Another Interesting Connection

As we have seen we can use the analytic formula n/log n – 1) to estimate the frequency of primes up to n.

Then we can use the complementary analytic formula n(log n – 1) to calculate the corresponding frequency of natural factors up to n.

Therefore if we obtain the ratio of the (estimated) frequency of the natural factors and primes respectively (both up to n) it will be given as (log n – 1)².

For example the actual frequency of natural factors (to n) = 357 and the actual frequency of primes 25.

Therefore the ratio = 357/25 = 14.28.

This compares fairly well with the estimated ratio = (3.60517)² = 13.00 (correct to 2 decimal places).

Alternatively we could express the ratio of the (estimated) frequency of primes to natural factors (to n) as 1/(log n – 1)².

Then when n is very large the two ratios would approximate closely to (log n)² and
1/(log n)² respectively.


Again as we have seen, we can use the holistic formula t/2π(log t/2π – 1) to calculate the frequency of Riemann (Zeta 1) zeros to t on an imaginary scale (where n = t/2π).

We can then "convert" these zeros in an analytic manner on the real scale (by setting n = t/2π).

Thus the formula for the "converted" Riemann zeros i.e. n(log n – 1), then serves in a direct manner as a means for estimating the frequency of natural factors.

Therefore, we can equally express the ratio of the formula for calculating the frequency of "converted" Riemann zeros to that for calculating the corresponding frequency of primes as (log n – 1)².

Alternatively, we can express the ratio of the formula for calculating the frequency of primes to that for calculating the frequency of "converted" Riemann zeros as 1/(log n – 1)². 

In conclusion, I would like to highlight the truly complementary nature of both the Zeta 1 and Zeta 2 zeros.

The zeta function for calculating the Zeta 2 zeros is of a finite nature and given as:

ζ₂(s)  =  1 + s¹ + s² + s³ + … + s^{t – 1} = 0 (where t is prime).

The extended infinite version of the formula is then given as:

1/2 = 1 + s¹ + s² + s³ + .......

For example, we can perhaps easily see why in the simplest case where t = 2,

 ζ₂(– 1) = 1 – 1 = 0.

However with the infinite extended version of this formula,

1/2 = 1 – 1 + 1 – 1 +.......

Remarkably, for all other prime roots (except 1) representing the Zeta 2 non-trivial zeros, the expected value of the infinite series = 1/2.

This then readily provides an important connection with the corresponding Zeta 1 function i.e

ζ₁(s)  =  1 ^{– s}  + 2 ^{– s} + 3 ^{– s} + 4 ^{– s} + ...... = 0, where all solutions for s are of the form 1/2 + it and 1/2 – it respectively.

Thus the requirement that all the Zeta 1 non-trivial zeros lie on the line through 1/2 (which is true if the Riemann Hypothesis holds) is directly linked with the Zeta 2 zeros.

The Zeta 2 zeros are then in turn directly linked with the Zeta 1 zeros is enabling all natural numbered solutions for

ζ₂(s)  =  1 + s¹ + s² + s³ + … + s^{t – 1} = 0 (i.e. where t can be a natural number > 1).

So in dynamic interactive terms, the Zeta 1 and Zeta 2 are truly interdependent in a holistic manner.

Therefore the Riemann Hypothesis cannot be proved in the standard analytic manner of Conventional Mathematics (which assumes independence of polar reference frames).

Put another way, the Riemann Hypothesis directly relates to a key requirement for the consistency of both the cardinal and ordinal aspects of the number system (in quantitative and qualitative terms). This cannot be proved using mere quantitative notions based on the cardinal interpretation of number. 

Conventional proof in Mathematics thereby is already built on the implicit assumption that the Riemann Hypothesis is indeed true. However this represents essentially an act of faith in the ultimate consistency of the mathematical system (which cannot therefore be proved or disproved within this system).   

Complementary Formula for Zeta 2 Zeros (5)

Once again the Zeta 2 zeros relate to the roots of the natural number members of the primes (considered as dimensional groups).

So 5, for example as a prime group is comprised of its 1st, 2nd, 3rd, 4th and 5th members, or alternatively 1st, 2nd, 3rd, 4th and 5th dimensions.

It is represented as 1⁵.

This is the Type 2 (ordinal) definition of number.

Then we quantitatively express these ordinal notions, in a linear (1-dimensional) fashion, by obtaining the corresponding roots of 1.

Once again, strictly speaking 1 is not included as a non-trivial zero  as it is - by definition - non-unique in being always a root of 1.

So the key significance once again of the Zeta 2 zeros is that they provide a linear (1-dimensional) means of expressing ordinal notions, of a qualitative nature, indirectly in a quantitative manner.

Thus they provide a means of converting from the Type 2 aspect of number to the corresponding Type 1 aspect.

Though the Zeta 1 (Riemann) zeros are inherently more difficult to intuitively grasp, they operate in a dynamically complementary nature to the Zeta 2 zeros.

Now 5, in Type 1 terms, relates directly to the cardinal quantitative notion of this prime represented as 5¹.

As we know however, each prime is unique in having no factors (other than 1 and the prime number itself).

Thus in dynamic terms we can appreciate the extreme paradoxical nature of the primes.

From the cardinal Type 1 perspective, they uniquely serve (except 1)  as the building blocks of the natural numbers, However from the ordinal Type 2 perspective, each  prime is uniquely defined (except 1) by its natural number members (indirectly expressed through the various prime roots of 1).

Thus the Zeta 1 (Riemann) zeros directly relate to the mysterious qualitative transformations involved through multiplication of the various factors (that comprise the composite natural numbers) .

And as we have see it is the natural - rather than the prime factors - that are directly involved here!

However there is a critical problem with our very use of language in intuitively expressing such transformations.

When appropriately understood, it is eventually easy to see the Zeta 2 zeros as representing the ordinal - rather than cardinal - nature of number in a reduced linear manner.

So the very way we appreciate (ordinal) natural number rankings is with respect to a linear number scale!

However the nature of the Zeta 1 zeros operates in the opposite direction.

Once again with the Zeta 2, we essentially reduce higher dimensional notions in a linear (1-dimensional) manner (which is the very way we are accustomed to interpret mathematical reality).

However with the Zeta 1 we are moving in the opposite direction, from quantitative 1-dimensional, to higher dimensional qualitative notions.

In other words, just as the Zeta 2 zeros provide the means of converting from the Type 2 to the Type 1 aspect of number, the Zeta 1 (Riemann) zeros provide the corresponding means of converting from the Type 1 to the Type 2 aspect of number.

In other words, proper appreciation of the Zeta 1 zeros inherently requires appropriate qualitative appreciation of a dynamic holistic nature.

And as such understanding is not formally recognised within present Mathematics, this creates insuperable difficulties with respect to their adequate appreciation.

Now the Zeta 1 zeros all lie on an imaginary line (through 1/2). This means that the numbers are not directly real (i.e. in a rational conscious manner), but rather imaginary (i.e. intuitively unconscious) indirectly presented in a rational manner.

From an early age (around 10 or 11) I already had become severely disenchanted with conventional mathematical interpretation. So my one great strength is that I have been accustomed all my adult life to looking at mathematical reality in a dynamic interactive manner.

This is why I believe I can readily see many important issues regarding which professional mathematicians are still in complete denial.

In short it requires a totally new way of looking at mathematical relationships that is inherently dynamic and interactive to come to terms with the fundamental nature of the number system.

And this cannot be achieved though mere extension of the accepted (linear) analytic means of interpretation.
It will also require a (circular) holistic mode (unconscious and intuitive) that is utterly distinct in nature.
Finally, comprehensive appreciation will require both modes (analytic and holistic) to be combined in a properly balanced manner.

Complementary Formula for Zeta 2 Zeros (4)

I have been emphasising the truly complementary nature of the two ways of calculating the frequency of prime numbers.

Once again the standard analytic cardinal (Type 1) approach uses the well known formula n/log n – 1, where the frequency of primes is measured up to n on a linear scale.

What is interesting is that this linear measurement is with respect to just one independent set of numbers. So for example in measuring the frequency of primes to 100, the estimate is with respect to one unique set (made up of the cardinal numbers 2, 3, 5, 11,....)

What is equally interesting is that each prime (in this cardinal measurement) is composed of multiple part sub-units.

So 5 for example = 1 + 1 + 1 + 1 + 1.


By contrast in a direct complementary manner, the (unrecognised) holistic ordinal (Type 2) approach uses the formula (2t/π)/(log 2t/π – 1). Here the frequency of primes is measured up to t on a circular scale, with n = 2t/π.

As the very nature of the circular scale is to indicate interdependence with respect to number. Therefore any number point can be taken as the initial starting point with which all other numbers in a number group can then be consistently related with each other.

This means that this circular measurement can be taken with respect to potentially numerous sets of numbers. So for example in measuring the ordinal frequency of primes to 100, the estimate is with respect to - potentially multiple sets of numbers (with any of the 100 points valid as the initial starting point).

However - again in complementary manner - what is interesting about each ordinal prime is that it is composed of just one unit.

So the estimate of prime frequency here is with respect to the 2nd, 3rd, 5th, 11th ... members of the ordinal set of members of 100. So whereas the cardinal notion of 5 (as we have seen) is composed of multiple part units (in a quantitative manner), the ordinal notion of 5th has one unique meaning (in the context of a group of 100)

So once again, both the cardinal and ordinal measurement of the frequency of primes are related to each other in a complementary manner requiring analytic and holistic interpretation respectively.

And as the very notion of complementarity requires a dynamic interactive means of understanding, this implies once again that the fundamental nature of the number system in its two-way relationship as between the primes and natural numbers (in both cardinal and ordinal terms) is necessarily of a dynamic interactive nature.

More than anything else this is the crucial insight that needs to be taken on board by anyone following these blog entries which directly implies that the very nature of Mathematics is fundamentally different from what is customarily envisaged.


Now just as cardinal and ordinal complementarity applies in relation to the estimation of the frequency of primes, equally it also applies to the estimation of the frequency of (natural) factors.

So again n(log n – 1) measures the (combined) frequency of (natural) factors of the composite natural numbers up to n on a real scale.

This represents the standard (linear) analytic approach to such measurement.

However the complementary holistic formula t/2π(log t/2π – 1) as we have seen, provides a stunningly accurate calculation for the estimation of the frequency of the Riemann (Zeta 1) zeros up to t on an imaginary scale (where n = t/2π). The imaginary scale in this context provides the means of expressing what is properly of a qualitative circular nature, indirectly in an analytic manner!

Thus the Riemann zeros provide the alternative holistic manner for estimation of the frequency of prime factors.

So what does this precisely mean?

Well if we take 6 as an example we have 2, 3 and 6 as natural factors.

Now we can treat these factors as representing number quantities, which the formula n(log n – 1) directly measures.

However a unique qualitative aspect also applies, so that where multiplication of factors in involved, a (dimensional) qualitative transformation also takes place (which cannot be captured in a linear manner).

So the Riemann Zeros relate directly to the (natural) factors of the composite numbers with respect to their (dimensional) qualitative aspect.

In one way this finding is remarkable. We are accustomed to linking the Riemann zeros to the primes. However though it is certainly true that a complementary (opposite) relationship connects the primes and Riemann zeros, the direct relationship is between the Riemann zeros and the (natural) factors of the composite numbers.

Complementary Formula for Zeta 2 Zeros (3)

I keep pointing out the utterly complementary nature of fundamental mathematical relationships in dynamic terms.

When one properly appreciates such complementarity one begins to see the relationship as between the primes and the natural numbers in a completely new light.

Normal mathematical interpretation is strongly defined by the absolute nature of dualistic distinctions made.

Therefore though quantitative and qualitative constitute an extremely important complementary pairing in dynamic terms, Conventional Mathematics is based on the ultimately untenable dualistic assumption that we can seek to have quantitative knowledge as separate from the qualitative aspect.

Also though internal and external again comprise another fundamental complementary pairing in dynamic terms, Conventional Mathematics is again based on the ultimately untenable dualistic distinction that we can have external (objective) knowledge of mathematical relationships that can be abstracted from corresponding internal (subjective) interpretation.

So again Conventional Mathematics is strongly based on a linear (1-dimensional) approach where - in any relevant context - knowledge is based with reference to just one non-interacting polar reference frame.

This in fact - though rarely adverted to - is the most important thing we can say about such Mathematics.
Though it might seems as heresy to those accustomed to accepting the conventional wisdom, Conventional Mathematics - by its very nature - is crucially one-sided and therefore distorted with respect to the nature of truth thereby generated.

So for example we have been long conditioned to view the number system in an abstract manner as a set of absolute quantitative relationships frozen in space and time.

Such a mistaken interpretation stems directly therefore from the linear manner of interpretation involved, which thereby eliminates authentic dynamic notions.

In truth, the number system is inherently of a dynamic interactive nature. From one perspective, we cannot meaningfully form the external objective notion of number in the absence of corresponding mental constructs which - relatively - are of an internal nature.

Likewise we cannot meaningfully have independent quantitative notions of number in the absence of corresponding qualitative notions that relate to notions of shared interdependence

So in relation to the primes and natural numbers, we do not have just one static perspective that is of an absolute quantitative nature. Rather we have two aspects, quantitative and qualitative, that are in dynamic interaction with each other.

Of course just like the left and right turns at a crossroads, what is quantitative from one perspective is qualitative from the other; and what is qualitative from one perspective is quantitative from the other.

This interaction of quantitative and qualitative is expressed through the cardinal and ordinal nature of number in dynamic interaction with each other.

If we view the cardinal in quantitative terms, then the ordinal is thereby of a qualitative nature; however in reverse when we view the ordinal in quantitative terms, the cardinal is of a qualitative nature.

Thus depending on polar perspective, cardinal and ordinal have both quantitative and qualitative aspects.

However as the standard linear approach is geared merely to quantitative appreciation, we need a corresponding holistic manner of interpretation to convey the qualitative aspect of mathematical understanding.

So the two sets of zeta zeros (Zeta and Zeta 2) can be appropriately seen as the holistic counterparts to the quantitative analytic nature of both the cardinal and ordinal aspects of the number system.

In dynamic interactive terms therefore, the two sets of zeros perfectly complement in holistic manner the cardinal and ordinal aspects (understood in analytic terms).

However, again we can switch perspectives, so that the two sets of zeros now have an analytic interpretation with the cardinal and ordinal numbers directly complementary in a holistic manner.

Put more generally all mathematical relationships - when viewed appropriately in a dynamic interactive manner - possess both analytic (quantitative) and holistic (qualitative) aspects.

In fact coming back to yesterday's blog entry, I believe that I did not emphasise fully the complementary nature of the holistic (Type 2) formula for generating the frequency of primes.

With the standard cardinal approach the frequency of primes relates to just one possible set of prime numbers.

However with the corresponding ordinal approach, innumerable different sets can be chosen which equally approximate the number of primes.

So as in our example when t = 127, the formula estimates 24 primes up to n = 81 (where n = 2t/π).

However, strictly this does not relate to just one set of numbers. In fact if we keep choosing a random similar number of "converted" roots, their sum would approximate the same answer.

Thus though there is an obvious bias in the (linear) cardinal number system as between the primes and (composite) natural numbers, there is no such bias with respect to the (circular) ordinal number system, represented by the various roots of 1. This likewise extends to the "converted" roots of 1.

Thus if we for example were to repeatedly choose 40 "converted" roots at random from the 127, and then obtain the sum each of these selections of 40 roots, the answers would approximate close to each other (with the approximation improving as both population and sample size increase).

The reason for this once again is that the Zeta 2 zeros - in a similar manner to the Zeta 1 - properly represent in dynamic holistic manner, the paradoxical identity of opposite polarities.

So once again - now with respect to the ordinal natural number members of a prime group - notions of independence and interdependence, randomness and order, quantitative and qualitative etc. are mutually combined with each other. So from this perspective the very notions of prime and (composite) natural numbers as separate lose their meaning.

Thus when we estimate the number of primes using the holistic formula, (2t/π)/(log 2t/π – 1), any of the "converted" roots can be included. 

Complementary Formula for Zeta 2 Zeros (2)

Yesterday I indicated that an alternative ordinal (Type 2) approach exists for the estimation of primes up to a given number.

Again we saw that the well known formula for calculation of the Riemann (Zeta 1) zeros up to t on an imaginary scale is,

t/2π(log t/2π – 1),

This represents the "circular" holistic version of the corresponding simple analytic formula
n(log n – 1) for calculation of the combined frequency of natural factors up to n on a real scale (where n = t/2π).

We have a similar complementarity in evidence with respect to the (unrecognised) Zeta 2 zeros.

So (2t/π)/(log 2t/π – 1) represents the circular holistic formula for the corresponding frequency of the "converted" Zeta 2 zeros which can be used to calculate the frequency of prime numbers up to 2t/π on a real scale.

This can then be used as an alternative to the better known cardinal approach for calculating primes up to n on the same real scale i.e. n/log n – 1, where n = 2t/π.

Perhaps it would help to clarify the rationale with a practical example.

In earlier work, on the Zeta 2 zeros, I had manually calculated all roots (cos and sin part parts) up to p = 127.

So armed with this information I then set about calculating the prime numbered roots up to 127 with a view to calculating the sum of both sin and cos parts as an alternative means of measuring the frequency of primes.

The sum of the "converted" values of the cos part for all 127 roots of 1 = 80.85 (correct to 2 decimal places).

The sum of the "converted" values of the cos part for all prime numbered roots (up to 127) = 21 .55.

Therefore we can use these 2 measurements to estimate the frequency of prime numbers up to 81 to nearest integer) with both measurements rounded to nearest integer.as 22.

In fact in this case 22 is the exact number of primes up to 81!

I then calculated the corresponding sum for the "converted" values of all roots to 127 for the sin part, and then the sum of the prime numbered roots up to 127 for the sin part.

The sum of all "converted" natural number roots to 127 again = 80.25 (correct to 2 decimal places) with the sum of the prime numbered roots = 19.27.

This would suggest 19 prime numbers to 81 (which is 3 less than actual total).

However as the value of t increases, the sums for both cos and sin parts (with respect to natural and prime numbered "converted" roots) would converge closer and closer together so that  both estimates for prime frequency would be increasingly similar.


So once again, the key point about this exercise is that we have two complementary ways of calculating prime number frequency.

We have the standard cardinal (Type 1 approach) where we attempt to measure the number of primes up to a given number on the real scale.

However equally we have an (unrecognised) ordinal (Type 2) approach where we attempt to measure the "converted" frequency of the sum of prime numbered roots (for cos and sin parts) in relation to the overall sum of the natural numbered roots.

Now it is in relation to this 2nd approach that the formula,

 (2t/π)/(log 2t/π – 1) can be used.

So in relation to our example, t = 127.

Therefore this formula will measure the "ordinal" frequency of primes up to n = 2t/π on a "converted" real scale.

So 2t/π = 254/π = 80.85.

This confirms that the assumption that sum of both cos and sin parts for all natural numbered roots up to t = 127 approximates 2t/π is in fact already extremely accurate!

Our estimate for primes up to 81 from the formula = 24 to nearest integer (which represents an overestimate of 2 in this case).


So once again we have a linear (analytic) and holistic (circular) manner for calculating both the frequency of natural factors (to a given number) and the corresponding frequency of primes.

The linear (analytic) manner of estimating the combined frequency of natural factors up to n

= n(log n – 1).

The corresponding circular (holistic) manner fro calculating the combined frequency of natural factors up to t on an imaginary scale (using in fact the Riemann Zeta 1 zeros) is given as,

t/2π(log t/2π – 1) where n = t/2π.

Fascinatingly, we use an imaginary scale here, as the numerical measurement of factors (relating to dimensional transformation) is qualitatively distinct from number quantities (measured on a 1-dimensional linear scale).

So in fact we have established that a direct relationship exists as between the Riemann (zeta 1) zeros and the natural factors of the composite numbers.


We have also both linear (analytic) and holistic (circular) ways of calculating the corresponding frequency of primes.

The linear (analytic) manner for estimating the cardinal frequency of primes up to n,

= n/log n  – 1).

The corresponding circular (holistic) manner of estimating the ordinal frequency of primes up to t (where n = 2t/π),

= (2t/π)/(log 2t/π – 1).

So again we have established that a direct relationship exists as between the Zeta 2 zeros and the primes.


This however implies that a complementary relationship exists as between the Riemann (Zeta 1) zeros and the primes.

Equally it implies that a complementary relationship exists as between the Zeta 2 zeros and the natural factors (of the composite numbers).

Complementary Formula for Zeta 2 Zeros (1)

I have been discussing the "linear" and "circular" versions of a formula that can calculate both the combined natural factors up to a given number n (on a real scale) and a corresponding frequency of Riemann zeros up to a corresponding number t (on an imaginary scale) where n = t/2π.

Thus the frequency of  Riemann zeros here provides the equivalent holistic interpretation to the frequency of factors in analytic terms.

So the analytic formula for frequency of factors = n(log n  – 1) whereas the corresponding holistic formula for frequency of trivial zeros = t/2π(log t/2π – 1).

Now of course there is an even better known simple analytic formula that complements that for frequency of factors.

This is n/log n – 1), a more accurate version of  the well known formula (n/log n) for estimation of primes up to a given number.

Because of the complementarity as between analytic and holistic explanations, this suggests that a parallel  formula exists for measuring the frequency of the Zeta 2 zeros.

In fact, the dynamic use of complementarity can suggest the precise nature of this formula.

In the corresponding holistic formula for calculation of Riemann (Zeta 1), n (in the analytic formula)  is replaced by t/2π.

However in the two analytic formulae, complementarity applies in this manner.

If a = n and b = log n – 1, then ab is replaced by a/b.

So n(log n  – 1) for calculation of frequency of natural factors becomes n/log n – 1) for calculation of frequency of prime numbers.
Notice how natural factors complement prime numbers in this case!

However complementarity also applies to 2π with respect to the holistic formula for calculation of Riemann (Zeta 1) zeros, which now becomes 2/π with respect to the corresponding holistic formula for calculation of the Zeta 2 zeros.

Thus our equivalent holistic formula that complements n/log n – 1 is (2t/π)/(log 2t/π – 1).

So what does this formula precisely measure?

Well, I have mentioned before in previous blog entries how the Zeta 2 zeros correspond directly with the prime roots of 1 (with the exception of 1 which is always a root).

The significance of these prime roots of 1 (i.e. Zeta 2 zeros) is that - by definition - they comprise a unique set for all prime numbers. In other words these roots can never repeat themselves (where prime numbered roots are concerned).

Now clearly when we add up all the roots of 1 (including the trivial root 1) we always get zero.

However there is a fascinating way for converting these roots in a (reduced) real manner where we simply ignore both negative and imaginary signs treating both sin and cosine parts of all roots in a positive real manner.

So for example the 3 roots of 1 are 1, .– 5 + .866i... and – .5  – .866i  respectively. Strictly the latter two roots here comprise the Zeta 2 zeros as solutions to the equation,

1 + s¹ + s²  = 0. (Again, 1 is always a common root) .

Now the sum of the 3 roots (as always with the sum of the n roots of 1) = 0.

However, indirectly we can convert to a meaningful quantitative measurement by ignoring both real and imaginary signs.

So the "converted" real  values of the 3 roots are 1, .5 + .866, and .5 + .866

Now the sum of the "converted" cos parts = 1 + .5 + .5 = 2 and the sum of the "converted" sin parts =

.866 + .866  = 1 .732...

Now what is striking is that we when we then obtain the average of both "converted" cos and sin parts that they approximate 2/π = .6366... (with the approximation quickly improving as the value of t increases).

Already with just 3 values the average of the cos values = .666... and the average of the sin values (also dividing by 3) = .5773...

Now the actual average of "converted" cos values will always exceed 2/π while the corresponding average of sin values will always be less than 2/π.

Remarkably, the ratio of the (absolute) difference of cos and sin values quickly approximates .5, bearing comparison with the corresponding situation with respect to the Riemann zeros (as lying on the imaginary line through .5).

Indeed in this case (with just 3 values), the absolute ratio of differences = .03.../.0593... = .506...

So the approximation to .5 is already very close!

Because the average of both cos and sin parts approximates to 2/π, this means that if we sum up all t roots for both cos and sin parts they will approximate 2t/π in both cases.

If we now were to express the sum of the prime valued cos and sin parts (of t) in relation to the sum of all roots of t, we would use our required formula,

(2t/π)/(log 2t/π – 1) .

Inbuilt in this is the important assumption that the value of "converted" prime roots (both cos and sin parts) would be random and thereby unbiased with respect to all "converted" roots!

So we have now arrived at the fascinating conclusion that there are in fact two ways of expressing the relationship of prime number frequency.

We can, as in the conventional manner, express this in Type 1 cardinal terms as the frequency of the primes (in relation to the natural numbers) up to n.

However we can equally in a Type 2 ordinal manner express this relationship as the combined value of the "converted" prime roots (cos and sin parts)  of 1 in relation to the combined value of all t roots.

Thus for example if we take the 100 roots of 1 to illustrate we would get "converted" real values for both the cos and sin parts of all these 100 roots.

We would then estimate the combined value of the prime roots (2nd 3rd 5th, 7th,..., 97th roots ) in relation to the combined value of all 100 roots for both "converted" cos and sin parts.

This would then give an alternative "ordinal" measurement of the frequency of the prime numbers in relation to the natural numbers.     

New Perspective on Riemann Zeros

In earlier blog entries, Simple Estimate of Frequency of Riemann Zeros 1 and Simple Estimate of Frequency of Riemann Zeros 2, I demonstrated the very close link that connects the (natural) factors of the composite numbers with the Riemann (Zeta 1) zeros.

So again if we measure the frequency of the (natural) factors (on the real scale) up to n and then compare this with the corresponding frequency of the Riemann zeros (on the imaginary scale) up to t, where n = t/2π, the two totals will approximate very closely with each other.

Indeed we can then simply convert the Riemann zeros to measurements on the corresponding real scale by dividing each value (on the imaginary scale) by 2π.

So we now can draw direct comparison with the frequency of the (natural) factors of the composite numbers up to n, with the corresponding frequency of these "converted" Riemann zeros also up to n (both now measured on a real scale).

Once again, to illustrate. whereas the (distinct) prime factors of a composite number such as 12 are 2 and 3, the corresponding natural factors of 12 are 2, 3, 4, 6 and 12. Thus in compiling the natural factors of a composite number, we include all factors (including the number itself) except 1.

Now when it comes to prime numbers we do not consider any factors. So for example though a prime number such as 7 is - by definition - divisible by 7, we do not in this case include 7 as a factor precisely because it is not a distinct factor (i.e. is not distinct from the number in question) and has no factors.

By contrast in the case of 12 for example, though 12 is not a distinct factor in this sense, it is indeed comprised of sub-factors.

So in the light of these definitions, the prime numbers all share the same characteristic of possessing no natural factors, whereas - by definition - the composite numbers are comprised of at at least two factors.

This highlights the discontinuous nature of the number system, which alternates as between randomly occurring primes (with 0 factors) and non-random occurring composite numbers (with at least 2 factors).

Thus on the natural number scale we start with 1 (which serves as the unit measurement for cardinal numbers, both prime and composite). Then we encounter the 1st prime i.e. 2 (0 factors), then the next prime 3 (0 factors) then the 1st composite number 4 (2 factors), another prime 5 (0 factors) then the next composite number 6 (3 factors) another prime 7 (0 factors), the next composite number 8 (3 factors) then another two composite numbers i.e.  9 (2 factors) and 10 (3 factors).

Therefore the frequency of the total number of natural factors encountered in the 1st 10 natural numbers = 2 + 3 + 3 + 2 + 3 = 13.

This compares very closely with the corresponding frequency of the "converted" Riemann zeros (up to 10) = 14.

And, as I demonstrated in the earlier entries, the two measurements approximate ever closer to each other (in relative terms) as we increase the value of n.

So once again the formula for calculating the original (non-converted) Riemann zeros, is

t/2π(log t/2π – 1), which is remarkably accurate - not only in relative - but also in absolute terms, though I have suggested that t/2π(log t/2π – 1) + 1 provides an even more accurate measurement in the majority of cases. 

Now the corresponding formula for calculation of the "converted" Riemann zeros is,

n(log n – 1), which again is remarkably accurate (in both absolute and relative terms).

This also provides a very good estimate (though not of the same absolute accuracy) of the combined frequency of natural factors up to n.

In one way it is quite remarkable that we are here showing a very close direct relationship as between the Riemann zeros and the (natural) factors of composite numbers, when generally the connection is drawn as between the Riemann zeros and the primes!

Therefore though the relationship of the zeros with the (natural) factors is of a direct nature, the corresponding relationship of the zeros with the primes is of a complementary (i.e. dynamically opposite) nature.

This is why indeed I have frequently used the Jungian notion of the "shadow" to describe the relationship of the Riemann zeros to the primes in that the zeros in fact comprise the collective "shadow" number system to the natural number system (based on the product of primes).

Alternatively, we can express the Riemann zeros as the holistic number system counterpart to the analytic system of the (cardinal) natural numbers.

In other words, customary analytic interpretation of the (cardinal) natural numbers is based on quantitative notions of number independence. The deeper significance of this is that analytic notions imply linear unambiguous interpretations (based on using just one polar reference frame)

However holistic interpretation by contrast is based on the qualitative notion of number interdependence, where at least two complementary polar reference frames are simultaneously adopted in a circular paradoxical type manner.

So in analytic terms, the natural composite numbers are clearly contrasted with the primes. Thus, as we have seen, we can interpret the primes as comprising 0 (natural) factors whereas the composite numbers have least 2 or more factors.

We could also see this as the contrast between notions of randomness and order, with the primes comprising the random and the composites the ordered  aspects of the cardinal number system respectively.

Therefore the holistic appreciation of the Riemann zeros relates to the paradoxical notions of both prime and composite aspects (or equally randomness and order) being simultaneously combined in a dynamic interactive manner.

In my recent blog entries,  I dealt at length with the paradoxical notion of how the Riemann zeros can be seen as the points, where the notions of randomness and order are simultaneously combined with respect to the cardinal number system.

Indeed, a fascinating alternative way can now be used to suggest the same dynamic notion.

As we have seen in standard analytic terms, the primes and the composite natural numbers are clearly divided with once again the primes having 0 factors and the composites 2 or more.

However whereas the factors of the composite numbers occur in discontinuous blocks (i.e. of 2 or more) the Riemann zeros occurs always as single figures.

Now, remarkably all the Riemann zeros can thereby be defined as numbers with 1 factor. Therefore they fall neither into the category of the primes or composite numbers as separate but rather as combining what is common to both primes and composites. Thus though in analytic terms, the primes and composites are necessarily separate (with respect to factor composition) in a dynamic holistic manner they can be seen as approximating total identity with each other. And this recognition comes from the ability to simultaneously combine both cardinal and ordinal appreciation - which are the reverse of each other - in a common intuitive recognition.

This holistic recognition is what happens in a more obvious way, when we we able to see left and right turns at a crossroads as interdependent (so that what is left from one perspective is right from the other and vice versa).

So in this regard it is exactly similar with respect to the primes and the (composite) natural numbers when the two reference frames of cardinal and ordinal recognition are simultaneously combined.

However, quite remarkably there is no formal recognition whatsoever within Conventional Mathematics of this vitally important holistic manner of dynamic interpretation.

Put more bluntly, we have now been training ourselves for millennia to look at mathematical relationships in a fundamentally distorted manner.

More on Randomness and Order (7)

We have seen that the modern physical interpretation of the Riemann (Zeta 1) zeros is to consider them as representing a quantum chaological system.

So this idea combines both the notions of the quantum behaviour of particles (at the subatomic level) and also those of chaos were the behaviour with respect to an event becomes ultimately highly unpredictable due to extreme sensitivity to initial conditions.

However when we look at this more closely, this corresponds very well with the dynamic two-way relationship of both randomness and order within the number system.

So once again from the cardinal perspective each individual prime appears of a highly random nature, whereas the collective relationship of primes (to the number system) is of a complementary highly ordered nature; then, in reverse from the ordinal perspective, each individual prime is of a highly ordered nature (i.e. with respect to the arrangement of its individual natural number members) whereas the collection of primes from this context is now of a highly random nature.

So when we interactively combine in a dynamic experiential manner both the cardinal and ordinal aspects of number, what is random from one perspective is ordered from the other and what is ordered from one perspective is random from the other.

Thus, in dynamic interactive terms a double coincidence of randomness and order pertains with respect to the number system (represented through the two sets of non-trivial zeros - Zeta 1 and Zeta 2 - respectively).

Quantum Chaology in fact represents the physical expression of this coincidence of randomness and order.

Quantum Mechanics bears direct comparison with the cardinal behaviour of primes in that each individual sub-atomic event is highly random yet an extraordinary order or regularity attaches to the overall collective behaviour of particles. Thus we can predict such overall behaviour to an extremely high level of probability!

The Theory of Chaos (in its various manifestations) is somewhat the reverse. Here a highly level of predictability attaches to an individual event in isolation. But then its ultimate collective effects (in a wider environmental context) becomes highly unpredictable due to extreme sensitivity to initial conditions.

So in contrast to quantum physical behaviour, we have order (or regularity) with respect to initial conditions ultimately giving rise to behaviour of a somewhat random unpredictable nature.

This complementarity as between both fields can also be appreciated in another interesting manner.

Whereas the quantitative behaviour of Quantum Mechanics is expressed through equations of a linear nature, its qualitative description is decidedly non-linear.

Then in reverse, the quantitative equations with respects to problems of chaos are expressed in a non-linear manner; but the qualitative interpretation of such behaviour is along classical (linear) lines.

However, there is much confusion I believe in evidence with respect to the relationship as between the number system and physics, with mathematicians expressing amazement that such seemingly strange links should arise.

So the search is on to find the physical or quasi-physical system whose energy states exactly correspond with the (Zeta 1) zeros.

However quite simply, this long sought after system is already well-known. In fact it is the number system.

Therefore the reason why the trivial zeros so closely match some operator of a quantum chaological nature, is because the number system is itself quantum chaological in nature (when appropriately understood in a dynamic interactive manner).

So rather than the number system mimicking so well as it were the behaviour of established physical systems, rather the inherent behaviour of these physical systems expresses the fundamental nature of the number system. So both quantum mechanical and chaotic behaviour are inherent aspects of the number system itself!

At a deeper level, these connections put paid to any notion of the number system as rigid and absolute, that can be abstracted from everyday experience (physical and psychological). Rather the number system represents the deepest encoding of all created phenomena as their inherent nature (in both quantitative and qualitative terms).
The reason we have placed so much belief in the abstract nature of the number system therefore reflects, at root, a distorted interpretation (where external and internal aspects of understanding are formally separated).

Indeed we could suggest a small refinement to the physical notions of Quantum Chaology with respect to the number system. Properly speaking we should have a double coincidence of such notions. This will require however explicit recognition of the Type 2 (ordinal) as well Type 1 (cardinal) aspects of the number system and their corresponding Zeta 1 and Zeta 2 zeros.

So what might appear as quantum behaviour from one perspective, appears as chaological from the other and vice versa.

Therefore the number system possesses both features of Quantum Chaology and Chaological Quantumness that ultimately approximate an ineffable state.

Of course, as in dynamic terms both physical and psychological aspects are complementary, we equally should emphasise the psychological counterparts of  both these types of behaviour in what might be described as "Qualtum Chaology" and "Chaological Qualtumness".

When we look at reality from a psychological perspective, we have the inevitable interaction of both conscious and unconscious aspects of personality (operating with respect to both cognitive reason and affective sensibility respectively).

Now the conscious rational mind can be used be impose a certain type of order on the more random behaviour of the senses. However this then can lead to an undue repression of feeling and instincts from the unconscious).

So the unconscious aspect can be used in a reverse manner to create a different kind of spontaneous order that is deeply based on refined instinctive promptings. So here rational behaviour at the conscious takes on a more random unpredictable nature that is brought into a new harmony through the integral guidance of the unconscious.

So in psychological terms, true maturity requires the recognition of two kinds of randomness and two types of order respectively. So if we take afffective events as random and unpredictable, order can be imposed in a conscious (cognitive) manner through disciplined reason.
However if we take conscious events now as random and unpredictable (through relinquishing the attempt at cognitive control), then a distinctive type of order can be imposed in an unconscious (affective) manner through spontaneous refined feeling.

Now clearly  the full development of personality requires an equal and balanced emphasis on both aspects (where ultimately both conscious and unconscious are fully integrated with each other).


And when appropriately understood from the psychological perspective in a holistic manner, the two sets of zeros (Zeta 1 and Zeta 2) represent the precise manner in which - from the two opposite directions - both conscious and unconscious aspects of the personality can be successfully harmonised with each other.

More on Randomness and Order (6)

As we have seen, the non-trivial zeros (both Zeta 1 and Zeta 2) are appropriately interpreted in a dynamic interactive manner, entailing the simultaneous recognition of complementary reference frames for interpretation. In this context they can be fruitfully understood therefore as representing the closest approximation possible to states that are simultaneously of both an ordered and random nature.

Now it has to be clearly recognised therefore that our experience of number can operate as between two extremes.

The conventional extreme is based on single independent polar frames of reference where the external (objective) aspect of recognition is abstracted from the internal (subjective) aspect and where the quantitative (analytic) aspect is likewise abstracted from the qualitative (holistic) aspect.

This leads therefore to the dualistic recognition of number as representing fixed forms. Indeed this is frequently associated with the ultimately untenable absolute view that number is ultimately of a totally abstract nature, independent of our customary experience in space and time.

However the non-trivial zeros in truth - which comprise an equally important aspect of the number system - represent the opposite extreme based on the interdependence of complementary polar reference frames. So here both external and internal aspects and also quantitative and qualitative aspects of recognition are so closely related that experience of number is increasingly of a nondual nature.

So quite literally at this extreme, the experience of number becomes so highly intuitive in a refined manner that it approximates to the experience of pure psychological energy states.

Now again it has to be  remembered that our actual recognition of all mathematical relationships, including of course number, entails both (conscious) reason and (unconscious) intuition.

However in formal mathematical terms, interpretation is based merely on rational type recognition (which ultimately distorts the true dynamic nature of mathematical understanding).

So we can readily see therefore how Mathematics as a formal discipline represents the specialisation of the conscious extreme of mere rational recognition (that is abstracted from the true dynamic experience of mathematical understanding).

However the appropriate interpretation of the non-trivial zeros (Zeta 1 and Zeta 2) requires specialisation in the - formally unrecognised - holistic intuitive mode of recognition, which is directly of an unconscious nature.
And once  again the essence of such recognition is the ability to simultaneously combine multiple reference frames in a dynamic interactive manner.

And when can successfully appreciate in this manner, phenomena of form increasingly lose their rigidity - arising from recognition based on abstracted single frames of reference - so that ultimately the experience of number approaches pure energy states (of a psychospiritual intuitive nature).

So the comprehensive dynamic interpretation of number lies between two limiting extremes.

At one end we approach the rational extreme based on mere conscious interpretation of a dualistic analytic nature. This is best represented by the the primes and the natural numbers.

At the other end we approach the purely intuitive extreme based on unconscious recognition of a nondual nature. This is represented by the two sets of non-trivial zeros (Zeta 1 and Zeta 2).

So properly understood in a dynamic manner, both the primes and natural numbers and then the two sets of non-trivial zeros are fully complementary with each other. The zeros therefore can be fruitfully seen in Jungian terms as representing the unconscious shadow of  our conventional number system (that is explicitly understood in a conscious manner).

Of course, in dynamic interactive terms, both physical and psychological are of a complementary nature.

Therefore just as the non-trivial zeros have a psychospiritual expression as intuitive energy states at the "higher" level of contemplative type awareness, equally they have an expression as physical energy states at the "lower" level of subatomic activity.

Now in fairness increasing recognition of the this physical expression of the zeros has been taking place in recent years, with the further refinement that they correspond to a quantum system with an underlying basis related to chaos.

However precious little recognition yet exists with respect to the psychospiritual expression of the trivial zeros that I have outlined. And ultimately both of these are necessarily complementary in nature, so that we therefore can find an equivalent type explanation in psychological terms that combines both
"qualtum spiritual" and unpredictable chaotic aspects.

Indeed as my own route to appreciation of the non-trivial zeros has come more from psychospiritual recognition  - though I have long recognised that a physical quantum basis for the zeros would also necessarily exist - I have spent much time attempting to refine what this psychological basis actually entails in terms of experience.

One of the earliest connections that I made was to see that the very notion of prime (as used in prime numbers) bears an important relationship with what is "primitive" in terms of unconscious experience.

So when we have not sufficiently mastered the unconscious aspect of our personalities, we always remain victims to primitive type instincts which surface into conscious experience in an involuntary manner.

Therefore without this unconsciousness mastery - which formerly was seen as the preserve of the advanced contemplative traditions - random disturbances from the unconscious will always threaten to undermine any disciplined order (imposed by the conscious mind).

Thus obtaining mastery of unconscious primitive projections, so that they can be properly integrated with conscious experience in a voluntary manner, thereby assumes key importance in terms of successful personality development.

When this is achieved - which always entails relative approximation in a dynamic interactive manner - both the random nature of (unconscious) instincts can be successfully incorporated with the ordered nature of (conscious) disciplined behaviour.

So just as in the last blog entry I pointed to the fact that the zeta zeros represented the dynamic identity of the notions of both order and randomness with respect to the number system, here we have the matching equivalent in terms of psychological behaviour.

Thus if we were to look at the "ideal" of a person who had achieved a high degree of integration of the personality, this would entail that underlying all engagement in conscious activity would be a highly refined unconscious mind that would be successfully integrated with the conscious aspect though a continuing flow of intuitive signals, the response to which would continually ensure that overall balance is seamlessly achieved.

So these intuitive signals would thereby serve as the psychospiritual equivalent of the non-trivial zeros.

Such a person then would have to capacity to act in a highly creative manner due to freely allowing the random instinctive unconscious to operate, yet also have the capacity to be extremely productive,  due to successfully combining creative instincts with the rational disciplined order imposed by the conscious mind.

Too often we find in life that most people fail to achieve this happy balance. So again at one extreme we have those who attempt to exercise a great deal of conscious discipline in their lives (e.g. business executives). Unfortunately however they can then wind up repressing unconscious instincts to a considerable extent, and then become the unwitting victim of such instincts when they inevitably surface in experience.

At the other extreme we have the case of those such as many artists who always are waiting for the "right inspiration" to start their work, but who through lack of sufficient discipline and order in their lives may achieve very little.
Thus from the psychospiritual perspective, true understanding of the non-trivial zeros is inseparable from the successful reconciliation of both the conscious and unconscious aspects of the personality.

More on Randomness and Order (5)

We have seen that randomness and order are complementary notions that can only be appropriately understood in a dynamic interactive manner. So they are not capable of any absolute definition in isolation, but are necessarily of a relative approximate nature (that implicitly imply each other).

We also saw that when approached analytically (i.e. through single independent frames of reference) from the cardinal and ordinal perspective respectively, what is random and what is ordered in each case are the reverse of each other.

So in cardinal terms, the behaviour of each individual prime appears highly random, whereas the behaviour of the overall collection of primes (withing the number system) appears highly ordered; then in ordinal terms, the behaviour of each natural number member (of a prime group) appears highly ordered, whereas the overall collection of primes (representing groups of individual members) appears highly random.

Therefore when we bring both reference frames together (i.e. cardinal and ordinal) holistically through the simultaneous recognition of interdependent reference frames, the very notions of randomness and order are rendered paradoxical in terms of each other.

This holistic recognition is then vital for the direct appreciation of the true nature of the zeta zeros (Zeta 1 and Zeta 2).

Thus from both perspectives, the zeta zeros imply the - seemingly paradoxical - situation where the notions of randomness and order approximate perfect identity (in relative terms) with respect to the number system.

In the case of the Riemann (Zeta 1) zeros , in terms of the number system as a whole (uniquely derived as the product of primes) we have an unlimited series of points (through 1/2 ) on an imaginary number scale (with matching positive and negative values) where each point  represents approximation to  a state that is equally random and ordered in number terms.

Then in the case of the Zeta 2 zeros, within each prime, we have an arrangement on natural number members in ordinal terms, where each individual member can be quantitatively chosen at random, while the overall qualitative relationship between the various members maintains a perfect order.

Thus the double paradox as between notion of randomness and order in this case arise from the two-way relationship as between the corresponding fundamental polarities of whole and part in both their individual and collective manifestations.

So there are a number of ways of expressing this holistic interdependence of the number system (expressed through the zeta zeros).

As we have seen the zeros represent the two-way approximate identity of the notion of randomness and order in dynamic relative terms.
Equally we could express this as the two-way identity of independent and interdependent, of cardinal and ordinal, of analytic and holistic, of quantitative and qualitative notions with respect to the number system.

There is also another complementary aspect involved in this relationship that is of equal importance.

As well as necessarily involving the dynamic two-way relationship as between whole and part, the zeta zeros equally entail the two-way relationship as between external and internal polarities.

In other words our experience of numbers in an objective external manner necessarily implies corresponding mental constructs that are - relatively - of an internal nature.

Thus strictly speaking the absolute notion of number - as some timeless entity existing in abstract mathematical space - is without foundation, for the very assertion of this position is not possible in the absence of corresponding psychological constructs of an internal nature.

Therefore all mathematical understanding necessarily takes place in a dynamic interactive context, entailing both objective notions (as external) and subjective mental notions (as - relatively - internal).

We cannot therefore have objective knowledge of number in the absence of corresponding mental interpretation (both of which are necessarily of a dynamic relative nature).

Crucially therefore, the very belief in an abstract unchanging world of number, reflects an absolute type  interpretation that is strictly untenable in terms of the actual dynamics of mathematical understanding.

Of course there is considerable value in exploration of the quantitative extreme where number relationships correspond well to such rigid assumptions. But rather like Newtonian Physics this represents an approximation that is invalidated at a deeper level of mathematical experience.
And just as Quantum Mechanics shows up the shortcomings of so many Newtonian assumptions, likewise the zeta zeros shows up the severe shortcomings of the the very paradigm that underlines Conventional Mathematics (as we know it).

Quite simply, proper appreciation of the zeta zeros will require a greatly enlarged mathematical paradigm.

This will entail (at a minimum) three distinct areas.

1) Conventional (Type 1) Mathematics. This represents the merely linear (1-dimensional) quantitative approach in analytic terms to mathematical relationships, based on single independent frames of reference.

2) Holistic (Type 2) Mathematics. This represents the greatly unrecognised circular qualitative approach in holistic terms to mathematical relationships based on multiple frames of reference that are simultaneously appreciated. Though this qualitative aspects is as equally important as the quantitative, we have not even begun yet to form an appreciation of its enormous potential in opening up in a completely new manner, vast fields of unexplored mathematical territory. Indeed its very existence is still resolutely denied at the formal level of accepted mathematical inquiry!

3) Comprehensive (Type 3) Mathematics. This entails the balanced dynamic interpenetration of  both Conventional (Type 1) and Holistic (Type 2) aspects of mathematical understanding.

In the most preliminary manner, I have been attempting to give some consistent indication of what this Type 3 approach might entail in all my blog entries on "The Riemann Hypothesis".

So I say again with considerable conviction that we are fast reaching by far the greatest watershed in our intellectual history, where the very nature of Mathematics will undergo profound change and with it, appreciation of all the sciences, the natural environment and our social relationships with each other.