Zeta 3 Zeros - Key Significance (4)

Just to briefly recap!

When we start out (internally) with each individual prime, the corresponding ordinal notion of that prime remains holistically undefined by the Zeta 2 zeros.

Then, when we start out (externally) with the collection of all the primes, the corresponding cardinal notion of each individual prime remains holistically undefined by the Zeta 1 (Riemann) zeros.

Therefore in truth the two sets of zeros (along with both the cardinal and ordinal nature of the primes) are intimately related in bi-directional fashion. And it is this combined appreciation of the dynamic interaction of both sets of zeros, that I refer to as the Zeta 3 zeros.

In other words, just as internally the two-way relationship as between prime and natural numbers, indirectly expressed though the Zeta 2 zeros, arises in a synchronistic manner (that is ultimately ineffable), likewise externally, the two-way relationship as between prime and natural numbers, now indirectly expressed through the Zeta 1 zeros, likewise arises in a synchronistic manner (that is ultimately ineffable).

So therefore the internal synchronicity (expressed through the Zeta 2 zeros), dynamically implies the external synchronicity (implied by the Zeta 1 zeros).  Likewise the external synchronicity (expressed through the Zeta 1 zeros) dynamically implies the internal synchronicity (expressed through the Zeta 2 zeros).

Thus in respect to (internal) Zeta 2 zeros, we start with the initial requirement that each number is prime so that only the prime roots of 1 (except 1) are extracted.

Therefore in order to extend the Zeta 2 zeros to encompass all the composite natural numbers (as well as primes) we require the external relationship implied by the Zeta 1 zeros (where the primes are seamlessly integrated in collective manner with the natural numbers).

And in turn the "circular" nature of the Zeta 1 zeros intimately depends on the initial default definition (with respect to the standard unit radius) in Zeta 2 terms.   

Therefore the great fundamental question of how both quantitative and qualitative aspects of understanding can be consistently combined in the  understanding of number - and indeed every other possible mathematical relationship - ultimately dissolves in total mystery.

So underlying the entire mathematical edifice, inescapably lies an initial act of faith in the subsequent consistency of all its relationships.

This prior consistency - as between its quantitative and qualitative aspects - cannot be proven from within its axioms. Rather the subsequent use of such axioms depends on this initial prior assumption of consistency.

And this is as close as we can come in the mathematical realm to a pure TOE. However it is one that renders all - merely - analytic attempts to grasp the ultimate nature of reality as futile.

Though it has been greatly forgotten in recent centuries, the true goal of Mathematics - as the ancients better realised - is to combine both the analytic and holistic aspects of understanding in the marriage of reason and contemplative wisdom (as refined intuition). Here, science ultimately ends in comprehension of the pure mystery of all existence.

So mathematical symbols, especially with respect to number - rightly interpreted - ultimately represent the last phenomenal veil that bridges the phenomenal world of form from pure ineffable emptiness.

However - and this point requires to be repeatedly stressed in our present mathematical culture - all such symbols have both an analytic (quantitative) and holistic (qualitative) meaning so that strictly, in dynamic interactive terms, these symbols entail the relationship of both aspects.


I have commented before e.g. "Buddhist Mathematics" on the famous heart sutra;

"Form is not other than Emptiness
Emptiness is not other than Form"

Now translating this in holistic mathematical language;
Oneness (1) is not other than Nothingness (0)
Nothingness (0) is not other than Oneness (1)

In other words, the unity of all form i.e. in the holistic appreciation of what is spiritually inherent in all creation, is ultimately inseparable from the corresponding appreciation of (spiritual) emptiness (as not being identifiable with phenomenal forms).

Now an important feature of the present IT revolution is that it owes so much to the rational analytic interpretation of 1 and 0 as separate binary digits.
However the corresponding holistic realisation is that 1 (as oneness) and 0 (as nothingness) are ultimately identical (in a spiritually intuitive manner). Therefore, just as all information can be potentially encoded using the analytic interpretation of the binary digits, likewise all (dynamic) transformation processes can likewise be potentially encoded using the holistic interpretation of the same two digits (relating to linear and circular type logic respectively).

When one thinks about it for a moment, the two-way relationship of the primes and natural numbers does not include - what I refer to as - the two original numbers i.e. 1 and 0.

So in an important sense, at an even more fundamental level of mathematical reality, 1 and 0 themselves originate in a synchronistic manner and then remain deeply implicit in all other number notions such as the primes and remaining natural numbers.

Thus one obvious weakness with the conventional approach to explaining the natural number system is that in postulating the primes as the independent "building blocks" of that system, 1 (the most important natural number) is automatically excluded, even though using the Peano-based approach, the entire natural number system can be derived from 0 (through the successive adding of 1).

So a clear benefit of the dynamic approach I advocate, is that this necessarily entails the two-way interaction of the Peano-based addition and the prime-based multiplication approaches respectively.

Indeed it is only within this dynamic interactive approach that the true complementary relationships as between addition and multiplication can become evident.

However already underlying the additive and multiplicative approaches to derivation of the natural numbers, is the prior two-way relationship as between the original numbers 1 and 0.

In fact in its most basic form, 1 and 0 are necessarily always involved in the dynamic interaction of the quantitative and qualitative aspects of number.

For what is distinctly appreciated as 1 (in a quantitative manner) entails the corresponding realisation that it thereby represents 0 (in a qualitative fashion).
In this sense, what is distinctly recognised as a unit (1) in a quantitative analytic manner, thereby cannot be distinctly recognised as a corresponding unit (1) in qualitative holistic terms.
So what represents 1 in analytic terms, represents 0 in a holistic manner.

And likewise what is then generally recognised as a general unit (1) i.e. in the overall appreciation of oneness in a qualitative holistic manner, is thereby not distinctly recognised as a part unit (1) in quantitative analytic terms.
So, in reverse manner, what represents 1 in holistic terms, represents 0 in an analytic manner.

Therefore, in experience, one continually switches in two-way fashion as between its quantitative and qualitative aspects through the mutual two-way switching as between 1 and 0.

And this is intimately tied up with the proper appreciation of number.

Now I have repeatedly emphasised that there are two aspects to number, which I refer to as Type 1 and Type 2 respectively.

So the number "2" for example is defined as 2¹ in the Type 1 and 1² in the Type 2 aspects respectively, with 2 being identified as the base in the former and the dimensional number in the latter aspect respectively.

Now, from an additive approach, we can represent 2 as 1 + 1 (with respect to both aspects).

However, what is not recognised in conventional mathematical terms is that the very meaning has now switched from a - relatively - analytic perspective, in the Type 1, to a holistic perspective in Type 2 terms.
In other words, Type 1 relates to independent units, whereas Type 2 relates to interdependent units respectively. Put another way - reminiscent of quantum mechanics - we have thereby switched from a particle to a wave definition of number.

So when the use of 2 (as base number) refers to separate individual units (in an analytic manner) the corresponding use of 2 (as dimensional number) properly refers - in dynamic relative terms - to the holistic relationship of these units with the whole.

For example - say - in a more concrete manner, we divide a cake into 2 (equal) slices. Each of the slices now represent independent units in analytic terms. However the overall relationship of these (part) slices to the (whole) cake, represents interdependent units in a holistic manner.

So the proper use of 1 in this context - referring to the qualitative whole - is 0 (in dynamic relation to a quantifiable slice).

Likewise the use of 1 - now referring to each quantifiable part - is 0 (in dynamic relation to the qualitative whole).

So again just as 1 and 0 - as statically understood in analytic terms - are so important in terms of encoding information, 1 and 0, as dynamically understood in holistic terms, are equally important in terms of encoding - potentially - all transformation processes. This then serves as the fundamental means through which switching in both physical and psychological terms takes place as between quantitative (differentiated) and qualitative (integrated) phenomena respectively.

Zeta 3 Zeros - Key Significance (3)

In discussing the Zeta 2 zeros, I was at pains to clarify the distinction as between the analytic and holistic interpretation respectively of the ordinal notion of number.

In the context of a given number t - which initially is defined as prime - the corresponding t roots of 1 provide an indirect quantitative means of expressing the natural number ordinal members of that prime (considered as a group of related members).

So once again, for example, in the context of 3 (as prime) the 3 roots of 1 express - in an indirect quantitative manner - the qualitative notions of 1st 2nd and 3rd respectively which in terms of the Type 2 aspect of the number system are represented as 1^1/3, 1^2/3 and 1^3/3 respectively.

However the last root here representing the notion of the 3rd of 3 (with the two other positions fixed) = 1, is quite distinct from the other roots. And likewise this is always true of the last ordinal position (with the other positions fixed).

So this last root in fact represents the conventional analytic interpretation of the ordinal notion, whereby it is directly identified with the corresponding quantitative definition of the number in cardinal terms.

Thus in this analytic context, 1st is identified as the last unit of 1 (which of course = 1). Then with this position fixed, the 2nd is identified with the last unit of 2 (= 1). And finally in the case of our example of 3, the 3rd is identified with the last unit of 3 (= 1).

So the cardinal definition of 3 = 1 + 1 + 1 then happily coincides with the corresponding ordinal definition of 3 = 1st + 2nd + 3rd.

However when we strip out this "trivial" case where the root = 1, we are then left with the remaining roots that directly relate to the Zeta 2 zeros.

So again in the case of 3, we obtain the solution of

1 + x¹ + x² = 0, which yields the other 2 roots (of the 3 roots) of 1.

And these roots representing the notions of the 1st of 3 and 2nd of 3 (where positions are not fixed) - again in an indirect quantitative manner -  express the true holistic meaning of these ordinal notions.


However, once again we are left with a paradox.

Though we start with a prime (as  dimensional number in the Type 2 aspect) the ordinal notion relating to this prime, remains undefined by the Zeta 2 zeros.

So again though we have obtained, in the case of 3, an indirect quantitative expression of the holistic notions notions of 1st and 2nd (with respect to 3), this does not apply to 3rd (which reduces down to its conventional analytic meaning).

And this applies in turn to each of the primes which cannot holistically be given an ordinal interpretation (in relation to the corresponding group of prime members).

So we can give a holistic meaning to 3rd, for example in the context of 5 members; however we cannot do this in relation to 3 members!

So the Zeta 2 zeros relate to a holistic notion of internal order (i.e. with respect to the individual members - excepting the last - of a prime group of members).

However we also have, in complementary dynamic fashion, a corresponding holistic notion of external order (i.e. with respect to the collective relationship of the primes with the natural numbers).


Now it is puzzling in a way that the term "ordinal" in a mathematical sense is solely identified with the Peano based additive approach to the number system i.e. where each natural number is obtained through the addition of 1 to the previous natural number. So the notion of order (1st, 2nd, 3rd,...) is here identified successively with each of the natural numbers in a strictly linear fashion.  

As we know, there is a corresponding way of deriving the natural numbers based on the prime numbers and the operation of multiplication. Thus from this alternative perspective, each natural number represents a unique product of prime factors. 

However, surprisingly though we readily associate corresponding ordinal notions with the first additive approach to the natural number system, this is greatly lacking with respect to the second multiplicative approach.

Now clearly we are talking about a different kind of order here in relation to the primes, which is not of a neat successive nature. However it is an extremely important kind of order nonetheless!

Thus the primes while maintaining their distinct individual identity in quantitative terms, can be collectively combined with each other in an apparent seamlessly integrated fashion.

So when we talk about the "ordinal nature of the primes", we are referring to this qualitative relationship of factor interdependence, which uniquely enables the consistent generation of the natural number system.

And just as the Zeta 2 zeros provide an indirect quantitative means of expressing the internal ordinal  nature of the individual natural number members of a prime number group (excepting the prime itself), likewise in complementary fashion, the Zeta 1 (Riemann) zeros provide an indirect quantitative means of expressing the external ordinal nature of  the unique collective relationship of prime factors with respect to the natural number system.

And again, as in the previous case, we have a paradox in that on this occasion the individual primes are themselves defined in an analytic manner (directly with respect to their quantitative value).

However, when these primes are then combined with each other as the unique factors of composite natural numbers the qualitative aspect of the primes then operates in a holistic manner.

Thus the Zeta 1 zeros strictly relate to this holistic qualitative nature of the collective order of the primes. In this way they are the mirror opposite of the primes.

So the individual primes have a recognised quantitative identity (as the" building blocks" of the natural numbers). Then the collective relationship of the primes through the unique products of prime factors (that consistently enable the generation of the natural numbers), expresses the (unrecognised) qualitative holistic identity of the primes. In fact the proper intuitive recognition of this - literally - represents the collective energy of the primes.

Then it operates in reverse in relation to the Zeta 1 (Riemann) zeros. Each individual zero represents a unique holistic point of qualitative identity (approaching a pure energy state). Like the two turns at a crossroads, this arises from the paradoxical recognition of the two-way relationship of primes and natural numbers (indirectly expressed on an imaginary scale), which coincides with these zeros.  

Then the collection of zeros represents their quantitative identity, whereby they can be used to correct the errors arising from the general estimate of prime frequency to a given number so as to eventually zone in on the correct absolute value.           

Zeta 3 Zeros - Key Significance (2)

I wish to comment again on the unavoidable paradox inherent in the attempt to define number in an absolute analytic type manner.

Once again, one may attempt to define a number, such as 3, as composed of independent homogeneous sub-units (with no qualitative distinction).

However, by definition this means than any of these units can be defined as 1st, 2nd or 3rd in an ordinal context. In other words, without some qualitative distinction, we have no means of distinguishing 1st, 2nd and 3rd in analytic ordinal terms.

Thus the starting notion of absolute independence in a quantitative cardinal manner, implies the opposite extreme notion of pure relative interdependence in a qualitative ordinal manner.

And within a rigid static framework, there is no way of coherently reconciling quantitative (cardinal) and qualitative (ordinal) number notions. Expressed equivalently in a psychological manner, there is no means therefore for the coherent reconciliation of holistic (intuitive) with analytic (rational) notions!

Thus  both aspects can be only be given given proper interpretation within a dynamic interactive framework, entailing both aspects of relative independence and relative interdependence respectively, through which we can provide a coherent approach for number.


This problem with number is closely related to the physical fact that we must always - again for coherent interpretation - view number as existing in a relative dimensional context of space and time.

In fact there is an extremely close connection - indeed they are ultimately indistinguishable - as between both the physical and psychological understanding of space and time and the holistic mathematical notion of number (in dimensional terms).

This indeed is another key reason why this dynamic approach to number, that I have been consistently proposing is so radical, in that it automatically leads to a completely new holistic manner of understanding space and time dimensions (in physical and psychological terms).

So the current rigid physical notions that we live in a world of three space and one time dimension automatically follows from the reduced analytic attempt to view all relationships with respect to their merely quantifiable aspects.

Then when modern approaches such as string theory propose a 11-dimensional world (10 of space and 1 of time) it carries no holistic relevance. In other words, we can make little or no no intuitive sense of such notions within the accepted scientific i.e. merely analytic, interpretation.

In fact, properly understood - from a holistic perspective - current mathematical and scientific interpretation is radically 1-dimensional and is strongly identified with the manner we understand time as flowing with respect to reality - identified thereby in "real" terms" - in a positive forward direction. The customary three space dimensions are then separated from time and rigidly identified with perceived features of objects (as having 3 dimensions).

And this view of the world is ultimately untenable, as the physical features of object phenomena have strictly no meaning independent of the psychological mental constructs we use to interpret such objects. However the accepted analytic approach continually attempts - even while admitting problems with this approach at the quantum level of investigation - to abstract in an absolute manner the (subjective) knower from what is (objectively) known about reality. 


So I will briefly deal here with the simplest case of how the true holistic interpretation of "2" (as number dimension) leads to a new understanding of the nature of space and time.

The holistic meaning of 2 relates to the understanding of 2 as representing interdependent - rather than independent - units. Now, clearly where complete interdependence is involved, we can no longer distinguish units as separate. So this would concur with the purely intuitive appreciation of 2 (as an energy state).

However in practice, the holistic (interdependent) aspect of number must be constantly balanced by the corresponding analytic (independent) aspect for meaningful interpretation. Thus notions of number interdependence are always of a relative nature, that also necessarily imply corresponding notions of number independence.

In this context, I have detailed at length how the two notions come together in the understanding of turns at a crossroads. So within a 1-dimensional frame of reference, when one approaches the crossroads - say - heading N, then left and right turns have an unambiguous meaning. Thus, these represent two separate independent turns (in analytic fashion).

However within a 2-dimensional frame of reference, when one envisages approaching the crossroads from N and S directions simultaneously, left and right turns have a circular (paradoxical) meaning. So what is left from one direction (heading N)  is right from the other (heading S) and what is right from one direction (heading N) is left from the other direction (heading S).

Equally we could say in this holistic context, that what is + 1 is also – 1; and what is – 1 is also + 1.
So this circular (paradoxical) language arises through attempting to express the 2-dimensional holistic dimesnional notion of "2" (representing the interdependence of the two units) indirectly in a 1-dimensional analytic fashion. 

And as we have seen, this concurs with the mathematical task of obtaining the 2 roots of 1.
So in correct dynamic terms, the two roots of 1, i.e. + 1 and – 1 have an analytic meaning as relatively independent; however they equally have a holistic meaning (when combined) as relatively interdependent. 
So the sum of the n roots of 1 (in this case 2) = 0, indirectly expresses the holistic interdependent i.e. qualitative nature of the number n as - literally - without quantitative meaning.


However to see clearly the implications here for the understanding of space and time, let us consider the two dimensions as relating to the external (objective) and internal (subjective) aspects of experience respectively.

Now when one becomes externally aware of a number object, it is thereby physically posited in space and time. One is thereby enabled to separate its individual identity in space and time.

Then, when one becomes internally aware of its corresponding mental perception, it is thereby psychologicially posited in space and time. So one is now likewise enabled to separate its individual identity, internally in  space and time.

In conventional terms, a very reduced interpretation is given, whereby the number object is assumed to possess an absolute identity (in an external objective manner). This implies that its corresponding internal mental perception - if indeed this is even recognised - is assumed to correspond in an absolute manner with the number object, which in view of the experiential relationship involved is strictly speaking absurd!

It is because of this assumed absolute correspondence as between (physical) objects and (psychological) interpretation that we believe in the linear nature of time and space.

In other words, movement is viewed to take place strictly in a positive (forward) direction. And - as we have seen - this is based on the 1-dimensional mode of analytic interpretation that is the bedrock of all accepted mathematics!.

However, once we recognise the truly relative nature of the two-way polar interaction of external and internal aspects with respect to experience, then space and time likewise assume a merely relative identity.

So again, if we identify the forward movement of space (and time) with the external recognition of the number object, then  the internal recognition of its corresponding mental perception must move - relatively - in a backward (negative) direction.

Then when reference frames switch to the internal mental interpretation of the number object, the forward movement of space and time is now identified with this psychological direction; this then implies that - relatively - the external number object is now associated with the backward (negative) direction.

Thus from a 2-dimensional holistic perspective, space and time are understood in relative fashion (in complementary physical and psychological terms) as both possessing 2 real dimensions (that are positive and negative with respect to each other).

And far from this representing some abstract notion, this inherently explains the true  nature of space and time (where one's experience is now sufficiently refined to explicitly operate at the 2-dimensional level).

In the past such experience has been largely identified with the great spiritual contemplative traditions (East and West), without however any sustained attempt to relate the implications of such refined intuition to accepted scientific and mathematical understanding.

So the point I am making is that the holistic mathematical notion of dimension equates exactly with the corresponding nature of how space and time are experienced at these "higher" dimensions. 

Thus the dimensions > 1 not alone express increasingly larger numbers in numerical terms. They equally represent increasingly "higher" dimensions of intuitive understanding (through which their holistic nature can be properly appreciated).  

And though again in the past, the possibility of such understanding was confined to the small group of people traversing an advanced contemplative path (as defined by the various religious traditions) future generations will slowly awaken to the immensely important mathematical and scientific implications of such refined intuitive wisdom.


The deeper realisation here is that - far from being abstract - the notion of number is embedded as its most fundamental encoding in all created phenomena (both physical and psychological).

Indeed from this perspective, all the great wonders of nature represent the subsequent phenomenal decoding of number (in both quantitative and qualitative terms).

However it must be again remembered here that we have now greatly enlarged the notion of number to ultimately embrace - not only the quantitative realms - but also in a manner, that is scarcely yet imaginable, all of the qualitative realms.


So to return again to our starting point regarding the paradox of number.
As I have stated, if the sub-units of each cardinal number were truly independent (lacking qualitative distinction) no means would thereby exist for the recognition of their corresponding ordinal aspect.

One cut also express this therefore by saying that number has no meaning apart from its relationship with space and time!

In fact it is only the implicit assumption of the spatial (and temporal) connections as between the sub-units of a number that enables their ordinal nature to be recognised (from the customary analytic persepctive).

So when we represent - say the number 3 - on a number line as 3 units, these are extended in linear fashion in space.

With respect to time this likewise entails that recognition must take place in a clear unambiguous sequence  In other words, the first unit is identified with the unit that spatially starts at 0; then the 2nd unit is identified with the unit beginning at 1; then finally the 3rd unit is identified with the unit beginning at 2. Thus for number to be ordered in this accepted linear fashion, it requires an ability to spatially distinguish the 3 units in a restricted manner.

Therefore, the units are not truly independent but intimately depend on a certain arbitary relationship with space and time dimensions.

And of course the very notion of such a relationship, implies - literally - a qualitative dimension to number!

Zeta 3 Zeros - Key Significance (1)

I use the term Zeta 3 Zeros to refer to the combined interactive recognition of both Zeta 1 and Zeta 2 zeros.

Now if we go back to the Zeta 2 zeros, we will recall that we they were initially defined with respect to each prime (representing a unique group of individual members) .

So one again using the prime number "3" to illustrate, the conventional quantitative definition treats the individual sub-units in a homogeneous independent manner (as without qualitative distinction).

Thus in simple additive terms, 3 = 1 + 1 + 1 (where each of the independent units are indistinguishable from each other).

And in more detailed fashion, this quantitative approach - that corresponds with cardinal interpretation - represents the Type 1 aspect of number i.e. where number is defined with respect to a default dimensional value of 1 (which typifies the interpretation of such numbers as points on the number line as - literally - 1-dimensional in nature).

Thus in Type 1 terms 3¹ = 1¹ + 1¹ + 1¹.


The Zeta 2 zeros then in complementary fashion - which can only be properly understood in a dynamic interactive context - represent the corresponding attempt to understand "3" in qualitative fashion.

Just as the assumed independence of the 3 units forms the basis for the quantitative interpretation, the corresponding interdependence of these units forms the basis for qualitative interpretation.

Now this qualitative interpretation of "3" (as "threeness") is directly understood in an intuitive manner (just as the quantitative interpretation is directly understood in a rational manner).

This is then expressed in a Type 2 terms as 1³ = 0 (i.e. where the qualitative notion of "threeness" - literally - lacks any quantitative distinction.

However, if we were to leave it at this, there would be no means for coherently relating quantitative and qualitative notions of number.

So the key importance of the Zeta 2 zeros is this context is that they provide an indirect quantitative means of expressing the qualitative notion of number in a rational manner.

Seen from the opposite perspective, they provide an indirect qualitative means of expressing the quantitative notion of number in an intuitive manner

Therefore by taking the 3 roots of 1 we can break up as it were this notion of "threeness" in a relatively independent manner (as three individual components) while then relating these again in a relatively interdependent manner. So each individual root has a relatively independent status (as a quantitative value) while the collective sum of roots expresses their relatively interdependent nature (which in quantitative terms = 0).
And through this conversion, we are thereby enabled in two-way fashion, to relate the cardinal notion of number (as quantitative) with the ordinal notion of number (as qualitative) respectively.


So the key importance of the Zeta 2 zeros  is that they provide thereby a means of expressing ordinal notions of the relationship between numbers (that are qualitative in nature) indirectly in a quantitative manner.

Thus again with respect to "3", 1st, 2nd and 3rd as ordinal notions are qualitative in nature. However indirectly these can be "converted" in quantitative terms using the 3 roots of 1.

Likewise in reverse 1, 1 and 1 as the sub-units of the cardinal notion of "3" are quantitative in nature.

However, indirectly these can be likewise expressed in a qualitative manner. So 1 as the - necessary - last unit of 1) is deemed the 1st. Then 2 (as the last unit of 2) is deemed the 2nd, 3 as the last unit of 3 is deemed the 3rd and so on.

In this way the three sub-units of  the cardinal notion of 3 can be converted in an ordinal manner as 1st, 2nd and 3rd units respectively.

And this is how conventionally ordinal and cardinal notions are combined in conventional mathematical terms.

However this implies a merely analytic type interpretation (where the qualitative aspect is reduced in terms of the quantitative in a fixed manner).

So once again 1st (in this context) always refers to the last unit of 1; 2nd refers to the next additional unit i.e. as the last unit of 2; 3rd refers then to the next additional unit (as the last unit of 3) and so on.

Thus from this perspective 1 + 1 + 1 = 1st + 2nd + 3rd

However a paradox arises here. For if the 3 units are in fact indistinguishable, then 1st 2nd and 3rd can equally be identified with any unit.

However this requires a new holistic way of looking at the relationship which now implies the interdependence (rather than independence) of each unit.

This means in effect that any of the 3 units can be potentially deemed as 1st (in a ranking context).
Then when one particular unit is then actually fixed as 1st,  either of the 2 remaining units can potentially be picked as 2nd . Then when one is actually picked as 2nd, then the 1 remaining potential unit becomes synonymous with the 3rd actual unit.

Thus with respect to the last unit, potential (holistic) automatically reduces to the actual (analytic) interpretation. And this again is how in conventional mathematical terms, ordinal notions of number - which are properly of a qualitative nature - are successfully reduced in a quantitative manner.

So again here 1st has the fixed actual meaning of 1st (of 1); 2nd then has the fixed actual meaning of 2nd (as the additional last unit of 2); 3rd has the fixed actual meaning of 3rd (as the last additional unit of 3) and so on.

This equates in mathematical terms as identifying each ordinal number in Type 2 terms with the t th of the t roots of 1 i.e. 1^t/t = 1¹ (in a Type 1 manner).

However the other t – 1 roots of 1, represent in an indirect quantitative manner, the true qualitative holistic nature of the ordinal numbers.

So again with reference to 3, the 3 roots of 1 are 1^1/3, 1^2/3 and 1^3/3, = – .5 +.866i, – .5 –.866i and 1 respectively.

These represent 1st, 2nd and 3rd respectively (in the context of 3 members). However the true potential i.e. holistic meaning,where a necessary interdependence exists as to what  might be chosen as 1st and 2nd, only applies to the first 2 positions; for by definition what potentially is 3rd,when the first two positions have been filled in this circumstance, is now fixed to what actually is 3rd!  

So the Zeta 2 zeros are given in general terms by,

1 + x¹ + x² +.... + x^{t – 1} = 0 (where t is prime).

Thus when t = 3, the two solutions are given by 

 x² + x¹ + 1 = 0.

One now realises that the ordinal notions of 1st, 2nd and 3rd, .... t th, have a purely relative meaning (depending on holistic context). So for example, 1st (of 2) is distinct from 1st (of 3) which again is distinct  from 1st (of 4) and so on.

Thus the Zeta 2 zeros from this important perspective, express - indirectly in a quantitative manner - the holistic ordinal nature of each prime number.

So again the Zeta 2 solutions for the two non-trivial roots of 3, express - in quantitative terms - the holistic natural number ordinal meaning of 1st and 2nd (in the context of 3 members).

However the paradox here is that the prime number "3" itself is not defined by these solutions. Though 1st and 2nd (of 3 members) are - indirectly - defined by the Zeta 2 zeros, as we have seen the notion of 3rd (as the last of 3 members) reduces down to its analytic meaning, where it is identified in fixed terms as 1 unit. 


And then, this is paradox is true for each of the cardinal primes, where its ordinal counterpart remains holistically undefined by the Zeta 2 zeros.

This then switches attention to the Zeta 1 (Riemann) zeros, where this key problem is addressed.

Zeta 1 Zeros - Key Significance (10)

The imaginary part of all the Zeta 1 (Riemann) zeros is believed to be of a transcendental nature.

So in this entry I will probe the holistic mathematical significance of this fact.

As we have seen, the imaginary notion, in holistic terms, represents an indirect analytic means of conveying truth that is directly of a holistic nature.

From a psychological perspective, this then represents an important way of conveying meaning, that is properly of an unconscious nature, indirectly in a rational conscious manner.


It also has an important physical relevance with respect to the behaviour of sub-atomic particles.

As is well-known in this highly dynamic state, particles begin to lose any strict notion of an independent objective identity. Rather their fleeting nature gradually becomes inseparable from an overall holistic environment representing the collective interdependence of particle interactions.

And this is then replicated in mathematical terms by the use of complex equations (containing an imaginary component) to describe such behaviour.

So again as we have seen each zeta zero represents a highly interdependent holistic state (ultimately representing the two-way simultaneous identity of both prime and natural number aspects). And this holistic energy state is then indirectly expressed in rational analytic terms through the imaginary - rather than real - number notion.

Also, the imaginary part of each zero is believed to be transcendental.


The holistic mathematical significance of the transcendental notion can best be approached through initial focus on the most famous of all transcendental numbers i.e. π.

In quantitative terms, π represents the ratio of the circumference of a circle with respect to its line diameter.

In corresponding qualitative terms, π (as a transcendental number) represents the pure relationship as between circular and linear understanding. Circular in this context relates to the (indirect) rational attempt to translate the holistic aspect, whereas linear relates to the (direct) rational attempt to translate the corresponding analytic aspect of understanding.

Now all transcendental numbers represent a certain unique relationship as between holistic and analytic understanding respectively.

For example e is the next nest-known transcendental number.

In quantitative terms, when we raise e to x (representing a power or dimension) and then differentiate the expression, the answer remains unchanged.

Then in reverse when we integrate this same expression, it again remains unchanged.

There is fascinating holistic equivalent of this behaviour where the (analytic) differentiation of individual phenomena in experience becomes inseparable from the corresponding (holistic) integration of all phenomena.

So this represents a very rarefied state where (conscious) analytic and  (unconscious) holistic aspects of experience are equally recognised and also where a dynamic balance can be maintained as between both so that neither aspect is given undue prominence in experience.

And this is what transcendental means from a holistic mathematical perspective!

Then an imaginary transcendental event implies the additional ability to immediately disentangle psychological projections (emanating from the unconscious) so that any undue identification of such projections with conscious objects is quickly dissolved.

In other words from a psychological perspective, an imaginary transcendental state is as close as one can come to the true marriage (in full freedom) of both conscious and unconscious aspects of personality. 

So one here approaches a purely intuitive psycho-spiritual energy state that directly borders on the ineffable.


What is important to understand is that this holistic appreciation of each zeta zero, in dynamic experiential terms - which is the truly appropriate way to interpret number - represents the complementary opposite of corresponding analytic interpretation.

The very basis of analytic interpretation is to separate the (rational) conscious as much as possible from the corresponding intuitive (unconscious) aspect.

In conventional mathematical terms, this separation is misleadingly interpreted in absolute fashion, so that numbers appear then to possess an unchanging quantitative identity (independent of a qualitative relationship with them).

However in correct dynamic terms, this separation always remains of a strictly relative nature. Then as one focuses more and more on the - now - relatively - independent analytic aspect, interpretation of number approaches ever closer to an absolute limit (of an unchanging quantitative identity).

However in this correct dynamic context, the Zeta 1 (Riemann) zeros approach the opposite extreme of a purely - relatively - interdependent identity, where the bi-directional paradoxical - ultimate synchronous - nature of both the quantitative and qualitative aspects of number is appreciated.

And the appreciation takes place in a directly intuitive manner, where the formal identity of number becomes so transparent and ephemeral that it approaches a pure energy state (in physical and psychological terms).

So therefore the full dynamic interpretation of number entails the balanced combination of both analytic and holistic aspects. One can now see number as ranging over a spectrum, where at the analytic extreme it approaches an absolute formal identity of an unchanging quantitative nature and then at the opposite holistic extreme, a purely relative qualitative identity (that - ultimately - is of an ineffable nature).

Therefore when one starts by identifying the primes as the independent quantitative "building blocks" of the natural number system in a rigid analytic manner, appropriate interpretation of the Zeta 1 (Riemann) zeros should approach, in complementary fashion, a purely formless relative state of paradoxical interdependence, that is directly of a holistic intuitive nature.    


However, as always with respect to experience, reference frames can switch.

Therefore, from the opposite perspective, if one starts by identifying the interdependent qualitative relationship  between the primes (as the unique factors of the natural numbers) in a holistic manner, then the complementary opposite interpretation of the Zeta 1 (Riemann) zeros should approach a rigid analytic form (in a quantitative manner). And of course, it is this latter interpretation of the zeros that can be used to correct the general prediction of the frequency of primes up to a certain number so that one can eventually zone in on the correct answer!


However, the huge unrecognised problem with current interpretation of the number system, is that both the primes and the Zeta 1 zeros are viewed in an absolute manner (as representing merely quantitative identities) whereas the true relationship between them is properly of a dynamic relative nature representing the two-way complementary interaction of both analytic (quantitative) and holistic (qualitative) aspects.  


Just finally in this entry, I will comment on the holistic significance of the fact that the imaginary aspect of the zeros always occurs in pairs (that are the positive and negative of each other).

What this implies is that a very high degree of integration of both refined conscious and unconscious appreciation is required for appropriate comprehension.

Once again the imaginary aspect represents an indirect quantitative means of "converting", what is properly of an unconscious qualitative nature in a conscious (rational) manner.

However if one mistakes this imaginary projection (from the unconscious) as real i.e. through rigidly identifying its meaning in conscious terms, then clearly a confused understanding results.

In fact with respect to general experience, this problem is extremely common through shadow projections. Thus the hidden dark side of our our own unconscious is frequently projected externally on to conscious events (which we then mistakenly identify as the problems that require resolution).

In fact this issue is even more profoundly true of Mathematics, where an almost total blindness remains among the profession as to the hidden unconscious aspect of understanding of all its symbols and relationships.

Thus a typical reaction to a blog such as this would be one of blunt dismissal as totally irrelevant with respect to what is conventionally understood as Mathematics.

Unfortunately, to a considerable extent Mathematics can thus be seen as remaining tightly confined in a hermetically sealed chamber of its own limited assumptions, which blindly refuses to address any serious questioning of these assumptions (which - by definition - does not represent the accepted mathematical wisdom).

So the close pairing of both positive and negative aspects of the imaginary part of the zeros, holistically represent the fact that immediate negation of any rigid conscious identification of imaginary meaning as - mistakenly real - real, continually takes place. And because numbers as mathematical phenomena of experience no longer assume a permanent fixed identification (in analytic terms) one is thereby increasingly enabled to appreciate their true holistic meaning (in a directly intuitive manner).

In fact there are very close parallels here with the notion of virtual particles in sub-atomic physics that come in pairs (with positive and negative signs) that are very closely related. So a virtual (i.e. imaginary) particle can only enjoy a very fleeting short-lived identity in "real" terms, as it immediately becomes annihilated in the form of physical energy by its opposite anti-matter counterpart.

So its is very similar here with the appropriate understanding of each zeta zero, where the imaginary part quickly becomes eroded (in unconscious understanding) by its negative counterpart in the form of intuitive (i.e. psycho-spiritual) energy. And this is how the extreme of a purely holistic interpretation of each zero is obtained.

So when correctly understood, this holistic understanding of the Zeta 1 (Riemann) zeros lies at the opposite extreme of rigid analytic understanding (where number approaches an absolute unchanging form in quantitative terms).

In fact, correctly understood the zeros exist as the last thin partition as between the (finite) phenomenal and infinite (i.e. formless or ineffable) realms.

So just as it is becoming dimly recognised in physical terms that the Zeta 1 zeros are intimately tied up with the energy states of certain quantum reactions, in like manner, future generations will recognise that the Zeta 1 zeros have an intimate bearing on the "highest" psycho spiritual states (of advanced meditation).

So in this sense, their true significance greatly surpasses what - presently - constitutes Mathematics.   

Zeta 1 Zeros - Key Significance (9)

Presuming the truth of the Riemann Hypothesis, all Zeta 1 (Riemann non-trivial) zeros come in pairs with the complex conjugate form 1/2 + it and 1/2 – it respectively.

So again the first pair of zeros are 1/2 + 14.134725i and 1/2 – 14.134725i respectively (correct to 6 decimal places).

In fact it is believed that the imaginary part of all zeros is of a transcendental nature.

So therefore each zero as a complex number, given the truth of the Riemann Hypothesis, contains a real part = 1/2 and an imaginary part of a transcendental nature.

In previous work, I spent some time exploring the precise holistic mathematical significance of all the number types (of which those belonging to the imaginary transcendental set are the most rarefied).

As we have seen, in this context, the significance of the imaginary notion is that it offers an indirect analytic means of conveying meaning that is directly of a holistic nature.

So the true nature of the interdependent  relationship between the unique prime factors of the natural numbers is directly of a qualitative holistic nature. And the individual zeta zeros represent this relationship (which directly complement the opposite notion of the independent quantitative nature of the primes). However indirectly this relationship can then be expressed in an imaginary quantitative manner.

And just as all the real numbers are presumed to lie on a line (i.e. the number line), in like fashion, the zeta zeros - as the dynamic complementary expression of this line - are presumed therefore to lie on an imaginary line (which again is necessarily true if the Riemann Hypothesis holds).


So we can now perhaps begin to appreciate the true significance of the Riemann Hypothesis (which is completely overlooked in conventional mathematical terms).

Because of its limited absolute nature (where qualitative notions are reduced to quantitative) it is conventionally assumed as axiomatic that all real numbers lie on the same number line!

However, when one properly recognises that numbers necessarily possess a qualitative aspect (of relational interdependence) as well as a quantitative aspect (of individual independence) the number system, in dynamic terms, is thereby relative in nature, whereby quantitative notions of relative independence and qualitative notions of relative interdependence ceaselessly interact with each other in a bi-directional manner.

So the key issue that arises, in this dynamic interactive context, is the consistency of both quantitative and qualitative aspects (which are distinct) in terms of each other.

This thereby entails that the linear notion of consistency, relating to the primes - that is assumed to hold in analytic quantitative terms as individual independent "building blocks" with respect to the real numbers on this line - must be balanced by a complementary linear notion of consistency that is assumed to hold in a holistic qualitative manner with respect to these same numbers (through their interdependent relationship with each other). And as we have seen this complementary notion, relating to the  holistic qualitative nature of the number system, is indirectly expressed - in an  analytic quantitative manner - through the mathematical notion of the imaginary.

Therefore, this now entails that the (direct) quantitative notion of consistency with respect to numbers on the real number line intimately depends on the corresponding (indirect) quantitative notion of consistency with respect to the zeros on the imaginary number line i.e. that all such zeros should lie on the imaginary line.

And because of the requirement that objective reality in external terms be - ultimately - exactly matched by subjective interpretation in internal terms (which we dealt with in the previous entry) this poses the additional requirement that this imaginary line be drawn through 1/2 (on the real axis).

In other words, the fundamental significance of the Riemann Hypothesis is that its truth is a necessary requirement for the presumption of quantitative consistency with respect to the real number system.

In other words, from a proper dynamic interactive perspective, we are not entitled to presume quantitative consistency without reference to the complementary qualitative aspect of number.

Equally - and very importantly - we are not entitled to presume qualitative consistency without reference to the complementary quantitative aspect of number.

So the truth of the proposition that all the zeros line on the imaginary line (through 1/2) depends on the corresponding truth that all real numbers likewise lie on a line.

And the proposition that all real numbers lie on this line equally depends on the truth that all the zeros lie on the imaginary line (through 1/2).

And there is a necessary uncertainty principle in operation here.

In order to approach the quantitative extreme of rational absolute type understanding of mathematical relationships, the qualitative aspect must be rendered so "fuzzy" as to be no longer even identifiable. And this is the position that characterises conventional mathematical understanding, where its distinctive qualitative aspect is no longer even recognised (in formal terms).

And likewise to approach the opposite extreme of a purely relative type understanding of mathematical relationships, the quantitative aspect in turn must be rendered so "fuzzy" as to be no longer even identifiable. And this represents purely intuitive understanding in a psycho spiritual manner, where the ultimate relationship as between the primes and natural numbers is now directly "seen" in such a synchronous manner (that neither can maintain a distinct identity).

So all mathematical activity must lie between these two extremes, where purely absolute and purely relative understanding are respectively approached, implying always both quantitative notions (of relative independence) and qualitative notions (of relative interdependence) respectively.


Because both of these possible extremes (analytic and holistic aspects) are two sides of the same coin and - ultimately inextricably interdependent with each other - we cannot hope to prove either proposition (in a conventional analytic manner). So the Riemann Hypothesis is not capable of proof (or disproof) though in itself this is of much less importance than its (unrecognised) significance.

In other words, the true fundamental issue underlying all Mathematics, relates to the prior consistency as between its quantitative (analytic) and qualitative (holistic) aspects.

And this is the issue to which the Riemann Hypothesis - when properly understood - directly points.

Therefore, we must assume the truth of the Riemann Hypothesis to - literally - maintain our faith in the subsequent consistency of all quantitative relationships. Equally - though not properly appreciated - we must assume the truth of the number line (i.e. that all real numbers lie on this line) to maintain faith in the subsequent consistency of all qualitative relationships.

And because of their distinctive nature, neither of these aspects can be proved in terms of each other.

So underlying the interpretation of all mathematical relationships is - implicitly - an initial massive act of faith in the subsequent consistency (in both quantitative and qualitative terms) of the whole enterprise. 

Zeta 1 Zeros - Key Significance (8)

As is well known, each Zeta 1 zero - which is true if the Riemann Hypothesis holds - is postulated to lie on the imaginary line through 1/2 (on the real axis) with each zero of the form 1/2 + it existing with a conjugate zero of the form 1/2 – it.

So for example the first of the zeros 1/2 + 14.134725i has its conjugate zero as 1/2 – 14.134725i  (correct to 6 decimal places).

In this blog entry, I will endeavour to convey the important holistic significance of the real part (i.e. 1/2). 

The initial clue to the significance of 1/2, in this context, where 1/2 represents the real part of a complex number serving as a dimensional power (or exponent) can be obtained from considering the simplest case where 1 is raised to 1/2 and as we have already seen, the answer = – 1.

And again, from a holistic mathematical perspective, this implies the unconscious negation of what has already been posited as + 1 in a conscious manner.

And this unconscious negation in dynamic interactive terms, relates directly to a holistic intuitive psycho spiritual energy state, where both positive (+) and negative (– ) directions of experience are revealed as purely relative (and thereby rendered as paradoxical from the customary rational analytic perspective).

Looking at this psychological interaction in more detailed terms, it implies the ready recognition that the understanding of the number "1" (externally in objective terms) is dynamically inseparable from the corresponding understanding of the number perception "1" (internally in a subjective manner).

So the psychological fusion of both external and internal directions of understanding leads to the holistic intuitive recognition of number such as "1" (as "oneness" in this example) ultimately representing a pure energy state. 

And because the physical and psychological aspects of experience are likewise complementary, in dynamic interactive terms, this equally entails that all numbers can be equally given a pure holistic identity, as energy states, in physical terms.

So we are at the other extreme of understanding here. The analytic extreme - identified with pure reason - enables one to understand number as approaching an absolute phenomenal identity of unchanging form.

However the other extreme of holistic understanding - identified with pure intuition - enables one to understand number as approaching a purely relative identity (with no remaining phenomenal form) as an energy state, in both physical and psychological terms.


So when we start from the standard analytic perspective of identifying the prime numbers as approaching an absolute unchanging identity (from an independent quantitative perspective), the Zeta 1 (Riemann) zeros then correctly represent the complementary holistic perspective at the other extreme, where each zero is intuitively understood as approaching a purely relative status of qualitative interdependence, thus representing an energy state (in physical and psychological terms).

However because reference frames ceaselessly interchange in the dynamics of experience, we can equally start from the standard analytic perspective of identifying the Zeta 1 zeros as approaching an absolute unchanging identity (from an independent quantitative perspective).

The primes then in reverse manner correctly represent the complementary holistic perspective at the other extreme where they (as the unique factors of the natural numbers) are now understood as approaching a purely relative states of  qualitative interdependence, representing the integration of distinct energy states, in physical and psychological terms. 

And it is this latter holistic qualitative appreciation that properly constitutes "the music of the primes".

The complementary dynamic balancing of external and internal polarities of experience, can be usefully interpreted as the division of customary 1-dimensional experience into two related aspects. So now we - literally - get two halves (of 1/2 in each case) which are directly represented by the conjugate pairings in which the zeta zeros occur.

Once again in standard linear (1-dimensional) terms, a number such as "1" is given an external identity, where it is thereby treated as an object, independent of mental understanding. 

Likewise, when a mental construct is used - which relatively is internal in direction - again it is treated as if it enjoys an absolute existence, independent of its external direction. So in effect both directions are reduced in absolute terms in conventional mathematical terms i.e.where - literally - what relatively exist as positive and negative directions are ignored. So we have the holistic meaning here of what is absolute in mathematical terms (which closely parallels in dynamic fashion its analytic meaning).

However, when both external and internal directions are clearly realised as relative, then in effect, both are given equal weight as constituting 1/2 of the total understanding. And this requires that unconscious negation be employed to a high degree. 

So one momentarily posits the external direction in recognition of the mathematical object, before quickly negating it in an unconscious manner. 

This then causes a switch to the internal direction in recognition of its corresponding mental construct, which is likewise momentarily posited in experience before again becoming quickly negated in experience. 

So as dynamic switching as between external and internal directions becomes more refined, the holistic intuitive realisation of the interdependence of both poles increasingly grows.


And this is vital in appreciating in turn the true holistic nature of the Zeta 1 (Riemann) zeros.

So what we have here is the strong intuitive realisation that what is dualistically understood as having an unambiguous objective existence, in truth has no meaning apart from corresponding mental interpretation which - relatively - is subjective in nature.

So in truly appreciating the nature of these zeta zeros, one intuitively achieves in holistic fashion, the marriage of the objective identity of the primes with their corresponding subjective interpretation (which strictly have no meaning apart from each other).

This is a great unrecognised problem with respect to Conventional Mathematics. The truths that it contains reflect just one limited interpretation of reality (that is 1-dimensional in nature).

This leads to an absolute view of relationships, which from a dynamic experiential perspective is quite distorted and unbalanced.

However there exists an unlimited set of other dimensional interpretations - yielding a limited partial view of mathematical reality - that are all inherently dynamic in nature. 

And the Riemann zeta function - and its accompanying Riemann Hypothesis - require these relative - rather than absolute - interpretations for meaningful interpretation. 

Zeta 1 Zeros - Key Significance (7)

The Zeta 1 (Riemann) can be seen to operate in a complementary fashion from the Zeta 2 zeros, which is just another way of saying that from a dynamic interactive perspective, both sets of zeros are intimately related.

Whereas with respect to the Zeta 2, each zero can be  (indirectly) given a relatively independent quantitative identity, the collective sum of zeros (together with the default root of 1) reveals their true qualitative nature.

So for example with respect to "3" as a prime, – 1/2 + .866i , – 1/2 –  .866i and 1, represent the 3 roots of 1. These indirectly express in a quantitative manner, the notions of 1st (in the context of 3), 2nd (in the context of 3) and 3rd (in the context of 3) respectively, with the first two representing Zeta 2 zeros. 
Thus the 3 roots have a relatively independent identity. However the collective sum of these roots = 0, reveals the true qualitative nature of their relative interdependence with each other, through yielding a result with no quantitative significance!    

Now it works somewhat in reverse with respect to the Zeta 2 zeros. Here each individual zero, again indirectly expressed in a quantitative manner, reveals its true qualitative nature.

To be precise, to must see each zero as part of a pairing where in general terms 1/2 + it always implies 1/2 – it likewise as a zeta zero.

So in the simultaneous recognition of both zeros, the imaginary parts cancel out to leave 1.

And just as 0 deeply symbolises (in an indirect additive manner) the qualitative aspect of understanding, likewise 1 deeply symbolises (in an indirect multiplicative manner) the qualitative aspect.


However, associated with the collection of Zeta 1 zeros is - in complementary - terms a relatively independent quantitative aspect. and it is this aspect, which comes into play in correcting the deviations from the general formula to estimate the frequency of primes to a given number, so that ultimately an absolutely correct estimate can be approached.

However it is still important in this context to recognise that such calculations using the wave forms associated with the Zeta 1 zeros, as corrections on a general estimate of frequency, remain strictly of a relative nature.  

In other words, regardless of how many zeros are used in the correction process, the answer will never give the exact frequency of primes (which is an absolute number). Rather the answer will eventually approximate so close to the exact frequency, that one can then easily estimate the correct answer with a high degree of confidence.

However, before moving on, let us try to probe more deeply the exact nature of the Zeta 1 zeros.

As is well known, the exact distribution of the primes is represented by a discontinuous step function.

So for example at 1, we have yet encountered no prime. However then at 2 the frequency of primes increases to 1. Then at 3 it increases again by 1 to 2. However at 4 no change takes place, before again at 5 we have another discrete change of 1, as the cumulative frequency of primes increases again by 1 to 3. So the function representing cumulative frequency, remains completely flat and horizontal over the numbers which are not prime and then increases in a discrete manner - always by  1 - when a new prime is encountered.

It is somewhat similar with respect to the corresponding function, representing the cumulative frequency of natural number factors. Here where a number is prime, the cumulative frequency increases in discrete fashion by 1. However where a number is composite it increases - again in a discrete fashion by a number > 1.

Since 1 is a factor of 1 and the common factor of all numbers, which we do not consider in this context, the cumulative frequency of factors = 0. Then at 2 (which is prime) it increases to 1 and at 3 (again prime) it   increases, again by 1 to 2. However at 4 (which is composite) it increases by 2 (with factors. 2 and 4); then at 5 (which is prime) it increases by 1, while at 6 (which is composite) it increases by 3 (with factors 2, 3 and 6).

So we have an even more complicated step function for the cumulative frequency of factors, where for each prime, as we have seen, the function steps up in discrete fashion by 1. However for the composite numbers it steps up - again in discrete fashion - by more than 1.

And it is this function, to which the corresponding cumulative frequency of Zeta 1 (Riemann) zeros closely relates.

So once again, the cumulative frequency of factors to n, closely approximates the corresponding cumulative frequency of Zeta 1 zeros to t (where n = t/2π).

Therefore the Zeta 1 zeros can be best seen as an attempt to smooth out in a continuous manner the discontinuous discrete nature of the step function (associated with the cumulative frequency of factors) so that - in a very important sense - the changes in factor frequency due to the primes are no longer distinguishable from the corresponding frequency due to the (composite) natural numbers. And the Zeta 1 zeros therefore represent those points where the factor frequency increases by 1 for both primes and composite natural numbers (which at these points are indistinguishable).   

Now, reminding ourselves of the crossroads situation, which can be simultaneously approached from N and S directions so that left and right turns have a circular paradoxical meaning, it is quite similar here.

Therefore, we cannot make dualistic sense, in a linear analytic fashion, of the notion that primes and (composite) natural numbers can be somehow identical; however we can indeed understand this in a nondual holistic manner (that is directly intuitive in nature) .

Therefore, understanding of the true holistic nature of the Zeta 1 zeros comes again from initial appreciation that the relationship as between primes and natural numbers can be defined within two relatively independent reference frames, which are - in my chosen terminology - external and internal with respect to each other. Then when one attempts to - literally - "see" this relationship simultaneously within both reference frames, the true holistic nature of the Zeta 1 zeros (as approaching both pure psychological and physical energy states respectively) is revealed. 

And in this direct intuitively inspired revelation, which indirectly can be interpreted  intellectually in a circular paradoxical manner, one realises the true ultimate nature of both the primes and natural numbers as perfect mirrors of each other in an inseparable manner.   

Once again, from a holistic mathematical perspective, the imaginary aspect simply represents an indirect analytic means of attempting to convey meaning that is directly holistic in nature.

And this is the simple the reason why all the Zeta 1 zeros appear to lie on an imaginary line. We will look at the deeper consequences of this in the next blog entry.