9. Summary and Conclusions (2)

9 - Summary and Conclusions (2)

The Riemann Zeta Function

The Riemann Zeta function is defined as:
ζ(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s +……   where s is a complex number = a + it.
Using the conventional (linear) method of interpretation and considering only real values for s, i.e. where t = 0, it is easy to demonstrate that the series for ζ(s) converges for all real values of s > 1.

However using the remarkable functional equation
ζ(1 - s) = 2^{1 – s} p^{– s} sin {(1 – s)p/2} (s – 1)! ζ(s) (Derbyshire P. 147),
where the angle for which sin is calculated is measured in radians it is possible to also calculate the zeta function for all values of s < 0.
For example if ζ(2) is known, then ζ(1 - 2) = ζ(- 1) can thereby be also calculated.
However an immediate problem arises, as in conventional terms the zeta function diverges for all negative values of s.

So this raises the important question as to what these numbers (i.e. for negative values of s) actually entail.

Remarkably – as I shall demonstrate further with respect to key specific numbers – these directly have a holistic (qualitative) rather than a conventional analytic (quantitative) interpretation.
The only other real values relate to those from 0 to 1.
We shall explain here why ζ(s) cannot be defined for s = 1, in either conventional or holistic terms.

As for the remaining values (0 ≤ s < 1), actual definition entails a composite mix of both conventional and holistic approaches.
When we now consider complex values for s, again the function is defined for all values except 1.
However here, because we are combining both real and imaginary values for s, that pertain directly to linear and circular understanding respectively, we require a radial method of interpretation (where both quantitative and qualitative aspects are combined).

In particular the Riemann Hypothesis is defined for certain complex values of s where a = ½.
So here through the functional equation, any value for ζ(½ + it) leads to a corresponding value for ζ(½ - it). And for appropriate values of t, both ζ(½ + it) and ζ(½ - it) = 0. The Riemann Hypothesis then postulates that it is only for the complex numbers where a = ½ that these so called non-trivial zeros exist.

ζ(1) Undefined

As we have seen, conventional (linear) interpretation is based on 1 as the default dimensional power of a number. So when a number is raised to another dimension (or power), its reduced quantitative value is given in terms of this default dimension of 1. Thus, from this perspective 2² = 4 (i.e. 4¹).

In effect this means that when a number is initially expressed with respect to a dimension (≠ 1) that two values can be given.

·        The conventional quantitative result (in terms of the default dimension of 1).

·        An alternative holistic qualitative interpretation (in terms of its actual dimension).

However - by definition - when the initial dimension = 1, then no alternative holistic result exists.
And as ζ(1) = 1/1¹ + 1/2 ¹+ 1/3¹ + 1/4¹ +……,   
diverges to infinity (in quantitative linear terms) an alternative interpretation cannot be given.


Positive Even Integer Values for ζ(s)

Euler successfully demonstrated that for positive even integer values of ζ(s) i.e. s = 2, 4, 6, 8,.. that the form of the result would imply a rational fraction multiplied by p^s.
For example - as we have already seen - when s = 2,

ζ(2) = 1/1² + 1/2² + 1/3² + 1/4² +……   = p²/6.

So in direct terms, ζ(2) has a conventional quantitative result (in linear terms).

However indirectly it also has a holistic qualitative interpretation (in circular terms).

Now p represents the relationship (i.e. ratio) of the circular circumference to its line diameter.

However, equally in qualitative terms, two-dimensional understanding represents the relationship of circular (paradoxical) to linear (unambiguous) interpretation.

We have already demonstrated this through the illustration of left and right turns on a road. Once again, we define the polar frame of reference (in terms of direction “up” or “down”), then left and right can be given a clear unambiguous meaning (in linear terms). However when we try to define direction holistically (without recourse to an arbitrary reference frame) left and right has a paradoxical circular meaning so that what is left in one context can be right in another (and vice versa).

The very essence of two–dimensional understanding is that it tries to reconcile arbitrary frames of reference (that have a limited partial meaning) with an overall holistic perspective (where both frames can equally apply).

So quite literally it represents therefore the refined relationship of circular (holistic) to linear (partial) understanding that requiring the interaction of both (circular) intuition and (linear) reason).

Thus when we try to express this relationship of complementary to separate opposites, it entails paradox.

To sum up therefore, though ζ(2) in direct terms has a finite value (from a linear quantitative perspective), indirectly the holistic qualitative nature of 2, as a dimension, is revealed in the precise form of the relationship.

A somewhat similar explanation can be given for all other positive even integer values of s.
Though the type of understanding associated with each qualitative dimension becomes more intricate and refined (as s increases), the same basic principle applies in that complementarity exists as between the root values (so those that are given a positive form can be exactly matched other roots that are negative).
Just as these as the roots (as quantitative expression of the corresponding circular dimensions) can be geometrically illustrated as a series of equidistant lines drawn from a centre to a circular circumference, in qualitative terms even dimensions express the balanced relationship as between circular and linear understanding.  

Negative Even Integer Values for ζ(s) - The Trivial Zeros

It can easily be shown through use of the functional equation that for corresponding negative even integer values of s, – 2, - 4, - 6,…. that ζ(s) = 0.

So this leads for all such values (negative even integers) to what are misleadingly referred to as the trivial zeros.

Indeed one good reason for this misleading label is directly due to the linear nature of conventional mathematics, which is not geared for their proper interpretation.
In qualitative terms, as we have seen, the positive sign literally indicates positing in a conscious rational manner.

Therefore when we try to express the qualitative nature of two-dimensional interpretation in a rational fashion, it leads to paradox i.e. in the attempt to explain the nature of complementary opposites in experience as both positive and negative.

However the negative sign literally indicates (dynamic) negation in an unconscious intuitive manner.

Thus, when one intuitively attains to the awareness that is implied by the (full) complementarity of opposites, this leads to the experience of nothingness (emptiness) as pure spiritual intuition (i.e. which is nothing in phenomenal terms).

+ 1 – 1 = 0 in conventional linear terms; likewise + 1 – 1 = 0 in qualitative holistic terms. (In other words, here phenomena that have been consciously posited as form are now dynamically negated in an unconscious manner, thus leading to the experience of pure intuition).
So here, even though we can indirectly represent the values of the zeta function in quantitative terms (= 0), the direct explanation of these zeros is of a qualitative holistic nature (relating to the pure negation of rationally interpreted two-dimensional understanding).

The philosopher Hegel is a good example of one who tried to use the positive expression of two-dimensional interpretation as the basis for his qualitative holistic understanding of reality. Indeed he even tried to a degree to apply such interpretation to mathematical symbols!

The Christian mystic St. John of the Cross represents a corresponding good example of one who largely used the negative expression of such two-dimensional interpretation as the basis for his unique expression of reality.

So the passive “dark nights” (of sense and spirit) that are given such cogent expression in his work, essentially relate to the negation of two-dimensional understanding (where attachment to holistic archetypes still resides).

Indeed he expressed the very goal of this journey as “nada” literally nothing (i.e. = 0 in qualitative holistic terms).

Once again, the other negative even integers refer to pure intuitive attainment with respect to more advanced configurations of reason (allowing for ever more varied directions or perspectives with respect to understanding). So, at each even dimension there is an appropriate refined cognitive appreciation (relating to the positive sign) and corresponding pure intuitive appreciation (relating to the negative sign).

And with truly balanced understanding both refined reason and intuition are equally developed (entailing both positive and negative dimensions).

Now the implication of these “trivial zeros” for prime numbers is in fact very significant. However, it relates directly to qualitative holistic - rather than linear quantitative - appreciation.

As the “higher” dimensions of understanding generally only attain their specialised development in a spiritual contemplative framework, it would be fruitful to look at this in greater detail.

Freedom from attachment - which is emphasised in the spiritual contemplative – conditions - basically relates to the harmonisation of primitive instinctive desires (whose roots lie in the unconscious) with the conscious experience of natural phenomena.

As we have seen - in relation to numbers - conscious experience relates directly to analytic linear interpretation. Unconscious experience - which literally involves the roots of such numbers - relates directly to holistic circular interpretation.

So put another way the contemplative, in seeking freedom from inordinate attachment, attempts to reconcile linear with circular experience.

And as the very essence of what is prime lies in the complete identification of these two systems, any failure with respect to their mutual harmonisation will be associated with corresponding problems relating to both primitive (unconscious) and natural (conscious) attachment. 

Thus we can look at the negative even integers (qualitative dimensions) as representing various levels of successful attainment with respect to such reconciliation. However because natural and prime phenomena are highly interdependent in experience, any extension with respect to natural involvement is associated by the reawakening of new primitive disturbances, thus requiring even higher dimensional equilibriums.

Indeed I have often pointed to possible shortcomings with (mere) contemplative attainment, in that one may seek too much security in overcoming primitive desire by safely managing the environment within a predictable range of conscious experience. And indeed is typical of monastic communities!

Indeed we can now show fascinating parallels with the quantitative number behaviour for the corresponding positive even integer values of s.

Thus when one looks at the denominators of the rational factors (associated with the even integer powers of p), a remarkable pattern becomes evident.

Quite simply for ζ(s), the denominator of the proper fraction (if not a power of 2) will be divisible by all prime numbers from 3 to s + 1 (and only such numbers).
In the special case where s can be expressed as a power of 2, then s will be divisible by every prime number from 2 to s + 1 (and again by only these prime numbers).

One can in spiritual terms attempt to achieve a lower equilibrium with very little involvement in natural affairs, or a higher equilibrium demanding greater activity.

The danger therefore with the pure contemplative life is that it may lead to an unduly low level of activity (thus limiting potential exposure to both conscious and unconscious phenomena).

Thus we can draw very important parallels regarding the quantitative behaviour of prime numbers and the qualitative adjustment to prime understanding i.e. that is primitive in origin (representing confusion of both conscious and unconscious).   

Positive Odd Integer Values for ζ(s)

We now look at the positive odd values for (s) i.e. 1, 3, 5, 7,….
Firstly, there is a key distinction here from the even numbered values.

Whereas the even values for the zeta function are - as we have seen - associated with corresponding powers of p, this is not the case for the negative values.

The key reason relates to the fact that the dimensions (powers) associated with odd values (unlike the even) do not arrange themselves in a complementary manner.

We can see this by looking at the corresponding root values.
So for example the 2 roots of unity + 1 and – 1 are complementary.
However the 3 roots of unity + 1, (-1/2 + √3)/2 and (-1/2 - √3)/2 do not form complementary pairings.

They point to a key difference as between odd and even numbered zeta values. And this carries a key psychological explanation!

All development involves both differentiation and integration. Differentiation is associated directly with the analytic appreciation associated with linear (rational) understanding; integration - by contrast - is associated with the holistic awareness associated with circular (intuitive) understanding.

This also provides an important insight into the manner in which prime numbers arise.
Clearly - apart from 2 - an even number is not prime. However the very process by which new primes are generated is with respect to such even numbers.

So even numbers represent a certain integral equilibrium state.

However odd numbers represent a disturbance of such equilibrium. So just as integration continually gives way to differentiation before a new “higher” integration can be restored, likewise even numbers (through reduction by 1) give way to odd numbers and thereby lead in many cases to the generation of new primes.

So in psychological terms we start with just one prime 2 (which is the basis for dualistic experience). However 2 can be combined with itself. So 2 in this way (through the higher dimensional integration of 2 whereby it is multiplied by itself) becomes 4. Then when reduced by 1, this gives way to a new linear differentiated state (3).

So we now have two starting primes which can be organised in various ways with each other to realise various integral forms. Then, through reduction by 1 this leads to the generation of further differentiated phenomena (and thereby the possibility of new prime numbers arising).

So just as the differentiation of new phenomena in experience arises with reference to already higher dimensional integral forms (through the attempted reduction of such forms), likewise larger valued primes actually arise with reference to composite numbers (that themselves have arise through the product organisation of smaller primes already differentiated).           


Negative Odd Integer Values for ζ(s)

Negative odd integer values of the zeta function are calculated through the transformation function from corresponding positive even integer values.
So ζ(1 - s) is calculated from ζ(s)

Therefore, for example, ζ(- 1) is calculated from ζ(2) with a value of – 1/12.
It can be easily shown that for odd integer values of s that ζ(s) is always a rational fraction.
Secondly the value alternates signs with successive odd numbers. So for the next number i.e. - 3, ζ(- 3) = 1/120.

As clearly, from a conventional linear perspective, the sum of the series for all odd integer values of s diverges to infinity, it is necessary to give an alternative holistic interpretation of the meaning of this finite rational result.

The clue to this is holistic interpretation is perhaps best illustrated with reference to the value of ζ(0).

Now I have already stated that for all values between 0 ≤ s < 1, that the value of ζ(s) entails a composite mix of both linear and holistic interpretation.

This can be perhaps initially seen by transforming the dimension 0 into the form
0 = + 1 – 1. This then entails the 1^st and 2^nd dimensions (which are the basis for linear (quantitative) and circular (qualitative) interpretation respectively.
Then one actual method of computing the value for ζ(0) entails using the eta function (entailing combinations of terms that are complementary opposites with respect to sign).

Then the series combining an even number of terms (entailing a continuing series of  complementary opposites +1 – 1) = 0.

However when an odd no. of terms is combined - leading to a linear result - we get + 1.
So. if we then average these two results we  get .5 for the eta and - .5 for the corresponding zeta series.

We can then attempt to give a satisfactory qualitative explanation of what this means by remembering that we have obtained it by simply averaging out the two approaches i.e. linear (rational) and circular (intuitive).

So quite literally .5 here implies in qualitative terms that understanding is half rational (implying that it is equally used with intuition).

To be fully accurate, the negative sign would imply internal (as opposed to objective) reason implying that the use of reason alternates between these two polarities.

In qualitative terms, odd number values for zeta function point to this separation of opposite poles which differentiation – as opposed to integration – requires. So at one moment, one differentiates conscious phenomena with respect to the objective world (in relation to the internal self (+) and then, this alternates to conscious phenomena with respect to the internal self (in relation to the external world). The integral view of complementary opposites – where these two polarities are united – is then associated with the even numbers.

We can now thereby attempt to give qualitative meaning to the rational values in the zeta function (for negative odd integers). 

ζ(-1) = - 1/12

Again in qualitative terms this would be associated with the differentiation of reason internally in conscious experience (at a relatively low level).
The implication here is that this represents the transition to the first of the spiritual equilibrium points (associated with contemplative spirituality). So this implies that the active use of reason is used more sparingly in a refined manner.

ζ(-3) = 1/120

The direction has now switched to positive. So this value qualitatively is now associated with reason (that differentiates in an external fashion). The very small fraction indicates a considerable further refining of such reason (which would be compatible with the continuing deepening of a contemplate state that reaches its next equilibrium at ζ(- 4)

ζ(-5) = - 1/252

So the direction has once again switched to negative implying a focus on reason used in an internal manner (with respect to subjective consciousness).
The fraction here attains its smallest value implying in a sense the greatest economy in the exercise of the rational faculties (which again would be compatible with an ever deepening contemplative state).
The fractional values then start to rise. For example values ζ(-7) =  1/240 and

ζ(-9) = -1/132.

Fro many years – long before I studied the Riemann Hypothesis - I held the view that pure contemplative development was consistent with development between 2 and 8 dimensions (considered both in their positive and negative sense). However for experience to remain healthy, further dimensions then needed to become radial (entailing a growing increase in active involvement).

Remarkably this is supported by this qualitative interpretation of the zeta values (for negative odd integers).

Indeed an even better correlation can be obtained through lagging the dimensions by 2 based on the fact that the denominators of the fractional values obtained from the zeta function ζ(s) (where s a negative odd integers) is always divisible by s + 2 (where s + 2  is prime).
This means that the denominators associated with  ζ(-1), ζ(-3),  ζ(-5), ζ(-9)      respectively are divisible by 3, 5, 7 and 11 respectively. For ζ(-7), s + 2 = 9, is not prime. However the denominator is still divisible by the highest prime factor 3, + 2 = 5.

With further negative odd values in the zeta function, the fractional values quickly rise above 1 and continue to rise for further s.

This would be compatible with the higher dimensions (experienced at the radial levels) where ever greater levels of conscious involvement with reality are developed (in line with developed contemplative ability).

Here experience becomes considerably more dynamic, as the larger values for the negative odd integers of zeta, act as ever more oscillating transition points on the way to the pure state of equilibrium (associated with corresponding negative even values).


Zeta Value for ζ(.5)

In a sense the appreciation of this zeta value provides sufficient insight to unravel the simple truth that lies behind the Riemann Hypothesis.

We have already described how the functional equation in effect converts real numerical values for the zeta function that have a conventional linear interpretation for s > 1 to corresponding negative values of s ( < 0) that have a corresponding holistic circular interpretation (in qualitative terms)

For all other values 0 ≤ s < 1, a composite mix of both conventional and holistic interpretations is involved.  

The importance of s = .5 is that 1 – s likewise = .5.

Therefore the only value that guarantees full compatibility as between the two approaches (linear and circular) = .5.

Thus if we start with ζ(.5) and then use the transformation function to obtain the corresponding transformed value for ζ(1 - .5) =  ζ(.5) the result will remain the same.

So here - and only at this real value for s - is the equality of both linear and circular interpretation of values for s maintained.

Put another way, only at .5 can full compatibility be maintained as between (standard) analytic quantitative interpretation of numerical values on the one hand and corresponding (philosophical) qualitative interpretation of these same values on the other.

And this finding is vital in terms of the appreciation of prime numbers as – by their inherent nature - they combine the identity of these two distinct logical interpretations.

So we will explain what this now entails both from matching quantitative and qualitative perspectives.

Let us firstly consider the unit circle (drawn in the complex plane). Now can literally view the line diameter of this circle (drawn along the real axis) when considered in linear terms as a unit = 1. Therefore the mid-point of this line = .5. And from this point we can circumscribe a circular using (in both directions) the common radius of .5.

In this way the central point of the circle (at .5) serves as that point where both the line (as diameter) and circle (as circumference) are identified.

When we give corresponding qualitative interpretation to this same geometrical representation, the midpoint of the circle represents the point where perfect balance is maintained as between opposite polarities of experience (external and internal).

So in psychological terms, this represents the golden mean as between opposite polar extremes.

Thus the task of developing a spiritual contemplative state (that results in pure intuitive awareness) requires that perfect balance be maintained as between the conscious rational polarities of experience. Without such balance, a degree of possessive attachment to the resulting phenomena of experience will result (thereby hindering the emptiness of pure intuitive equilibrium).

Of course in dynamic experiential terms, we only approximate - in a relative fashion - such a perfect equilibrium (which is finally attainable as a limit that can never be fully reached).

So one might ask again about the relevance of this to prime numbers!

Well in earliest infant experience, both conscious and unconscious remain fully entangled with each other (devoid of either linear differentiation or circular integration).

This is thereby what we call in holistic qualitative terms a totally prime (i.e. primitive state).

In like manner the original nature of prime numbers (in quantitative terms) entails this total entanglement of these two systems of interpretation (as object and dimensional quantities respectively) that are linear and circular with respect to each other.

The full disentanglement - again in qualitative terms of (linear) conscious and (circular) unconscious - requires a pure state of spiritual equilibrium (where rational conscious experience can be fully harmonized with intuitive unconscious awareness).

This can be looked on therefore as the qualitative solution to the prime number problem (i.e. in providing a qualitative method of understanding that is properly suited for interpretation of the nature of prime numbers).

So therefore armed with an appropriate method of interpretation (based on corresponding psychological development) one can thereby properly also examine the quantitative nature of prime numbers.

And the startling implication of the Riemann Hypothesis is that these two aspects of understanding (both quantitative and qualitative) are so closely related with respect to the inherent nature of prime numbers that this very recognition is in fact the very solution to the Hypothesis.   

So a solution which inherently entails the interaction of both systems (linear and circular) is - what I term - radial.

Therefore the Riemann Hypothesis represents a fundamental - indeed the most fundamental - axiom in Radial Mathematics.

Non-Trivial Zeros for ζ(s)

However – important though it is – the real value (.5) does not provide a solution for the Riemann Zeta Function.

So it is only with complex numbers that this is possible.

However given the central role of .5 (as real part of s) a quantitative solution will be of the form s = .5 + it. Therefore by the functional equation this implies that an alternative solution will be provided by 1 – s = .5 – it.

Thus the non-trivial zeros relate to the values of t that satisfy the zeta equation.

There are a great many of these, the first of which is 14.134…, and continually increase for higher range values of t.

The question arises as to what these solutions actually represent!

In quantitative terms - as we have seen - the general distribution of prime numbers can be calculated (though errors necessarily still exist)

So the simplest version of this distribution is given by n/log_en which gives the approximate number of primes in relation to the natural numbers n (for any size n).

This approximation was later considerably improved by Gauss (with his integral function) and then Riemann.

However even the improved Riemann Function leads to small errors in prediction, in that the function necessarily represents a certain continuous smoothing of data whereas specific primes arise in an unpredictable discrete fashion (giving rise to a step function).

The non-trivial zeros provide the bridge - as it were between - these two functions i.e. the smooth continuous function estimating the general frequency of primes on the one hand and the actual discrete step function that gives us the precise location of the primes.

So once again this bridge as between continuous and discrete, necessarily entails both linear and holistic elements (indicated by the complex numbers with real and imaginary parts) requires to estimate the zeros.

Now the formula for the frequency of the non-trivial zeros in the vicinity of any value of t is given by
2p/log(t/2p)

Now if we replace 2p (which is the circumference of the unit circle) by 1 (as the radial line) we get 1/log(t/1) = 1/log t.

Now this expresses the probability that t is prime or in terms of its reciprocal the average gap between prime numbers in the region of t.
In corresponding fashion the first formula expresses the average gap between non-trivial zeros in the region of t.

Fascinatingly however this formula is really a circular equivalent of the second (where the linear radius in the unit circle i.e. 1 is replaced by its circular circumference 2p).
Now it might be interesting to suggest perhaps why the frequency of occurrence of non-trivial zeros steadily increases with the value of t.

As I have states before the generation of primes the natural numbers is in fact interdependent throughout both number systems.
So we start with the first prime 2 (which is generated from 1 through splitting into polar opposites).
Then the product of this number with itself leads to the generation of the higher dimensional organisations as composite natural numbers and then through reduction by 1 to a further prime (i.e. Mersenne prime).

For example, the simplest product organization 2 X 2 = 4 (which is a composite natural number). Then through reduction by 1 we obtain 3 (as the next prime number). Initially this would be the only combination possible (as the number of product combinations would be limited to the value of the maximum prime already generated!

So now the possibilities for more varied product organisation grow whereby 2 and 3 can now be combined in up to 3 different product combinations. So this process leads to the generation of many more composite natural numbers and in turn (through subtraction of 1) to additional new primes. So the possibilities for ever more varied factor combinations steadily increases generating new composite numbers (and through subtraction by 1 new prime numbers also).

This would explain why the gap as between non-trivial zeros would steadily decrease. Even though the frequency of primes decreases as we ascend through the natural numbers, the possibility for multivaried combinations of existing primes greatly increases. So therefore the number of additional primes that will result from all these new composite numbers will steadily increase.

I would see this as the process by which primes are generated (with respect to existing composite natural numbers) to which the non-trivial zeros relate. So they serve as the bridge (through the accumulation of many small wave deviations) that harmonise the best general prediction of prime number frequency (based on a smoothed continuous distribution) and actual prime number distribution (related to a discrete step function).

However as always, the quantitative behaviour associated with the non-trivial zeros can be given a corresponding qualitative holistic interpretation with direct reference to psychological experience.

We have already identified the attainment of pure contemplative experience as psychologically akin to the general distribution of the primes. Traditionally in the past contemplative aspirants withdrew from the world to engage in an intense process of disciplined self awareness. When successful this eventually led to the successful harmonization of both conscious and unconscious aspects of personality.

We could correctly say that this process thereby represented the general mastery of primitive instinctive impulses thereby enabling spiritual intuitive energy to be wedded to conscious endeavour.

However despite such a great achievement, there can be a danger here of experience becomes too passive, so that the spiritual adept experiences a certain reluctance to become actively engaged in the world. In other words success in merely contemplative terms can lead to the restriction of everyday experience within well-defined limits (traditionally imposed through monastic rules)
In other words the potential range of natural experience (as associated primitive instinctive life) is thereby greatly limited.

However in the fullest expression of mystical development, both contemplative depth and extensive practical activity are combined in - what I refer to as - the radial life.
So in effect, through becoming increasingly engaged in commited active involvement, one becomes thereby continually exposed to the experience of new natural phenomena (and the underlying primitive instinctive impulses that are thereby activated).  

It is in this context that the trivial non zeros can be given a fruitful qualitative explanation.
As we have already seen, the central stance adopted is to maintain both internal and external polarities in as balances a manner as possible. In this way one avoids undue identification with dualistic expressions of either aspect in isolation (which would imply a degree of unwarranted attachment).

However in these circumstances, the equivalent of virtual particles keep surfacing from the unconscious in the form of short-lived imaginary projections.

Once again successful balance here requires that any momentary identification of a high-level projection (e.g. in the form of a universal archetype of form) is quickly counterbalanced by an immediate physical impulse (e.g. in the form of erotic fantasy).

In this way the “higher” imaginary spiritual projection from the unconscious with respect to the universal dimensional nature of experience is immediately counterbalanced by a corresponding “lower” imaginary physical projection in the immediate specific nature of an alluring physical object.

Thus, the vertical complementarity of extreme tendencies - with respect to both circular and linear systems of understanding - is continually reinforced.

Therefore, as we have already seen, experience keeps moving from integration (in temporary equilibrium points representing integration) towards differentiation (where that equilibrium is momentarily disturbed both in conscious and unconscious terms).

So from a psychological perspective, success - in speedily restoring equilibrium and a sense of spiritual peace - requires two related aspects.

(a) that the golden mean be maintained in relative terms as between opposite internal and external polarities i.e. in real (conscious) terms and

(b) that a corresponding golden mean be maintained as between opposite “higher” and “lower” projected phenomena i.e. in imaginary (unconscious) terms.

Thus, the temporary deviations (that unconsciously arise through prime disturbances) can be quickly corrected, thus enabling the ever growing dynamic organisation of natural phenomena (without undue attachment arising).
In this way, considerable involvement with active discrete phenomena thereby becomes compatible with the continual preservation of a deep sense of contemplative spiritual equilibrium.

Though I am not dealing with this specifically here, there is also strong evidence building for a direct physical expression of the Riemann non-trivial zeros at the quantum level (where the prime instinctive nature of material interactions operates).
Indeed when I first read about these developments, though the details were new, it came as no surprise, as it a basic tenet of holistic mathematics that every mathematical relationship has a potential application to reality in both physical (quantitative) and psychological (qualitative) terms, which are in fact complementary. 

Here the precise quantitative values of the zeros would seem to correspond with quantum energy states (in a way that has yet to be precisely demonstrated).

However, important and exciting as these findings are, I cannot see that their precise demonstration – even if possible - would prove the Riemann Hypothesis. What we would then have shown is a dramatic correspondence as between the energy states of a certain physical quantum system and the zeros in a mathematical hypothesis. And needless to say correspondence in this manner – regardless of how great - would not constitute standard mathematical proof.

Thus to recap, the functional equation in effect converts linear interpretation for the zeta function on one side to circular on the other (and vice versa).

And we have demonstrated that in real terms identity (in terms of such transformation) requires (a) that the real part = ½ and (b) that in imaginary terms, + it one side be balanced by – it on the other.  

Conclusion

I have not attempted here to give a detailed conventional quantitative account of the Riemann Hypothesis – a task for which I am certainly not equipped.

Rather my intention has been to suggest a radically different approach with respect to understanding the problem, which requires considerable revision in the accepted meaning of mathematics.

Basically I would see present mathematics as comprising just one main element (of three) in a more comprehensive overall framework.

So Conventional Mathematics is based on a linear method of logic that is directly geared to merely quantitative aspects of interpretation. Thus it can only proceed by reducing in any relevant context what is qualitative to what is quantitative

Holistic Mathematics - by contrast - incorporates a circular method of logic (that operates in conjunction with multi-directional linear understanding). It is directly geared to the use of mathematical symbols in an inherently qualitative manner.

Radial Mathematics interactively combines both aspects where a close correspondence is maintained as between both conventional and holistic aspects. 

The crucial point here is that prime numbers - by their very nature - incorporate a close marriage of both types of logic (linear and circular). So solutions to many fundamental theorems relating to prime numbers will thereby require a radial mathematical interpretation.

In particular this is true of the Riemann Hypothesis, which when interpreted in the appropriate radial context, can be seen to be directly representative of a fundamental axiom in this approach. 

There are many other important issues that I have not attempted to deal with in these explorations.

For example the many extensions of the Riemann Hypothesis raise fascinating new issues in their own right. However once again, the explanation of the central importance of the vertical line drawn through .5 will remain unchanged.

Also, the remaining complex values for the Riemann Zeta Function, which do not give rise to zeros also have an important radial significance (which I have not attempted to identify).

In conclusion I would like to use an interesting identity to highlight this fundamental connection as between linear and circular aspects (that is inherent in prime numbers).

In Chapter 7, I used the simple equation

xⁿ - 1 = 0 to suggest a complementary form of Riemann Hypothesis (where only real values for n are considered).

However we can also give imaginary values to n.

In particular through doing this we can derive the fascinating identity

(k/1)¹ where we are considering k is a natural no). = 1 ^{(log k)/(log 1)}

In this way we are able to draw a direct correspondence as between the number k in both the conventional (horizontal) quantitative system (where k is defined in terms of the default dimension 1) and k in the holistic (vertical) qualitative system (where the dimension log k is defined in terms of the default quantity 1)

So for example (4/1)¹ =  4  = 1^{(log 4)/(log 1)}

So we have replaced 4 and 1 respectively as quantity on the LHS with log 4 and log 1 respectively as (dimensional) quality on the RHS.

However 4 does not strictly represent a pure linear quantity (as 4 is really two-dimensional = 2 X 2).

Likewise log 4 does not represent a pure dimensional quality (as log 4 = log 2 + log 2).

So it is only with prime numbers that this pure correspondence - as between number as linear quantity and circular quality respectively – is maintained.

So for example when k = 2

(2/1)¹ = 2 = 1^{(log 2)/(log 1)} without the possibility of further reductionism on either side.

Likewise the truly circular nature of dimension (on the RHS) is clearly in evidence through the use of log 1 (as the divisor of log 2). 

For log 1 = 2πi, i.e. e^2πi = 1 (i.e. where the dimension 2πi, which in this context = 0,  represents the pure circular notion of dimension).

So this simple identity powerfully points to the inherent nature of all prime numbers as radial (i.e. that combine both linear and circular methods of interpretation simultaneously).