an imbalance in terms of granting equal physical recognition to the “inferior” specific aspect (attaching intimately to objects).

So while any residual attachment is being eroded - representing a failure to fully harmonise both linear and circular understanding - continual interaction as between the “higher” and “lower” (imaginary) unconscious will take place. Then with full reconciliation of both “higher” and “lower”, any dualistic distinction as between both is eliminated.  

So the dimensions of experience (and their corresponding objects) are literally circular and linear with respect to each other.

We have likewise seen that number qualities (as dimensions) and number quantities (as numerical objects) are likewise linear and circular (with respect to each other).

Thus we can readily see that in terms of the functional equation, the identity of linear and circular interpretations with respect to the real part of ζ(s) - where s = a + it - requires that a = .5.

Likewise it requires that with respect to the imaginary part that + it = - it

Now whereas any value of t will satisfy this requirement, it is only with a select set of values for t, that this will result in the non-trivial zeros.

Thus these non-trivial zeros have a special significance in terms of prime numbers (where a state of equilibrium is maintained as between both circular and linear aspects).

In experiential terms this entails the ability of a contemplative in the midst of activity (dependent on continual generation of prime objects and prime dimensions) to keep restoring a state of spiritual rest (literally free of primitive disturbances).

In numerical terms it entails the continual reconciliation of the specific location of “independent” prime numbers with the overall “interdependent” general distribution of the prime numbers (among the natural). 

Radial Mathematics

Thus the deeper significance of the famous Riemann Hypothesis points to the need for a binary system of interpretation for all mathematical relationships.

In other words just as we have two binary digits 1 and 0 in conventional analytic terms, we also have two binary digits 1 and 0 in qualitative holistic terms.

These digits here refer to the (conventional) linear and the qualitative circular systems of interpretation respectively.

Conventional Mathematics is based predominantly on the linear system of interpretation corresponding in qualitative terms to the binary digit 1.

Holistic Mathematics by contrast is based on the circular system of interpretation corresponding in qualitative terms to the binary digit 0.

Whereas Conventional Mathematics is primarily geared to the quantitative interpretation of mathematical relationships, Holistic Mathematics is directly geared to the qualitative (philosophical) interpretation of these same relationships.

However it is Radial Mathematics that properly combines both aspects (quantitative and qualitative aspects interact).

Because the interaction of both of these aspects is inherent to the very nature of prime numbers, their comprehensive understanding requires a radial mathematical approach.

Put another way just as we have real and imaginary numbers (in quantitative terms), likewise this is equally true in a qualitative manner.

Thus Conventional Mathematics – based directly on linear (either/or) interpretation of separate opposites - in qualitative terms comprises the real aspect.

Holistic Mathematics - also increasingly comprising circular (both/and) interpretation of complementary opposites - in qualitative terms comprises the imaginary aspect.

Radial Mathematics based on the interaction of both linear and circular interpretation in qualitative terms comprises the complex aspect (real and imaginary).

Thus complex quantitative interpretation - as befits the Riemann Hypothesis - requires a corresponding complex interpretation (in qualitative terms).

Ultimately the Riemann Hypothesis cannot be resolved in the absence of a binary (radial) approach.

Indeed when viewed in this appropriate context, it can be given a remarkably simple solution.