A Brief Addendum

As is well known log n or the more accurate log n – 1 provides a good approximation of the average spread or gap as between primes in the region of n.

However what is not properly known is that log n - 1 equally provides a good approximation of the average amount of natural factors of a number (also in the region of n).

And notice the complementarity as between both estimates! In one case we are referring to primes and in the other case to natural numbers; then again in the first case we are referring to the Type 1 notion of number (i.e. defined in a 1-dimensional manner); then in the second case we are referring to the corresponding dimensional notion of number (relating to its factor components).

This complementarity once again points to the truly dynamic interactive nature of the number system.

Thus the behaviour of the primes (with respect to the average gap as between primes in the region of n) is dynamically inseparable from the corresponding behaviour of natural number factors (this time with respect to their combined ratio in relation to n).

Put another way the behaviour in both cases is ultimately determined in a holistic synchronous manner.

Whereas great attention has been placed on one aspect of this behaviour (i.e. with respect to prime number frequency), precious little attention has been devoted to the complementary form of behaviour with respect to the frequency of the natural factors of numbers.

So in principle, just as it is is possible through use of the Riemann (Zeta) 1 zeros to eventually determine the exact frequency of the primes to any number, equally it should be possible to determine the exact cumulative frequency of natural number factors of the composites. through corresponding adjustments using the Zeta 2 zeros.

Another interesting fact!

n/log n – 1  approximates the frequency of primes to n.

The connection then as between the primes and natural numbers is of an additive nature,

n/log n - 1 equally can be used to measure the ratio as between n and the corresponding average number of factors of n.

However the connection here is of a multiplicative (rather than additive) nature.

Thus the two uses of the same formula illustrate the complementary relation as between addition and multiplication (which represent the Type 1 and Type 2 aspects of number with respect to each other).


Also it should be be observed that the average frequency of prime factors of a number is given approximately as log log n.

Therefore the ratio of natural number to prime factors is given as log n/ log log n.

If we now let log n = n₁, then this ratio can be given as n₁/log n₁.

Or if we wish to use the more accurate expression already employed, this would be given as n₁/log n₁ – 1 .

So now we can see clearly how a derivation of the formula for calculation of frequency of primes (with respect to the natural numbers) can be used for calculation of the ratio of natural number to prime factors.

And once again whereas the connection as between primes and natural numbers in the first (Type 1) is of an additive nature, in the second (Type 2) case it is of a complementary multiplicative nature.


Just one more observation!

log log n (or alternatively log log n – 1) in Type 2 terms can be used to approximate the average number of prime factors of n.

Therefore log log n equally can be given a Type 1 meaning.

Thus whereas log n measures the average gap between primes at n, log log n measures the average gap as between primes whose gap is log n..

So for example the average gap as between primes at n =1,000,000 = 13.815...
Therefore log log n measures the average gap of primes to 14 (approx).

Zeta Zeros Made Simple (12)

We have seen that in cardinal (Type 1) terms, each natural number can be uniquely expressed as a product of the primes.

Then in complementary ordinal (Type 2) terms, each prime can itself be uniquely expressed as an ordered sequence of natural numbers.

Thus again from the cardinal perspective, 3 is viewed as prime building block with respect to the natural number system.

However, when viewed from the corresponding ordinal perspective, 3 is now viewed as uniquely composed of its natural number ordinal members i.e. 1st, 2nd and 3rd that indirectly can be represented in a quantitative fashion. (Now again the final member which in general terms is the pth of p members is not unique as it is always equal to 1).

However all other expressions for 1st, 2nd, 3rd, 4th,......(p – 1)th members are indeed unique!.

Thus, when we consider both the (Type 1) cardinal and (Type 2) ordinal nature of number, then the relationship between the primes and natural numbers is clearly seen as circular (in a two-way fashion).

Thus again from the cardinal perspective (based on the multiplicative approach), the primes are seen as the unique building blocks of the natural numbers; however when seen from the corresponding ordinal perspective (based on the additive approach)  the natural numbers are seen as the unique building blocks of each prime!


Thus the very nature of the number system, though which both the primes and natural numbers are mutually generated, is of a dynamic nature based on the bi-directional interaction of complementary opposites (that necessarily are of a relative nature).

And as this system approaches perfect synchronicity, the primes and natural numbers are increasingly seen as perfect mirrors of each other in a formless ineffable manner.

Thus if we initially conceive - as is the standard practice - of the primes and natural numbers in an analytic fashion, then the two sets of zeros (Zeta 1 and Zeta 2) can only be properly understood in a corresponding holistic fashion (where the interaction of both quantitative and qualitative aspects are mutually harmonised).

So the very essence of analytic interpretation, once again, is that polar opposites such as quantitative and qualitative are clearly separated in an absolute type manner.

However the corresponding essence of holistic interpretation is that that these same opposites are now fully harmonised in a relative type manner.

So once again in the standard cardinal interpretation, approaching from the additive perspective, the natural numbers 1, 2 and 3, for example, are clearly identified in analytic fashion as absolute quantities (that are independent of each other).

However properly speaking in corresponding ordinal interpretation 1st, 2nd and 3rd are identified in holistic fashion as relative notions (and thereby interdependent with each other).

Thus we can give these a relatively independent quantitative identity (as represented by the 3 roots of 1). However the qualitative interdependence of these roots is then expressed through their collective sum = 0.
In other words, though individually each ordinal position can be given  a relatively independent quantitative identity, collectively all of these positions have but a qualitative identity = 0 (in quantitative terms).

Thus we can see that very interpretation of roots is here of a holistic nature.
And then the Zeta 2 zeros simply represent the same roots with the one common (i.e. trivial root of 1) temporarily excluded.


Then again in the standard cardinal interpretation, this time approaching from the multiplicative perspective, the primes 2, 3 and 5 for example are again clearly identified in analytic fashion as absolute quantities (that are independent of each other).  

However properly speaking the corresponding "ordinal" interpretation (whereby the primes are uniquely combined to form the composite natural numbers) are again identified in holistic fashion as relative notions (and thereby interdependent with each other).

Thus from one perspective, we have the individual primes, as independent of each other; on the other hand we have the collective relationship of the primes that uniquely generates each composite natural number throughout the entire system.

Thus from the analytic perspective, the primes again appear as absolutely independent of the composite natural numbers in a quantitative manner. However from the holistic perspective, the combined relationship of primes is now viewed as fully interdependent with the natural numbers in a qualitative manner.

So now in complementary fashion each Riemann (Zeta 1) zero wonderfully expresses a location (on an imaginary axis) where both the primes and natural numbers are relatively identical with each other in a qualitative fashion. Here now the collective sum of zeros relatively yields an independent quantitative identity that can then be used to completely unravel the distortion in the general (continuous) estimate of prime number frequency.


Therefore the very role of the zeta zeros (Zeta 1 and Zeta 2) is to dynamically enable in a fully coherent and consistent manner, the continual switching in the number system as between both quantitative and qualitative aspects, In other words, the zeta zeros enable this two way switching as between Type 1 and Type 2 aspects, from the notion of number as absolutely independent (of other numbers) to the corresponding notion of number as consistently related to all other numbers in the system (and vice versa).

And as we have seen this switching as between quantitative and qualitative aspects (and qualitative and quantitative) is mediated through the corresponding two-way relationship as between the primes and the natural numbers (and the natural numbers and primes).


In conclusion, I cannot stress strongly enough how reduced - and thereby distorted - is the conventional mathematical approach to the number system.

At its very heart lies the totally unwarranted attempt to exclude all qualitative type considerations from quantitative type relationships. So there is no explicit notion of a holistic aspect to Mathematics that is distinct from the analytic! Therefore there is likewise no realisation that ultimately both analytic and holistic aspects must be combined in an integrated fashion to provide the appropriate framework for either aspect (in isolation).

In truth, the quantitative obsession in Conventional Mathematics is utter madness and yet this this is the "rock" on which such Mathematics is built.

Some time in the future, it will be clearly realised that as quantitative and qualitative notions are fully complementary in mathematical terms, that any attempt to coherently understand the quantitative, must thereby likewise include the qualitative!

However this conversion urgently needs to start now, which can thereby eventually lead to an unimaginable transformation in the entire scientific and intellectual landscape.

Zeta Zeros Made Simple (11)

We have seen that in holistic terms, the various roots of 1 provide the appropriate means of expressing the unique ordinal nature of each number in a group (indirectly in a quantitative manner).

So once again, for example, in a prime group of 5 members, the 5 roots of 1, i.e. 1^1/5, 1^2/5, 1^3/5, 1^4/5, 1^5/5 holistically express the notions of 1st, 2nd, 3rd, 4th and 5th in the context of 5 members.

And of course the final root 1^5/5 = 1, expresses the default case of the 5th (in the context of 5), which directly relates to the last unit of 5, just as 1^4/4 represents the last unit of  4, 1^3/3 the last unit of 3, 1^2/2 the last unit of 2 and 1^1/1 the last unit of 1 respectively.

And these holistic expressions relate directly to a circular - rather than linear - notion of number, that geometrically is represented as the equidistant points on the unit circle (in the complex plane).


In conventional Type 1 terms, the primes are unique in that they provide the basic building blocks from which all other composite natural numbers are derived (in a quantitative manner) .

However, here in reverse complementary manner, we can see the alternative nature of primes, in that each prime is composed of a unique set of natural number roots (which cannot be replicated for any other prime).

Now strictly this statement needs qualification, in that one root i.e. 1^n/n is always non-unique as 1.
In fact, in this context, we could refer to this default root as the trivial root of 1.

So the t roots of 1 are expressed as 1 = s^t, i.e. 1 - s^t  = 0.

Therefore to find the expression for the non-trivial roots, we divide the expreby the trivial root i.e. 1 - s = 0.

This then yields the expression 1 + s¹ + s² + s ³ + ..... + s^{t – 1} = 0.

I refer to this as the Zeta 2 Function, which complements the Riemann (Zeta 1) Function,

i.e. 1 ^{– s}  + 2 ^{– s} + 3 ^{– s} + 4 ^{– s} + ......   = 0

Whereas the Zeta 1 Function is infinite, the Zeta 2 function is finite; then whereas the natural numbers appear as base numbers with respect to the Zeta 1, they appear as dimensional numbers with respect to the Zeta 2;  finally whereas the unknown s appears as a (negative) dimensional number in the Zeta 1, it appears as a (positive) base number in the Zeta 2!


If we initially, with respect to the Zeta 2, confine the value of t to the primes, then the solutions for s represent all the unique ordinal natural number solutions for s.

For example when t = 3, the Zeta 2 Function is

1 + s¹ + s²   = 0.

Therefore the 2 unique solutions (i.e. non-trivial zeros) in this case are

– .5 + .866i and – .5 –  .866i respectively.

And these solutions represent the indirect quantitative expression of the qualitative notions of 1st and 2nd (in the context of 3 members). Because quantitative and qualitative notions are thereby related, these properly require holistic mathematical understanding.

And these solutions  which uniquely express ordinal notions (in the context of different sized groups), I refer to as the Zeta 2 zeros.

So each non-trivial natural numbered root (within a prime group), can be given a unique quantitative expression (in a relative manner).

However - by definition - the sum of all roots (including trivial) = 0.

What this entails, in holistic mathematical terms, is that the collective sum of roots has no quantitative significance.

So notice the complementarity!

In cardinal Type 1 terms 1 + 1 + 1 = 3. Here, though the collective sum of components has a quantitative significance, each of the individual component parts (as homogeneous) has no qualitative significance.

Then in Type 2 terms 1st + 2nd + 3rd,  indirectly represented quantitatively as  – .5 + .866i  – .5 –  .866i + 1= 0.

Here though the individual components have an (indirect) quantitative significance, the collective sum has no quantitative significance. In other words it has a merely qualitative  interdependent significance (where all arbitrary relative connections as between quantitative variables cancel out).

Zeta Zeros Made Simple (10)

For many years I have been fascinated with the possibility of converting the Type 2 aspect of the number system in a coherent Type 1 manner.

So once again the natural numbers in the Type 1 aspect are expressed with respect to the default dimensional number of 1, i.e.

1¹, 2¹, 3¹, 4¹, ......

Then the natural numbers in the Type 2 aspect are expressed in a complementary manner with respect to a default base number of 1, i.e.,

1¹, 1², 1³, 1⁴, ......

Whereas the natural numbers in the Type 1, carry the direct connotation of cardinal identity in a quantitative manner, by contrast in the Type 2, the "same" numbers carry the complementary connotation of ordinal identity in a qualitative type manner.

So 1¹ in the Type 2 carries the connotation of 1st (in the context of 1) which clearly is the unit 1.

Thus if there is only one unit (in cardinal terms) no ambiguity can exist as to the 1st member of this group 

So the ordinal meaning of  1¹ is thereby identical in this case with its cardinal counterpart.

However when we move on to 2nd, 3rd, 4th and so each ordinal ranking cannot be unambiguously identified with one fixed position (unless the order is predetermined according to some arbitrary rule). In other words, in such cases ordinal ranking is of a relative - rather than absolute - nature.

However to express the notion of 2nd unambiguously with the last member of 2 we obtain. 

1^2/2. So 2nd is now effectively reduced to being identified with the 1 remaining unit of 2 i.e. the 1st of the remaining one unit).

1^2/2  reduces to 1¹ unambiguously in this manner.

And we can continue on in this manner so that 1^3/3 is  identified with the last member of a group of 3 and 1^n/n  with the last member of a group of n.

In this way, each of the ordinal positions can be unambiguously identified in a linear manner with their corresponding cardinal identities i.e. 1st with 1, 2nd with 2, 3rd with 3 and ....nth with n. 

Though it might appear trivial, an enormously important transformation has taken place, whereby in each case what was originally defined ordinally in a Type 2 manner has now been successfully converted in a cardinal Type 1 manner.

However this absolute type definition of ordinal positions represents just one limiting case where the nth member of a group is always identified with the last  member of the group n.

But there are innumerable other possible cases where the nth is identified with the group of n + 1, n + 2, n + 3 members and so on.

For example in the case of a cardinal group of 3 members, as well as the (default) case of the 3rd (of 3) we could also try and define the 1st of 3 and the 2nd of 3.

In fact this problem is intimately identified with the question of obtaining the 3 roots of 3.

Thus first in the context of 3 is defined in Type 2 terms as 1^1/3; 2nd in the context of 3 is 1^2/3; and 3rd in the context of 3 is - as we have seen - 1^3/3 .

What we have established here is in fact the Type 2 definition of rational fractions.

So for example with respect to a small cake in Type 1 terms, 1/3, 2/3 and 3/3 would have the recognised quantitative meaning of 1 (of 3 equal slices), 2 (2 of 3 equal slices) and 3 (of 3 equal slices) respectively

However in Type 2 terms, 1/3, 2/3 and 3/3 have the corresponding (unrecognised) qualitative meaning of 1st in the context of 3, 2nd in the context of 3 and 3rd in the context of 3 respectively.

Thus remarkably we can use the n roots of 1 to obtain all possible ordinal positions (in the context of a group of n).

And by obtaining these roots, we are in effect obtaining  a Type 1 transformation in quantitative terms of Type 2 ordinal notions that are strictly of a qualitative nature.

Zeta Zeros Made Simple (9)

We have seen that the ordinal nature of number relates directly to its holistic aspect. This is in contrast to the cardinal notion - which by contrast - initially appears as analytic in nature.

What this further means is that whereas the cardinal notion is based directly on the notion of number as independent (from other numbers), the ordinal notion is based directly on the complementary notion of number as interdependent (and thereby related with other numbers).

So for example we view the cardinal number "3" as independent from other numbers in a quantitative manner.

However the ordinal notion of 3rd is necessarily interdependent with a group of other numbers.

So 3rd in the context of the simplest group of 3 related numbers, implies a different relationship than 3rd in the context of  - say - 20 numbers!

Strictly speaking, analytic interpretation is always based on the clear separation of opposite poles.

Thus once again when one refer to a cardinal number in an analytic sense, this implies that the quantitative aspect can be clearly separated from the qualitative! This implies in effect that with such interpretation the qualitative aspect (insofar as it is recognised) is assumed to directly correspond with the quantitative and thereby is reduced to the quantitative.

A mathematician if sufficiently pressed, may even concede that the mental constructs that are internally necessary to interpret the number reality in an objective external manner are strictly of a qualitative nature.

However the subsequent problem of the relation of such psychological constructs with this objective reality is quickly explained away by assuming an absolute correspondence as between both aspects, so that qualitative aspect can thereby be completely ignored.


Holistic interpretation by contrast is based on the dynamic interaction as between opposite poles.

Thus with ordinal recognition, both quantitative and qualitative aspects are necessarily involved.

So, for example, to make the ordinal recognition recognition of 3rd, one must first identify a number group collectively in cardinal terms - say - 3. Then the notion of 3rd implies a qualitative relationship as between the 3 members of this group as 1st, 2nd and 3rd respectively.

Now in the dynamics of experience, both cardinal and ordinal recognition are mutually involved.
Thus cardinal recognition implicitly entails corresponding ordinal recognition; equally ordinal recognition implicitly entails corresponding cardinal recognition. Therefore, properly speaking, both the cardinal and ordinal aspects enjoy a merely relative identity as complementary partners.

However in conventional mathematical terms, the cardinal aspect is misleadingly given an absolute identity with the ordinal thereby reduced to the cardinal.

So for example the relationship of the primes to the natural numbers is viewed in conventional terms solely with respect to their quantitative cardinal identity.


There is indeed however one limiting case where ordinal meaning does indeed reduce directly to cardinal interpretation.

As we have seen, the cardinal definition of 3 = 1 + 1 + 1 (representing homogeneous independent units).

Now, if we continually identify each ordinal number solely with the last member of each number group to which it belongs, then ordinal reduces directly to cardinal meaning.

So for example the last member of a group of 1 unit is clearly that same unit.

So 1st in this context = 1.

If we continue on to identify 2nd with the last member of a group of 2, its meaning is again unambiguous as the last unit of the group.

Then when we identify 3rd with the last member of a group of 3, the meaning is again unambiguous as the last unit.

So 3rd in this context = 1.

Thus with reference to 3, 1st + 2nd + 3rd = 1 + 1 + 1 = 3.

Thus here, ordinal meaning - where each ordinal position is rigidly fixed with the last member of its number group - equates directly with cardinal meaning.

So 1st means 1st in the context of 1; 2nd means 2nd in the context of 2; 3rd means 3rd in the context of 3 and so on.

However, there are innumerable other ways in which each ordinal position could be defined. For example we could define 2nd in the context of 3 members or 2nd in the context of 100 members and so on. Indeed we can identify 2nd in this alternative way with every cardinal group > 2.

Thus an unlimited number of options thereby exist for each ordinal position with a merely relative identity.

If one obtains 2nd place in a competition confined to 3 entrants, this might not seem very impressive. However if it is 2nd in relation to say 10,000 entrants this now - relatively - appears a much greater achievement.

Therefore each ordinal notion (1st, 2nd, 3rd, etc.) can be viewed in two ways!

1) in an absolute fixed rigid manner - amenable to analytic understanding - where it is defined as the last member of its appropriate cardinal number group (i.e. 1st last of 1, 2nd last of 2, 3rd last of 3 and so on).

2) in a relative flexible manner - amenable to holistic understanding - where it can be defined in a relatively distinct manner with reference to an unlimited number of cardinal groups (greater than the minimum size required for the absolute definition).


Properly understood 1) is really just a special limiting case of 2).

Therefore the really big task arises as to how to give expression to the potentially unlimited number of relative options associated with each ordinal position.

We will look at this in the next blog entry, showing how it leads naturally to the complementary - though still largely unrecognised - notion of the Zeta 2 zeros.

Zeta Zeros Made Simple (8)

I have long been fascinated with the dimensional notion of number. Even as a young child, I found the conventional representation of multiplication unsatisfactory.

So far example 3 * 5 = 15 (in conventional terms).

More accurately this result could be written as 15¹.

In other words, in conventional mathematical terms, the reduced quantitative value of the multiplication operation (expressed in 1-dimensional terms) is solely considered.

However, it seemed obvious to me, even then that a qualitative transformation is equally involved.

Thus, as with a rectangular field, 3 * 5 would be represented in square (i.e. 2-dimensional) rather than linear (1-dimensional) units.

Therefore, when we multiply numbers both a quantitative and qualitative transformation is involved (with the qualitative aspect relating - relatively - to the change in the dimensional units involved).


The deeper investigation of this issue then led me to the realisation that there are two distinct aspects to number definition  - that are quantitative and qualitative with respect to each other - which keep switching in the natural dynamics of experience.

So we have the Type 1 aspect of number that is quantitative in the accepted sense that is defined in (default) 1-dimensional terms.

Thus in quantitative terms, all real numbers lie on the number line.

So, if again we take 3 to illustrate, its Type 1 definition is given as 3¹.
This of course equates with the cardinal notion of number which can be represented in quantitative terms as composed of independent individual units.

So 3 = 1 + 1 + 1.

It equally applies that because of the homogeneous nature of each unit, they are thereby lacking any qualitative distinction.


Now the Type 2 definition of number is directly related to its dimensional expression (as power or exponent) which is now defined with respect to a default base number of 1.

The idea here is when we raise 1 to a power (> 1) that clearly no quantitative change is involved. However the qualitative nature of the units does indeed change.

So for example if we raise 1 to the power of 3 (i.e. obtain the cube of 1), no quantitative change is involved with the result remaining as 1. However we are now dealing with 3-dimensional - rather than 1-dimensional - units which in this context - relatively - represents a qualitative rather than quantitative notion.

So this latter Type 2 definition of 3 is given as 1³.

So the vital point to recognise here again is that number has in fact two distinctive aspects (Type 1 and Type 2)  that are quantitative and qualitative with respect to each other.

Thus whereas the quantitative is based on the notion of independent units, the corresponding qualitative notion - by contrast - relates to number interdependence.

When on reflects on the nature of higher dimensions (> 1) this quickly becomes apparent.

Clearly to construct a 2-dimensional square figure the length and width cannot be considered as independent but in fact must be related to each other in a precise manner. So when we represent the unit line as its length, then the corresponding unit representing the width, must be now drawn at the end of the line (representing the length) at right angles to it.

Thus the length and breadth are thereby not independent of each other but rather related to each other in a definite manner.


Then we further go on to represent a cube (as the representation of 1³), the units representing length, breadth and height must now be related to each other again in a precise manner.


Thus the crucial distinction as between the quantitative (Type 1) and qualitative (Type 2) aspects of number is that the former relates to the notion of units which are independent, whereas the latter relates to the corresponding notion of units that are interdependent (and thereby related to each other).

Therefore, properly understood, number is a dynamic interactive notion with complementary aspects that are relatively independent and interdependent with each other.

Without the independent aspect, we would not be able to distinguish the very numbers (that are to be related to each other in subsequent operations); without the corresponding interdependent aspect, we would have no means of achieving a common relationship as between these distinct numbers.

So both aspects (i.e. quantitative independence and qualitative interdependence) are vital for all number operations, which can only be properly understood in a dynamic relative manner.

However Conventional Mathematics is riddled through and through with the most massive reductionism, whereby in every context, notions that are properly qualitative in nature, are reduced in a merely quantitative - and thereby greatly limited and distorted manner.


To conclude this entry, I mentioned earlier how the Type 1 (quantitative) notion of 3 can be represented as

3 = 1 + 1 + 1 (where each unit is independent).

However the corresponding Type 2 (qualitative) notion of 3 cannot be represented in this manner (as the various units are related and thereby interdependent with each other).

So the latter representation is given as:

3 = 1st + 2nd + 3rd (where the various units are now related in a qualitative type manner).

So we now see that both the cardinal and ordinal notions of number are relatively distinct in nature (pointing in turn to both its quantitative and qualitative aspects).    

Zeta Zeros Made Simple (7)

Once again we return to the important distinction as between analytic and holistic type understanding that, potentially, equally apply to (all) mathematical symbols and relationships.

Analytic interpretation essentially is based on the clear separation of polar opposites in experience.

So again with respect to the simple example of road directions, North (N) and South (S) are polar opposites.

When we then approach a crossroads heading in just one direction - say - N, then we can unambiguously identify left and right turns with respect to this clearly separated polar reference frame (i.e. N).

Equally when we approach the crossroads from the opposite direction heading S, again we can unambiguously identify left and right turns with respect to this clearly separated polar reference frame (i.e. S).

So in terms of either reference frame (considered separately in an independent fashion) left and right turns have an unambiguous meaning where a turn is either left or right in an absolute manner!

This represents in fact a good example of analytic type interpretation (as I define it) where meaning takes place within independent polar reference frames that can be clearly separated.


Now all mathematical understanding is necessarily conditioned by polar reference frames. For example all such understanding necessarily entails an interactive relationship as between external (objective) and internal (subjective) poles.

So analytic interpretation in this context is based on the assumption that the external (objective) aspect can be clearly separated from the corresponding internal (subjective) aspect in an absolute type manner. In this manner a direct correspondence is thereby necessarily assumed as between objective truth and subjective (mental) interpretation.


Likewise all mathematical understanding necessarily entails an interactive relationship as between individual (part) and collective (whole) notions that are quantitative and qualitative with respect to each other.

So once again analytic interpretation is based on the assumption that quantitative and qualitative aspects can be again be clearly separated in absolute type manner from each other.

Thus formally Conventional Mathematics is viewed as the objective interpretation of quantitative type relationships (in an absolute unambiguous type manner).

In other words Conventional Mathematics is formally associated with merely analytic (Type 1) interpretation of all relationships. I equally refer to this as linear type interpretation (which literally commences from the view that all the real numbers lie on a number line!)


However when we return to our crossroads example we can perhaps appreciate that a distinctive circular type of interpretation - that is paradoxical in terms of linear - equally applies.

Therefore when we consider the two polar directions of N and S simultaneously as interdependent with each other, then left and right turns at a crossroads are rendered as circular and paradoxical (in terms of conventional linear interpretation).

So what is designated as a left turn (heading N) is equally a right turn (heading S); likewise what is designated a right turn (heading N) is equally a left turn (heading S).

Thus when we view both polar reference frames simultaneously (as interdependent) each turn is understood as both left and right (depending on context).

So linear logic (based on independent polar reference frames) is associated with unambiguous either/or distinctions; circular logic (based simultaneously on interdependent reference frames) is based on paradoxical both/and distinctions.


Now it is this latter type of understanding, where the fundamental polar reference frames that necessarily govern all mathematical experience are viewed simultaneously as interdependent, that I term holistic.

So from one important perspective, in holistic terms we cannot hope to clearly separate the external pole of objective mathematical recognition from the corresponding internal pole of mental interpretation!

Likewise from an equally important perspective, we cannot hope to clearly separate the quantitative pole from corresponding qualitative pole of mathematical understanding.

In other words there are base and dimensional aspects to all number recognition which are quantitative and qualitative with respect to each other.

Thus when I define 3 for example in a quantitative base fashion, it already assumes the default dimensional aspect of 1.

So 3 (in Type 1) terms is expressed as 3¹.

However when I define 3 - relatively - in a dimensional qualitative fashion, it already assumes the default quantitative base of 1.

So 3 (in Type 2) terms is expressed as 1³.

Thus all numbers - such as 3 in this example - continually switch as between their Type 1 and Type 2 aspects (that are quantitative and qualitative with respect to each other).

Like with the turns at the crossroads, when we attempt to understand either aspect (Type 1 or Type 2) in isolation, unambiguous analytic type interpretation is possible.

However, when we attempt to understand these two aspects as interdependent, then just as with the turns at the crossroads, interpretation now becomes deeply circular and paradoxical in a holistic type manner.


Now just as comprehensive interpretation of the turns at a crossroads ultimately combines both analytic (linear) and holistic (circular) type understanding equally this is so in  mathematical terms.

In fact, properly understand, the Type 1 and Type 2 aspects correspond to the distinction as between cardinal and ordinal interpretation of number (that are quantitative and qualitative with respect to each other).


Ultimately with respect to the crossroads example, we come to the realisation, by combining both analytic and holistic type interpretation, that left and right turns have merely a relative meaning (depending on an arbitrary context).

Likewise all mathematical interpretation of relationships - in what I term Type 3 understanding, combines analytic and holistic type interpretation - has a merely relative meaning (depending on context).


In conclusion, in this entry, I wish to demonstrate the paradoxical nature of the Riemann (Zeta 1) zeros from the holistic mathematical perspective.

Once again, using analytic type distinctions, in Conventional Mathematics, a clear absolute distinction is made as between the primes and the composite natural numbers which are connected with each other through multiplication.

So the primes have no factors (other than 1 and the prime number itself) while the composites necessarily contain two or more prime factors.

This of course also also entails that the composites necessarily contain two or more natural number factors.

Then I make the following definitions

With all primes both 1 and the prime itself are excluded as factors; however with composites, though 1 is again excluded, the natural number itself is included as a factor.

So from this perspective, 5 as prime, has no factors, while 4 as composite has 2 natural number factors (i.e. 2 and 4).

Therefore in general terms, while all primes have no factors, then all composites have 2 or more factors!

In this way a clear analytic distinction separates the primes and the composites.

If we now raise the intriguing issue of numbers that contain just 1 factor (as it were) this then confounds the analytic logic that clearly separates the primes and composites.

However, from a holistic perspective, this has an unexpected meaning, whereby the Riemann (Zeta 1) zeros are now seen as the entities that paradoxically bridge the divide that separates the primes from the composites.

So again in analytic terms, a number is either prime or composite; however in holistic terms a number is both prime and composite (when Type 1 and Type 2 aspects of number are related).

Now of course we cannot hope to understand this in a conventional mathematical manner as it is based solely on Type 1 analytic type interpretation. However we can hope to understand this in Type 3 terms (which requires the ability to recognise both the Type 1 and Type 2 aspects of number definition and then to simultaneously relate them as interdependent).


Essentially what this entails is the ability to recognise that the relationship of primes to natural numbers (and natural numbers to primes) is directly opposite with respect to both reference frames (Type 1 and Type 2).

Therefore in holistic terms (where both are now seen as interdependent) the primes and composites are seen as perfect mirrors of each other and thereby ultimately identical in an ineffable manner.


This is then beautifully reflected in the fact that the Riemann (Zeta 1) zeros, in holistic terms can be seen as those numbers with just one factor (thereby bridging the analytic divide as between the primes and composites).


Let us see more clearly how this operates.

Once again in analytic terms we start with the primes 2 and 3 (with no factors) before encountering 4 (with two factors). Then we encounter the prime 5 (with no factors) and then 6 as composite (with 3 natural number factors). Next we encounter 7 as prime (with no factor) and then 8 as composite (with 3 factors).

Thus there is considerable discontinuity here as between the primes (with no factors) and the composites (with multiple factors).

The Riemann (Zeta 1 zeros) can then be fruitfully seen as the attempt to reconcile these factor disparities as between primes and composites.

Now for proper comparison we divide each Riemann (Zeta 1) zero by 2π.

Thus the positions of the first 8 zeros (to 2 decimal places) are,

 2.25,  3.35,  3.98,  4.84,  5.24,  5.98,  6.51,  6.89

Now compare this with the corresponding accumulation (up to the number 8) of the first eight natural number factors (of the composites).

So when a composite has factors we repeat that number (in accordance with its total number of factors). So 4 has 2 factors, 6 has 3 factors and 8 also has 3 factors.

So we record this as

4,       4,       6,       6,       6,       8,       8,       8,

Though we have only considered the first 8 values here, one can see how they represent a sort of smoothing of the discontinuous nature of the primes (with no factors) and the composites (with multiple factors).

So whereas the primes (with 0 factors) and the composites (with multiple factors) alternate in a somewhat uneven manner, the Riemann Zeros represent the perfect harmonisation, with each zero occurring just once.

When we match the Riemann (Zeta 1) zeros up to t and the corresponding accumulation of natural number factors up to n (where n = t/ 2π), two remarkable features are in evidence.

1) the frequency of Riemann Zeros matches very closely the corresponding frequency of natural number factors. For example up to t = 628, we have 361 zeros, whereas up to n = 100 (where n = 628/2π) we have the accumulation of 357 natural number factors.

2) the accumulated sum of the product of each composite number by its total number of factors (up to n) matches very closely the corresponding sum of the linearly adjusted Riemann zeros up to t (where again n = t/ 2π). For example, I have calculated that that this accumulated sum up to n = 100 is 201367, whereas the corresponding sum of linearly adjusted Riemann zeros (up to 628) is 20133 (an accuracy of nearly 99%).

See "Estimating Sum of Riemann Zeros (2)".

Thus this offers valuable empirical evidence for the fact that the Riemann zeros represent in fact a smoothing out of the discontinuous manner in which factors occur as between the primes and the composites.


So to repeat, the crucial feature of the holistic interpretation of the Riemann (Zeta 1) zeros is that they represent paradox in terms of analytic mathematical notions.

Thus again in analytic terms, we clearly separate the primes and composites; with the Riemann zeros, these features are holistically reconciled. Alternatively in analytic terms we clearly separate randomness and order with respect to the number system. Again through the zeros, these features are holistically reconciled. Likewise in analytic terms we separate the operations of addition and multiplication. Again through the zeros, these two operations are dynamically reconciled.

Zeta Zeros Made Simple (6)

The simplest way to understand the Riemann (Zeta 1) zeros is with respect to the dynamic notion of complementarity.

In Jungian terms, the complement of a psychological function, that is consciously revealed in personality, is its hidden shadow that remains unconscious.

So for example the (unconscious) complement of sensation (as consciously revealed) is intuition and vice versa in that the (unconscious) complement of intuition (as in turn consciously revealed) is sensation.

What this means is that these two functions are dynamically necessary for the expression of each other, which can only be achieved harmoniously when equal recognition is given to both conscious and unconscious aspects.

However when the conscious is solely recognised in explicit fashion, it creates an imbalance whereby it assumes a distorted absolute type significance, with its complementary aspect thereby remaining hidden in the unconscious (as its unrecognised shadow).


Now if we substitute the somewhat equivalent notions of analytic and holistic for conscious and unconscious respectively, then we can say that the Riemann (Zeta 1) zeros represent the holistic complement to the conventional analytic notion of the prime numbers.

Equally from the reverse perspective, the primes represent to holistic complement to the conventional analytic notion of the Riemann (Zeta 1) zeros.

Thus, just as the psychological functions, through which we interpret reality, have both conscious and unconscious aspects in dynamic interaction with each other, equally the number system has both analytic and holistic aspects in dynamic interaction with each other.

So once again, if we understand the primes with respect to their conventional analytic aspect, then the Riemann (Zeta 1) zeros represent the holistic complement to such understanding; and again from the opposite perspective, if the Riemann zeros now are understood with respect to their conventional analytic aspect, then the primes now represent in turn the corresponding holistic aspect.


So properly understood, all mathematical symbols can be given both (Type 1) analytic and (Type 2) holistic interpretations, with both continually switching through the dynamic interactive nature of experience.

However Conventional Mathematics is interpreted in a solely (Type 1) analytic type manner in absolute terms, which explicitly recognises the merely conscious aspect of rational understanding.

Quite simply however, the true relationship of the primes to the natural numbers (and natural numbers to the primes) cannot be properly understood in this manner.

So just as conscious and unconscious should be recognised in psychological terms as necessary partners, likewise this should also be true in terms of a comprehensive mathematical understanding with both analytic (Type 1) and holistic (Type 2) interpretations available for all its symbols and relationships.


We can in fact fruitfully probe further this relationship between the primes and Riemann (Zeta 1) zeros.

The primes are viewed in conventional terms as the independent building blocks of the natural number system.

So an independent building block in this sense has no factors!

Therefore the complement of this independent notion of prime numbers is corresponding notion of the composite numbers that are now comprised of two or more factors.

And in complementary terms we consider the natural number - rather than prime - factors of each composite number.


So on the one hand, we have the sequence of prime numbers that increases in a step like discontinuous fashion.

So we start with 2 to encounter the first prime. Then at 3 the step function again increases discontinuously by 1 (now yield an overall  frequency of 2 primes).
Then at 4, there is no change and we continue the function parallel to the x axis, before increasing once more by 1 at 5 (to now accumulate 3 primes). Then we have now change at 6 before increasing discontinuously by 1 at 7 (attaining a frequency of 4 primes).

And we can go on and on in this fashion to represent the exact frequency of primes.


Now we could use a similar type of step function to represent the complementary notion of natural number factors.

Now in measuring factors 1 is not included! With respect to the number itself (which is necessarily a factor) this is included only for composite numbers. Thus primes from this perspective have no factors. However a composite number such as 12 would have 5 natural number factors  i.e. 2, 3, 4, 6 and 12).

Thus once again we use the vertical axis to measure frequency. So we start with 2 (which is prime) and move along the axis horizontally to 3 (which is also prime) then at 4 we encounter 2 factors so we step up vertically by 2 at 4 (represent the accumulate frequency of natural number factors). Then we move from 2 (on the vertical axis) at 4 (on the horizontal axis) to 5 (which is prime) then we move horizontally across to 6 where we encounter 3 factors. So we step up by 3 at 6 (to now accumulate 5 factors on the vertical axis). Then we move from here horizontally, parallel to x axis, to 7 (which is prime) then to 8 (where again we step up by 3, to accumulate 8 factors on the vertical axis) and so on.

So we have on the one hand a step function to represent the exact frequency of primes (to any number) and on the other, a complementary step function to represent the exact frequency of natural number factors.

Now if we attempt to measure in the first case the general frequency of primes (up to any number) using a continuous smooth function, we could employ the simple formula  n/(log n – 1).

For example the exact frequency of primes to 1 million is 78,498.

The estimated frequency using our simple formula is 78,030 (which is already 99.4% accurate).


Now the corresponding formula n(log n – 1) can be used to accurately approximate the accumulated number of natural factors (up to a given number).

For example I manually counted 357 such factors up to 100 and the estimated measurement using the formula suggested is 361!

Thus whereas n/(log n – 1) provides a good general measurement of the frequency of primes (to a given number), the complementary formula, n(log n – 1) likewise provides a good general estimate of the accumulated frequency of natural number factors (to a given number).


Finally if we replace in this latter formula, n by t/2π, we now get the circular version t/2π{(log (t/2π) – 1}, which is the well known formula for calculation of the Riemann (Zeta 1) zeros to a given number.

In other words the Riemann (Zeta 1) zeros and the accumulated frequency of the natural number factors of composite numbers are intimately related.

Thus alternatively, to estimate the accumulated frequency of natural number factors to n, we estimate the frequency of trivial zeros to t (where n =  t/2π).

So just as the general formula can be used to smooth out as it were the discontinuous occurrence of actual primes, the trivial zeros themselves represent a complementary notion with respect to the general formula used for their estimation.