L-Functions

My comments on L-functions will be necessarily brief, as I do not have sufficient type analytic knowledge of the various types (which is mainly confined to simpler Dirichlet functions) to comment in more detail.

However, I do believe that the general holistic conclusions that I have reached with respect to the Riemann zeta function have a similar validity regarding all classes of L-functions.


The Riemann zeta function represents the simplest of L-functions encoding in the most general manner the relationship of the primes to the natural numbers.

However, it is important to remember that from a dynamic interactive perspective, this entails the two-way relationship of primes and natural numbers in both a quantitative and qualitative manner.

All other L-functions can be viewed as representing the corresponding relationship of certain unique configurations of primes, of varying complexity, with corresponding configurations of the natural numbers.

However it is important to again remember that from a dynamic interactive perspective, this entails the two-way relationship of such configurations in both a quantitative and qualitative manner.


We showed in dealing with the Riemann zeta function, that there are in fact two complementary zeta functions i.e. Zeta 1 and Zeta 2. Thus, whereas Zeta 1 expresses a general infinite function (in sum over integers and product over primes expressions), each individual term, can then be expressed through an appropriate infinite version of the Zeta 2 function.

For example, the first term, i.e. 1/(1 – 1/p^s), when s = 2 (with p = 2) in the Zeta 1 (product over primes expression) is 4/3.

And this can be stated in terms of the (infinite) Zeta 2 function,

i.e. 1 + x + x² + x³  +… =  1/(1 – x) where x = 1/p²  = 4/3.

Thus, logically we can extend the related complementary nature of Zeta 1 and Zeta 2 functions that is true for the Riemann case to all L-functions (i.e. L1 and L2).

Thus whereas the L1 function represents the twin sum over integers and product over primes expressions, the corresponding L2 function represent each of the individual terms of these expressions (which equally can be defined consistently in an alternative manner).

So above, we have seen that 4/3, representing the 1st term in the product over primes expression for ζ(2) (where p = 2), can be expressed as a simple geometric series where r = 1/p².

However, this value can equally be given an alternative expression through the following series 1 + 1/5 + 1/15 + 1/35 + …, with binomial coefficient C(n, 4).

And here p²= 4.

Illustrating just a little further, 9/8 represents the 2^nd term in the product over primes expression for ζ(2) (where p = 3).

Thus in terms of the former L2 expression this is given as

1 + x + x² + x³  +… = 1/(1 – x) where x = 1/p² = 9/8

Then in terms of the alternative L2 expression, it is given as

1 + 1/10 + 1/55 + 1/220 + …  = 9/8, with binomial coefficient C(n, 9) so that p² = 9.

Thus both L1 and L2 functions can be given twin alternative expressions reflecting the complementary nature of addition and multiplication respectively.

And the significance of this mathematical finding is that both the cardinal nature of the number system with respect to the relationship of primes (with special unique properties) to the natural number system as a whole and the ordinal nature of each such number are dynamically complementary with each other.

So strictly, from a dynamic interactive perspective, all L-functions express the two-way relationship, externally and internally, of both L1 and L2 aspects.


Again, all L-functions - with respect to the L1 aspect - can be expressed both in terms of an (infinite) sum over the integers and corresponding (infinite) product over the primes.

The significance of this from the holistic perspective is that addition and multiplication are analytic (quantitative) and holistic (qualitative) with respect to each other. Therefore when a function can be equally represented in both ways, this entails that number is fully defined (for that particular function) with respect to both its particle (analytic) and wave (holistic) expressions.

The complementarity here of both quantitative and qualitative aspects can be made more explicit in a revealing manner.

We have seen in relation to the Riemann Zeta function that,

ζ(2) = 1 + 1/2² + 1/3² + 1/4² + …   = 4/3 * 9/8 * 25/24 * …

With respect to the product over primes expression (on the RHS), 4/3 is related as we have seen to 2 (as the 1^st prime number).

Now if we choose to omit 4/3 from the RHS multiplicative expression, then we must remove every term where 2 is a factor from the corresponding LHS additive expression.

So 1 + 1/3² + 1/5² + 1/7² + …    = 9/8 * 25/24 * 49/47 * …

We have already seen in dynamic interactive terms, how when 2 is used as a factor it is now expressing the dimensional aspect of number, which is then - relatively - qualitative with respect to the corresponding base aspect (which is thereby in this context of a quantitative nature).

And, when appropriately understood, this is the key underlying message of the equality of both the sum over integers and product over primes expressions (which characterise all L-functions).

In other words, they operate in a manner (through addition and multiplication respectively) whereby they demonstrate the balanced two-way relationship of both the quantitative and qualitative aspects of number.


It is likewise similar with respect to the individual L2 functions.

When we look at the former expression of each term (through a simple geometric series) the natural numbers are involved (representing now the dimensional aspect of number)


Then the latter expression in fact is based on a simple process of addition (with respect to the base aspect of number).

So we start with the series 1, 1, 1, 1, …

Then by now representing the n^th term of a new series as the sum of the previous n terms we get

1, 2, 3, 4, … (the reciprocal of which terms comprises the harmonic series) with binomial coefficient C(n, 1).

We continue on in this manner to construct a new series, where again the n^th term represents the sum of the previous n terms. This then gives

1, 3, 6, 10, … with binomial coefficient C(n, 2).


Continuing on in the same manner (where nth term of new series = sum of previous n terms of previous series) we obtain,

1, 4, 10, 20, …, with binomial coefficient C(n, 3).


And illustrating further in the same manner to obtain one further series, we obtain


1, 5, 15, 35, …, with binomial coefficient C(n, 4) which we have already encountered.

So in the former L2 expression, each natural number (representing the dimensional aspect as the power of number) can be represented by the repeated multiplication of 1.

In the latter L2 expression each natural number as denominator (representing the corresponding base aspect of number) is derived through compound addition starting with 1’s.

So we can see clearly therefore, when understood in a dynamic interactive manner, that the relationship between both L2 expressions is complementary (with contrasting quantitative and qualitative aspects).


This complementarity (with respect to both L1 and L2 aspects) is closely associated with the fact that L-functions tend to be very symmetrical in nature.

And from a dynamic interactive perspective, such symmetry is associated with a high degree of number synchronicity (due to matching particle and wave aspects).

Such synchronicity is thereby a key feature defining the dynamic behaviour of all L-functions.

Thus, though recognisable quantitative and qualitative aspects are associated with both analytic and holistic expressions, these are dynamically related in a two-way complementary manner.

This then becomes automatically associated with a characteristic functional equation, where for values of the function for s, corresponding values of the function can be given for 1 – s.

And as we have seen, the functional equation represents a means of switching - relatively - as between particle (analytic) and wave (holistic) expressions of number.

Thus when an intuitively meaningful value of the function for s is obtained, this can be identified with its analytic (particle) interpretation.

Then when through the functional equation, a corresponding value for 1 – s is obtained that appears non-intuitive from the standard analytical perspective, this is because the value now properly relates to its holistic (wave) interpretation.

So, values of all L-functions for 1 – s, are reflected across the symmetry (critical) line, from corresponding values of s, through .5.


Therefore in general, the functional equation for every L-function can be seen as a means of mapping analytic (particle) expressions of numerical values with their complementary holistic (wave) expressions and vice versa.

And once more, this can only be meaningful within a dynamic interactive appreciation of the functions, entailing both analytic (particle) and holistic (wave) aspects.  

Now again universally for all L-functions, it is postulated that the critical imaginary line is drawn through .5 and that the all the non-trivial zeros for the function lie on that line.

Once again it requires a holistic perspective to see why this is an invariant feature of all L-functions.

In relation to the Riemann zeta function, I have already explained how - from a dynamic interactive perspective - the value of .5 arises from the need to fully balance the two opposite polarities of number, as (external) object and (internal) interpretation respectively. This serves as a requirement for holistic understanding (entailing appreciation of their twin interdependence) to be properly incorporated with the refined analytic appreciation (associated with the relative independence of each pole).

In other words, true holistic appreciation of number results from the realisation that its objective reality cannot be meaningfully separated from corresponding mental interpretation of such reality. And when both aspects are fully merged, this results in formless intuitive appreciation (representing a pure energy state).

All L-functions are similar in this respect.

So, their postulated zeros entail the corresponding holistic extreme - in the pure formless intuitive appreciation of number as energy states - to the analytic view of primes (and varying configurations of primes) as understood in the standard absolute formal manner. 

Likewise again the imaginary line in this context represents but an indirect analytic way of expressing holistic type appreciation (of a directly unconscious nature).

Therefore from a holistic perspective, it is easy to see that the same general features that apply to appreciation of the Riemann zeta function should apply likewise to all associated L-functions (L1 and L2). In short, these general features relate to the similar dynamic interactive manner in which all these functions should be rightfully interpreted.

So the zeros of an L-function (other than the Riemann), could thereby for example be incorporated into an explicit formula that could predict the exact number of a particular class of primes - leaving for example a remainder of 1, on division by 4 - up to a given number (on the real scale).

However, equally this particular class of primes could be incorporated into an explicit formula to exactly predict the corresponding number of zeros to n (on the imaginary scale).

So again, even in quantitative terms, it is apparent that these primes and zeros are interdependent (implying a dynamic interactive relationship).

Now it might be suggested that perhaps by searching for common characteristics with respect to L-functions that this might open the way to proving the Riemann Hypothesis (as a member of this class).

However from the dynamic perspective, these common characteristics of L-functions, rather suggest the opposite, implying in holistic terms precisely the same kind of problem that affects the Riemann Hypothesis.

In dealing with the Riemann Hypothesis, I concluded that its truth is necessary to justify the reduced assumption that all real numbers can be placed on the same number line. However clearly this cannot be proved through conventional mathematical axioms, as the assumption is already made that this is the case (that all real numbers lie on the same line).

And this is the very same issue that besets attempted proof with respect to any of the L-functions.

In the case of each particular L-function, the assumption that all its respective zeros lie on the same imaginary line is necessary to justify the corresponding assumption once again of the same number line relating to special configurations of primes.

What this would entail is that each special configuration of primes implies a unique relationship of interdependence with the natural numbers. And the consistency of this relationship with respect to both quantitative and qualitative aspects is thereby required before the standard reduced assumptions of number behaviour (in merely quantitative terms) can be made.

Now one might argue that once the Riemann Hypothesis is assumed to be true that therefore from this perspective all the other L-functions can likewise be assumed to be true.

But I would not see it quite like that!  From a dynamic interactive perspective, it is not so much that the truth of the Riemann Hypothesis implies the truth of all other L-functions or alternatively that the truth of all these other L-functions (in their zeros lying on the relevant critical lines) implies the Riemann Hypothesis, but rather that they mutually imply each other in a dynamic synchronous manner, which is ultimately ineffable.

In this way, the root nature of the number system is seen to be utterly mysterious. For already built into this system at its very origins is an unfailing capacity to perfectly synchronise all number relationships with respect to both quantitative and qualitative characteristics (and by extension all phenomenal relationships in nature), so that everything in particular can potentially be given its own special unique identity, yet - ultimately - fully integrated in collective manner with everything else in creation.

And the core of this mystery relates to the inter-dynamic nature of the Zeta 1 and Zeta 2 zeros with both the primes and natural numbers in regard to the number system. 

However once again this represents an act of faith in the ultimate consistency of the number system rather than an acceptable proof, for clearly, proof in this regard is not possible.

From my perspective, it would indeed be a tragedy for mathematics if the Riemann Hypothesis somehow could be proved, for this would reinforce the present absolute interpretation of number, when a dynamic relative approach is urgently required.

Therefore I would see that continued inevitable failure to prove the Hypothesis, will eventually lead practitioners to seriously question their underlying assumptions regarding number, thus gradually bringing about the intellectual revolution that is so necessary.

In this way, failure to prove the Hypothesis, could act as an all-important necessary catalyst in enabling the emergence of a much more comprehensive vision of mathematics.    

In conversation with Karl Sabbagh on P 210 of “Dr. Riemann’s Zeros” Alain Connes is quoted as saying saying “I believe, I have found a very nice framework but this framework is still awaiting the main actor. So there is the stage - it is perfectly well arranged and so on - but we are still expecting the heroine to come and complete it”.

I would rather suggest that the heroine has been centre stage all along, without unfortunately being noticed.

And this is due directly to the reduced analytic interpretation of symbols that presently defines accepted mathematical research.

In other words mathematics as we know it is completely defined in formal terms by the masculine principle of abstract reason. 

However when properly understood, mathematics contains an equally important feminine principle relating directly to unconscious holistic intuition (that indirectly is rationally conveyed in a circular paradoxical manner).

Unfortunately, though the feminine dimension was to a degree recognised in former times (as for example with the Pythagoreans), it subsequently has become almost totally suppressed (especially through the increasingly specialised abstract mathematical developments of the last century)

And without doubt this is the greatest single issue facing the future development of mathematics.

Though admittedly enormous progress has been made in abstract terms, the profession has completely ignored the hidden unconscious aspect of mathematical knowledge in steadfastly refusing to explicitly explore its vitally important feminine dimension.

And it is only when both masculine and feminine principles are properly recognised, in the equal incorporation of both analytic and holistic type appreciation that a fully integrated mathematical understanding can finally emerge.

Future Vision of Mathematics

Before briefly making a few observations on L-functions let me outline some reflections on a future “golden age” of Mathematics.

Here I would see three distinct areas (where only one currently exists).

1) The first - which for convenience - can be referred to as Type 1 Mathematics, relates to the traditional analytic approach based on the reduced quantitative interpretation of mathematical relationships.

At present, mathematics i.e. certainly with respect to formal recognition, is exclusively identified with this type.

As its methods have become increasingly specialised in an abstract rational fashion, admittedly, enormous progress has been made. And this will continue into the long distance future with many new significant findings for example with respect to our current topic of the Riemann zeta function (and associated L-functions).

However an important present limitation, as we have seen, is the manner in which exclusive identification with the analytic, blots out the holistic aspect (with which creative intuition is more directly associated).

So while not wishing in any way to prevent the further progress of analytic type developments, eventually I believe it will be accepted that the Type 1 represents just one highly important aspect, which should not be exclusively identified as mathematics.

So a strictly relative - rather than absolute - interpretation will thereby eventually emerge for all its relationships.


2) The second - which I customarily refer to as Holistic Mathematics - represents the Type 2 approach.

My personal development has been somewhat unusual in this regard. Though very interested in mathematics as a child, my first serious reservations with the standard treatment of multiplication arose at that time. Therefore, from that moment I was already reaching out for a new holistic dimension, not catered for in formal terms.

So mathematics for me has very much represented a solo voyage of uniquely personal discovery.

And in adult life my abilities - such as they exist - have been largely centred on elaborating the hidden holistic dimension of mathematics. Then in the last 15 years or so, using these holistic insights, I have turned my attention to topics such as the Euler Identity and the Riemann Hypothesis with a view to providing a radical new perspective as to their true inherent nature.

Though retired from work for nearly five years, it feels only now that I have come full circle in being able to finally resolve (at least to my own satisfaction) those childhood reservations regarding multiplication.

However it is very difficult attempting to convey holistic mathematical insights to a professional audience (that does not formally recognise their existence).

So my most successful communication has been with talented generalists (with some mathematical background) seeking to integrate various intellectual disciplines in a more holistic manner.

However even here there has been considerable resistance to the belief - reflecting the deep-rooted nature of conventional assumptions - that mathematics itself is in need of radical re-interpretation.


Bearing the above comments in mind, I will now try to convey some of the flavour of my holistic mathematical pursuits.

As it is directly concerned with the qualitative nature of mathematical symbols, much of this work has related to the holistic clarification of the various stages of psychological development through the use of these symbols.


In terms of the physical energy spectrum, natural light forms just one small band on the overall spectrum of physical energy.

In like manner, the mental structures (based on accepted common sense intuitions underpinning linear logic) represent just one small band on the overall potential spectrum of psychological development.

Conventional mathematics represents specialised understanding with respect to this one small band.

However just as there are “higher” forms of physical energy (besides natural light) equally there are “higher” forms of psychological energy (besides the accepted intuitions of conventional mathematics).

These “higher” bands on the psychological spectrum have been extensively investigated by the major esoteric religious traditions East and West, where they are identified with advancing levels of spiritual contemplation of an increasingly formless nature.

However what has not yet been realised - except in a perfunctory manner - is the important fact that these bands are likewise associated with new forms of holistic mathematical interpretation (utterly distinct from conventional type notions).

In analytic terms, the study of higher mathematical dimensions entails understanding of an increasingly abstract rational nature (where the object aspect is increasingly separated from its subjective counterpart).

However in holistic terms, the study of higher dimensions by contrast entails appreciation of an increasingly intimate intuitive nature, indirectly transmitted in paradoxical rational terms, where both object and subjective aspects are seen as ever more interdependent with each other.

Now one of the extraordinary findings arising from these investigations is that all psychological (and indeed physical structures) can be given a distinctive holistic mathematical rationale.

And the holistic notion of number is intimately tied to these structures.

So from the holistic perspective, accepted formal mathematical understanding is 1-dimensional (in qualitative terms). This simply means that the qualitative aspect is formally reduced in quantitative terms.

However associated with every other number (≠ 1) is a distinctive means of interpreting mathematical symbols with a partial relative validity. So the absolute type understanding that we accept as synonymous with valid mathematical interpretation represents just one special limiting case of a potentially infinite set.

And this insight was later to prove of inestimable value in relation to a true dynamic appreciation of the Riemann zeta function.

Also, the holistic relative notion of number is intimately connected to a corresponding new holistic appreciation of the nature of space and time with a direct relevance in physical and psychological terms.

For example, the holistic notion of “4” relates to a dynamic appreciation of the corresponding 4-dimensional nature of space and time (with positive and negative directions in real and imaginary terms).

From a psychological perspective, this entails that all conscious (real) experience has - relatively - both positive (external) and negative (internal) aspects and also that unconscious experience, indirectly projected in a conscious (imaginary) manner, has likewise both positive (external) and negative (internal) aspects.


At a deeper level, the imaginary directions explain - in an indirect conscious manner - how in the dynamics of experience, whole switch to part (and in turn part switch to whole) notions.

In brief, akin to directions on a compass, all other holistic dimensions can then be looked on as providing unique configurations with respect to the dynamic relationship as between wholes and parts (in objective and subjective terms).

The famous Swiss psychologist C. J. Jung offered some broadly similar insights, while implicitly formulating his notions in a manner amenable to holistic mathematical interpretation.

So when one of his disciples Marie Louise Franz was later to say that “Jung devoted practically the whole of his life's work to demonstrating the vast psychological significance of the number four”, it was this holistic (circular) - rather than analytic (linear) - notion of “4” that she had in mind. 

Just one final example I will offer, though all this represents but the tiniest glimpse into a potentially vast new field of investigation, is the holistic counterpart to the accepted binary system!

So the two binary digits 1 and 0 - given an independent interpretation in the standard analytic manner - can be potentially used to encode all information processes.

However, equally the two binary digits 1 and 0 - now given a holistic interdependent meaning as linear (1) and circular (0) type understanding respectively - can be likewise used potentially to encode all transformation processes

So for example, in this contribution, I have argued that the number system - and indeed all mathematics - should be interpreted as representing a dynamic interactive transformation process, entailing both quantitative and qualitative aspects.

And these two aspects relate directly to 1 and 0 respectively (in a holistic manner).

In fact, at the most general level, ultimately all differentiated and integrated processes both in physical nature and psychologically in human terms, are encoded in a holistic binary digital manner.


3) This, which I commonly refer to as Radial Mathematics or more simply the Type 3 approach, represents potentially the most comprehensive form of mathematical understanding, entailing the coherent integration of both analytic (Type 1) and holistic (Type 2) aspects.

When appropriately understood, all mathematics is intrinsically of a Type 3 nature (though not yet recognised because of the deep-rooted acceptance of reduced assumptions).

Indeed it is only in the context of Radial Mathematics (Type 3) that the other two aspects Conventional Mathematics (Type 1) and Holistic Mathematics (Type 2) can reach their fullest expression. So perhaps some day in the distant future, Type 3 will become synonymous with all mathematics.

However even within this category, I would distinguish three important sub-types, (a), (b) and (c) respectively.


Though rooted to a certain degree in the holistic aspect of Mathematics, sub-type (a) is mainly geared to the derivation of exciting new analytic type discoveries (with creative insight playing a key role).

There is no doubt that at least implicitly, Ramanujan belongs to this category. At a deeper level, I believe Riemann also belongs leading to highly original discoveries relying on a strong holistic dimension. However in neither case was the holistic dimension of mathematics explicitly recognised.

So in the future, even greater creative analytic discoveries in various fields will be possible, when mathematical talent is backed up with a truly mature holistic appreciation of symbols.


The second sub-type - while requiring appropriate analytic appreciation (the degree of which depends on the precise context of investigation) - is mainly geared towards the holistic interpretation of mathematical objects.

I would classify my own recent efforts as a most preliminary version of sub-type (b), operating necessarily at a very rudimentary level.

However this is still adequate for example to provide the bones of a distinctive dynamic appreciation of the number system with which to radically re-interpret the nature of the Riemann zeta function (and Riemann Hypothesis).

Thus, subtype (b) is not geared directly to analytic discovery, but rather a coherent dynamic interpretation of mathematical relationships. However because it is most creative at a deep level of enquiry, indirectly it can facilitate exciting new directions for analytic discoveries.


The final subtype (c) entails the most balanced version of both (a) and (b), opening up possibilities for the finest form of mathematical understanding, that is at once immensely productive and highly creative and readily capable for example of synthesising various fields of mathematical study.

One of its great benefits is that it can also provide the ready capacity to appreciate the potential practical applications of new mathematical discoveries.

The reason now why so much abstract mathematical analysis seems irrelevant in practical terms is precisely because it is understood in a manner that completely lacks a holistic dimension.

However with both aspects (analytic and holistic) properly incorporated, the practical applicability of abstract mathematical findings would be more readily intuited.

And from an enhanced dynamic perspective, all mathematical findings (which, when properly understood, are experientially based) have practical applications!

In this regard as a general principle, I would imagine that what is considered most important in abstract mathematical terms, is likewise potentially of greatest significance from an applied perspective.


Another great advantage of subtype (c) is that by its very nature, mathematics can now become readily integrated with the rest of human experience, allowing for the fullest expression of personality development.

Thus if we want a vision of what mathematics might look like at its very best, we would choose subtype (c) with respect to the Type 3 aspect.

However, we are still a very long way indeed from realising this wonderful reality, with the great lack yet of an established holistic dimension to mathematics, serving as the chief impediment.

Holistic Interpretation of Riemann Hypothesis

In the following section, I will attempt to provide a little more holistic perspective on the Riemann Hypothesis, with a view to representing its true nature.

All the (non-trivial) zeros are postulated as lying on an imaginary line, drawn vertically through .5 on the real axis.

So, one might ask what then is the key significance here of .5?

Well, in analytic terms, clearly .5 is half-way between 0 and 1.

When 0 is added to any number, its identity thereby remains unchanged.

And when any number is multiplied by 1, again its identity remains unchanged.

Thus the revealing point here is that in lying half way between 0 and 1, that .5 represents a point of balance, where in a sense both addition and multiplication attain an equal status with respect to the number system.

Then in more holistic terms, 1 represents the linear (analytic) approach and 0 the circular (holistic) approach to number respectively.

Thus once again from this perspective, .5 can be seen again as maintaining the golden mean in a balanced equality of both linear (analytic) and circular (holistic) understanding.


An even deeper rationale for the key significance of .5 (in holistic terms) can be offered.

In standard analytic terms, the external polarity of experience is abstracted from the internal.

So “1” as a number object is identified fully with its objective existence (as somehow possessing a validity independent of (internal) mental interpretation.

However when one equally allows for (external) objective reality and (internal) mental interpretation, then .5 again becomes the point where both polarities are fully balanced (as half external and half internal respectively)


Strictly in experience therefore, we cannot have external number reality without corresponding mental interpretation (of such reality) and vice versa.

Thus crucially, the significance of .5, points to the fact that for true interpretation of the zeta zeros, both external and internal aspects (as opposite poles of understanding) must be fully balanced with each other. And this in turn is necessary, so that holistic aspect of experience (which entails the complementary balancing of opposite poles) can itself be properly incorporated in experience.

Thus the fullest appreciation of the zeros requires a highly refined understanding, where analytic and rational aspects are equally balanced in a psycho-spiritual manner approaching pure ineffability.

Put another way, such understanding provides the most complete marriage of refined reason (directly in linear and indirectly in circular terms) with a contemplative vision (of a purely intuitive nature).

So here, objective reality with respect to number and the mental interpretation of such reality are so closely allied as to be ultimately inseparable.


The next key point relates to the fact that all of these zeros are postulated to lie on an imaginary line (through .5).

The holistic significance of the imaginary notion, as I have stated, is that it provides an indirect analytic expression of what is, directly, holistic in nature.

In this way, important notions of unconscious intuitive origin can be incorporated with standard analysis (of a rational conscious nature).

When seen from this perspective, the over-riding significance of the imaginary line (on which all the non-trivial zeros are postulated to lie), is that it indirectly serves as the unconscious shadow counterpart of the real number line, which is given a directly conscious interpretation in mathematical analysis.

In fact this is vitally needed to provide a proper answer to the important query regarding multiplication that I have already posed.

Once again, when we multiply numbers, a qualitative dimensional change takes place, which directly relates to the interdependence of such numbers with each other.

So we might indeed start by considering primes such as 2 and 3 as independent (and thereby “building blocks” of composite natural numbers).

However once we associate 2 * 3 with each other - and the individual units within each number - in a multiplication operation, then a qualitative dimensional aspect is introduced through such interdependence.

Now effectively, this is all ignored in conventional mathematical interpretation, where the result of 2 * 3 is given a merely abstract quantitative identity.


A unique qualitative aspect equally arises when we add numbers. For example, when we add 1 + 1, we may conclude that the answer is 2 in a merely quantitative manner, but without establishing the interdependence (as well as assumed independence) of both units, it would not be possible to appreciate the new collective whole identity of 2. 

However a deeper question thereby arises, which unfortunately from a conventional perspective, is completely overlooked.

Given that a qualitative identity necessarily arises when numbers are associated with each other through multiplication (and addition), how can we then assume the consistency of the reduced - merely quantitative - interpretation of results given in conventional mathematical terms?

And the answer remarkably, which ensures such consistency, lies at the very heart of the Riemann Hypothesis!

Thus, for the consistency of the real line to hold (in conventional mathematical analysis) it is necessary that the assumption regarding the imaginary line likewise holds with respect to the Riemann Hypothesis.

Therefore, for the consistency of the real line to hold, all the non-trivial zeros must lie on the vertical imaginary line (through .5).

So, when interpreted appropriately, we are in an important sense enabled to effectively distinguish the quantitative aspect of number independence from the corresponding qualitative aspect of number interdependence. 

Conventional interpretation indiscriminately mixes notions of number independence with corresponding notions of number interdependence, blindly assuming that both aspects are consistent with each other with respect to the real number line.

However proper recognition of the true qualitative aspect of number requires that it is treated in a distinct holistic manner as indirectly relating to an imaginary (rather than real) line.

Thus, if all non- zeros - indirectly expressing, in a quantitative manner, the qualitative interdependence of the primes with natural numbers - also lie on a straight imaginary line, we are then entitled to assume the consistency of qualitative with quantitative interpretation.

Expressed in a more psychological manner, if all the non-trivial zeros lie on the imaginary line (through .5), then we are entitled to assume, in mathematical terms, that the unconscious (holistic) aspect of number understanding can be seamlessly integrated with its corresponding conscious (analytic) aspect.

However in the highly limited approach that formally defines conventional mathematical interpretation, the unconscious aspect (though utterly distinct) is blindly ignored and thereby reduced in mere conscious rational terms.

So the key holistic issue, to which the Riemann Hypothesis points, does not even arise in this reduced context. And again, this is why the Riemann zeta function remains undefined in holistic terms for s = 1, i.e. undefined for the conventional mathematical approach!


I will attempt to deal even more briefly with a couple of further issues.

The non-trivial zeros occur as complex conjugates, so that each zero of the form a + it is matched by a corresponding zero of the form a – it.

So there is a complementary situation in place for the imaginary aspect of the zeros (as positive and negative with respect to each other).

Now again a close analogy with physics might help to suggest the true holistic nature of this complementary pattern.

Virtual particles very close to the ground of matter have a highly transient existence, arising in closely related, matter, anti-matter pairs that immediately combine with each other in a fusion of physical energy. And in this context, virtual can be directly equated with the imaginary notion.

So the fact that the non-trivial zeros occur in imaginary pairs relates to the fact that the positive and negative polarities with respect to imaginary opposites are extremely closely related. From a psychological perspective, this implies that the short-lived separate existence of any number projection from the unconscious becomes quickly eroded through relationship to its opposite pole (thus approaching a purely intuitive appreciation of number as a psycho-spiritual energy state).

And this would be complemented at a physical level by the highly transient activity of virtual particles again approaching a pure energy state.

In fact it is highly revealing, when one adds non-trivial zero pairs.

For .5 + it + .5 – it  = 1.

Thus in a sense, these conjugate pairs represent the extremely short-lived break-up of conventional 1-dimensional appreciation into two highly dynamic inter-related realities.


Thus number is conventionally understood at the everyday conscious level of reality, as pertaining to the 1-dimensional level (i.e. where all real numbers lie on the number line).

However when one is able to see behind this reality into the refined intricate manner in which both conscious and unconscious are related with respect to number, then one can begin to see directly into the domain of the non-trivial zeros. Thus from this perspective, the zeros relate to an unseen highly dynamic background context (directly of unconscious origin) of a vitally important nature, that enable our everyday conscious assumptions regarding number reality to seem self-evident.

It also would appear that the imaginary parts of the zeros constitute transcendental numbers.

20 years ago, when I was investigating the holistic nature of the various number types, I concluded that the most elusive - and thereby closest to pure contemplative appreciation - are the imaginary transcendental numbers.

So in approaching the holistic nature of a transcendental number it may help again by considering the best known, i.e. π.

In analytic terms, this represents the relationship between the circumference of the circle and its line diameter.

And in holistic terms, all transcendental numbers express a certain relationship as between circular (holistic) and linear (analytic) aspects of understanding, where both are seen as interdependent with each other.

The imaginary notion, in this context, reflects the projection (directly, of an unconscious holistic nature) indirectly, in a conscious analytical manner. So, to be incorporated successfully with experience, unconscious material must initially be projected in a conscious manner. Then to avoid the confusion of holistic with analytic meaning, one must immediately realise in experience the distinct nature of imaginary projections, therefore avoiding any rigid attachment.

When highly developed, imaginary transcendental understanding becomes so refined that phenomena no longer even appear to arise, thus approaching a true contemplative vision (in the pure psycho-spiritual release of intuitive energy).

In this way, the non-trivial zeros as energy states represent the holistic extreme, approaching pure relativity, to the conventional understanding of number as representing absolute unchanging forms. 


From the dynamic interactive perspective, it is readily apparent that the Riemann Hypothesis can be neither proved (nor disproved) in the accepted conventional mathematical manner.

In this sense the Riemann Hypothesis, by its very nature, transcends conventional mathematical interpretation, while already being immanent (i.e. necessarily assumed) in all such interpretation.

As we have seen, the conventional approach is based on the reduction of unconscious type meaning in a directly conscious manner.

However properly understood, from a dynamic perspective, the fundamental mathematical issue relates to consistency of the distinctive intuitive meaning of mathematical symbols at an unconscious level with their corresponding rational understanding in conscious terms.

And from this perspective, the Riemann Hypothesis can be interpreted as the condition ensuring the consistency of both aspects in the context of the overall number system.

So, for example as we have seen when we multiply 2 by 3, as well as the recognised quantitative transformation to 6, this likewise entails a qualitative dimensional change in the nature of the units employed.

Therefore the validity of reducing the result in a merely quantitative manner, so that 6 is now represented as a number on the (1-dimensional) number line, is based on the assumption of the Riemann Hypothesis that all the (non-trivial) zeros lie on the imaginary number line, where quantitative and qualitative aspects of number interpretation are fully reconciled with each other.


The intuitive unconscious can be seen as the shadow counterpart to the rational conscious mind.

Thus the assumption that all natural number quantities can be placed on the real number line requires the complementary (shadow) assumption that all the (non-trivial) zeta zeros lie on a corresponding imaginary line. 

In fact properly understood in dynamic interactive terms, we can no more guarantee that all the natural numbers lie on the real, than that all the non-trivial zeros lie on the corresponding imaginary number line.

So in dynamic interactive terms, both assumptions mutually imply each other.

An important limitation of conventional mathematical interpretation is that in reducing the (unconscious) qualitative aspect of mathematical understanding, in a (conscious) quantitative manner, it thereby already assumes the Riemann Hypothesis to be true.

So again, the blind assumption that all natural numbers can be placed on the real, already implicitly assumes that the non-trivial zeros lie on the imaginary line.

Put another way, the assumption that the Riemann Hypothesis is true, is already automatically implied by conventional mathematical axioms (through blindly reducing distinctive qualitative interpretation in an absolute quantitative manner).

We cannot thereby prove the Riemann Hypothesis within this limited mathematical framework (where what we are trying to prove is already implied by our axioms).  

Likewise, we cannot strictly disprove the Riemann Hypothesis from within these same axioms.

One might immediately point to a non-trivial zero that perhaps does not lie on the critical imaginary line as offering conclusive proof that the Riemann Hypothesis is false.

The discovery of such a zero would indeed be a momentous find, for it would undermine the consistency of our basic assumption regarding the number line.

In other words, we would no longer be entitled to assume that all the composite natural numbers (derived through multiplication of the primes) lie on the same number line as the primes.

So this in turn would strictly undermine the consistency of all mathematical proof with respect to the number system.

Therefore we would no longer be conclusively able to prove or disprove any mathematical proposition involving number from within such a flawed system.

Thus it is very important indeed that no non-trivial zero is ever found off the imaginary line, as this would rightly undermine our faith in the consistency of the whole mathematical enterprise!

Thus the Riemann Hypothesis, in the assumption that all the non-trivial zeros lie on the imaginary line (through .5) must remain a matter of faith, just as strictly, our corresponding belief that all natural numbers lie on the same real line, must likewise remain a similar matter of faith (with both implying each other).


So from one perspective, in dynamic interactive terms, we can accept that all the natural numbers lie on the real line if the Riemann Hypothesis is true; equally from the complementary perspective, if all the natural numbers indeed lie on the real line, then the Riemann Hypothesis is true. However, once again neither proposition can be proved (nor disproved) within the conventional mathematical framework of absolute quantitative identity.

I have been discussing above the holistic significance of the non-trivial zeros.

Equally they possess an important analytic significance in quantitative terms.

And it is this aspect of the zeros that is given exclusive attention within the accepted mathematical framework. 

Thus accepting the truth of the Riemann Hypothesis, it is possible to provide a formula (incorporating crucial adjustments based on the non-trivial zeros), enabling an exact quantitative estimate of the number of primes (to any given real value of n).

However, from the complementary opposite perspective, it is equally possible to provide a formula based on knowledge of the individual primes, to enable an exact quantitative estimate of the number of non-trivial zeros (to any given imaginary value of n).

So it should be clear - even from this quantitative perspective - that the primes and non-trivial zeros are mutually interdependent with each other, thus implying an important holistic - rather than analytic - connection.

However, from my reading of the conventional literature, I can see little recognition of this crucial point.


Before leaving this section, I wish to elaborate further on the true holistic nature of the (non-trivial) zeta zeros.

In this regard it may be helpful to remind ourselves once more of the earlier crossroads example.

We saw there that when the crossroads is approached from one direction (travelling N or S), then left and right turns can be given a separate independent meaning.

However, when one recognises an approach from both N and S directions simultaneously, then left and right turns are rendered paradoxical, with what is left from one perspective, right from the other, and vice versa. 

It is somewhat similar with respect to the overall relationship as between the primes and natural numbers.

When one approaches the primes from the quantitative perspective of independent “building blocks”, then the natural numbers appear to be unambiguously derived from the primes.

However when one now approaches the primes from the qualitative perspective of the unique combinations of prime factors, then the position of each prime appears to be determined through their overall collective relationship to the natural number system.   

Then when one simultaneously attempts to view the number system from both perspectives, clearly their relationship to each other appears paradoxical; so from the quantitative perspective of prime numbers, the natural numbers appear to depend on the primes; however from the - relatively - qualitative perspective of prime factors, the position of the primes (and thereby their quantitative identity) appears to depend on the natural numbers.

Now in holistic terms, the (non-trivial) zeros indirectly express in quantitative terms, this paradoxical identity of both the primes and natural numbers (as mutually interdependent with each other).

And as we have seen from the holistic perspective, in this context, the imaginary notion represents an indirect expression of the interdependence of the cardinal numbers. So each zero thereby represent a point on the imaginary line, where in a very true sense the identity of the primes is inseparable from the corresponding identity of the natural numbers.


So again from one perspective, we recognise that the primes have a unique individual nature. Then from the other, we can equally recognise that the collective distribution of primes among the natural numbers is predictable to progressively higher levels of relative accuracy

And the non-trivial zeros can be seen in this context as points that harmonise both the individual and collective identity of the primes (with respect to the natural numbers). This is borne out by the fact that the general formula for the frequency of non-trivial zeros (up to any number on the imaginary line) predicts results with an amazing degree of accuracy, not only in relative but also absolute terms.

So we could say that with each zero, the individual uniqueness of each prime is harmonised with the collective relationship of all the primes to the natural numbers. However because each zero still maintains a certain discrete identity, this can only be dynamically achieved in an extremely close approximate, rather than exact manner.

Equally we could say that at each point, as already stated, that the operation of addition is inseparable from that of multiplication. So it is through the zeta zeros that addition and multiplication are mutually reconciled in a fully consistent manner.

Perhaps most simply we could say that at each point the qualitative nature of number is inseparable from its corresponding quantitative identity. So here the analytic and holistic aspects of number mutually coincide in direct fashion.


So once more, we are at the opposite extreme from conventional mathematical interpretation, where the quantitative identity of each number is abstracted in absolute fashion from the qualitative (with the qualitative thereby reduced to the quantitative).

And this again is the key reason why one cannot hope to uncover the true meaning of the Riemann zeta function (and Riemann Hypothesis), while remaining rigidly within a mere analytic framework (based on absolute quantitative identity).

Looked at from yet another important perspective, the key role of the Zeta 1 i.e. (non-trivial) zeros is to serve as a seamless means of two-way conversion as between the Type 1 and Type 2 aspects of the multiplicative number system (as the unique combination of prime factors, that are quantitative and qualitative with respect to each other).

Likewise the key role of the Zeta 2 zeros - that we encountered earlier - is to serve as a seamless means of two-way conversion as between the Type 1 and Type 2 aspects of the additive number system (again where each number is given both a quantitative and qualitative interpretation)


In fact properly understood, the number system dynamically entails the complementary interaction of both the multiplicative and additive approaches.

So from the multiplicative perspective, when we multiply primes (as unique factor combinations) each prime is already defined internally from the additive perspective.

Thus as in our oft quoted example we multiply 2 by 3, both numbers are already defined in additive terms (with respect to individual units).

Likewise when we look internally at the individual members of a composite natural number from the additive perspective, this number already involves (through its unique prime factors) the multiplicative approach.

Therefore, to simply illustrate, though 6, as a composite number, can be clearly represented as the sum of its individual units, it already necessarily entails, through multiplication, a unique combination of prime factors (i.e. 2 and 3).

Thus again, both the internal and external distribution of the primes and natural numbers  (from both additive and multiplicative perspectives) are dynamically interdependent with each other, with both distributions ultimately determined in a holistic synchronous manner approaching pure ineffability.


Finally from this dynamic perspective, we can establish the truly fundamental relationship of the primes to the natural numbers.

Quite simply, the transmission mechanism of both quantitative and qualitative characteristics for the number system (and ultimately all created phenomena), is through the two-way complementary interaction of the primes with the natural numbers (and the natural numbers with the primes).

From a psychological perspective, this entails that the very manner in which both the conscious and unconscious aspects of personality interact is through this same fundamental relationship of primes and natural numbers (as quantitative as to qualitative and qualitative as to quantitative with respect to each other).

Some years ago I was struck by the deep similarity of the psychological meaning of the word “primitive” to my emerging dynamic appreciation of the nature of prime numbers in mathematical terms.

The essence of primitive instinctive behaviour is that a direct confusion exists as between the holistic nature of the unconscious and the specific form of conscious understanding.

So for example, in earliest infant behaviour, because of the degree of such confusion, the differentiation of distinct phenomena in experience is not yet possible. Thus, the infant is not yet able to provide an appropriate dimensional framework in space and time (relating to the holistic unconscious), with which to experience specific phenomena (in an analytic conscious manner).

Thus remarkably, solving the primitive problem, as it were, in psychological terms - in initially separating both conscious and unconscious aspects of personality with a view to their eventual integration - exactly mirrors the corresponding mathematical issue of solving the problem of the primes.

So here, one must successfully separate both the analytic (quantitative) and holistic (qualitative) aspects of mathematical understanding with a view to their eventual integration, where both aspects can thereby be seen to operate in a seamless interdependent fashion


So at some future stage in our human evolution, it will perhaps be realised that truly solving the problem of the primes mathematically (where both analytic and holistic aspects of understanding are property integrated), is inseparable from solving the corresponding “primitive” problem in psychological terms (where both conscious and unconscious aspects of personality can finally operate together in a seamless harmonious fashion)


And just as the Riemann zeros are now being linked to certain physical energy states (with respect to atomic behaviour), the Riemann zeros will likewise then be intimately linked to certain psychological energy states (representing the most advanced forms of spiritual contemplation). 


Thus once again, the enormous problem with respect to conventional mathematical interpretation is that it remains completely in denial regarding its holistic (unconscious) aspect.

A well-known physical analogy is very relevant in this context


In quantum physics, the uncertainty principle implies that we can never exactly determine both the position and momentum of a particle. So for example, if we attempt to improve our exact knowledge of a particle’s position, knowledge of its corresponding momentum becomes increasingly fuzzy.

Now an important uncertainty principle applies to all mathematical understanding regarding both analytic (quantitative) and holistic (qualitative) appreciation of relationships.

Thus, if we continually attempt, as is presently the case, to attain ever more precise analytic appreciation of relationships - through attempting to formulate them in an increasingly abstract manner - this then acts to completely block out recognition of the corresponding holistic aspect.   

And this, much more than even such a fundamental problem as the Riemann Hypothesis, is truly the key issue which requires to be now addressed by the mathematical community.

For despite the - admitted - stunning advances that have been made with respect to its present highly specialised approach, it represents but a limited though important unique case of an altogether more comprehensive vision of Mathematics.


Growing recognition of this much greater role of mathematics will inevitably have enormous implications for all the sciences and ultimately for society in general, representing perhaps the greatest revolution yet in our intellectual history.

We are now at an unprecedented stage of development where dramatic changes in technological, political, social, economic and environmental terms are set to take place.

And for successful adaptation, a major transformation in our basic mental mind-set is required that goes to the very core of mathematical understanding.