I have mentioned many times before that proper appreciation of the nature of the Riemann Hypothesis requires complementary (Type 1) analytic and (Type 2) holistic methods of mathematical interpretation.
Then Type 3 interpretation - which alone can reveal the true ultimate significance of the Hypothesis - entails the mutual dynamic synthesis of both Type 1 and Type 2 aspects.
In the same manner as in quantum physics where the wave interpretation of matter equally possesses a particle aspect (and the particle a wave aspect) the Type 1 analytic equally possesses a holistic aspect and the Type 2 holistic equally an analytic aspect of interpretation respectively.
Quite remarkably - and this is the most crucially important point of all - Conventional Mathematics gives no recognition whatsoever in formal terms to the Type 2 aspect of interpretation. Thus it inevitably attempts to interpret meaning that is properly of a holistic qualitative nature in a merely reduced analytic fashion.
The implications of this could not be more far reaching! For not alone is the Riemann Hypothesis not capable of proof in the standard Type 1 context; its very nature cannot be properly appreciated in this manner.
In other words the fundamental nature of the primes (to which the Riemann Hypothesis points) relates in number terms to the ultimate reconciliation of quantitative notions of independence with qualitative notions of interdependence.
While readily admitting its great achievements, the futility of present (Type 1) Mathematics resides in the continued attempt to deal with qualitative holistic notions (that properly relate to interdependence) in a merely reduced quantitative manner (where numbers are treated in an independent fashion). And nowhere is this problem of interpretation more exposed than with respect to the fundamental nature of prime numbers!
From the Type 1 perspective the natural numbers are viewed in cardinal terms (where integers literally reflect collective whole units). Here the primes are viewed as the (independent) building blocks of the natural numbers with every such number (except 1) representing a unique combination of prime factors.
However I have repeatedly pointed to the key limitation of this approach in that viewing numbers in this cardinal manner leaves us with no means of making ordinal distinctions.
Though this issue is repeatedly glossed over in conventional presentation, the making of ordinal distinctions as between numbers strictly reflects qualitative appreciation which does not come from their cardinal nature.
If we attempt to break down a cardinal number such as 3 in terms of its individual units we would represent it as 1 + 1 + 1. However because these units are both independent and homogeneous (i.e. lacking qualitative distinction) it would not be possible to rank them in ordinal fashion i.e. as 1st, 2nd and 3rd members of their collective group (cardinally represented as 3).
In other words the very ability to rank numbers in an ordinal fashion relates to qualitative notions of relational interdependence. And strictly as Type 1 Mathematics views numbers as separate independent units (in merely quantitative terms) it thereby can offer no satisfactory way of establishing relationships between numbers (which necessarily implies qualitative notions of interdependence). So quite simply once again Conventional (Type 1) Mathematics can only deal with such qualitative notions in a reduced (i.e. merely quantitative) manner.
The Type 2 approach as to the relationship as between the primes and the natural numbers completely inverts the nature of Type 1 appreciation.
Here each prime number - except 1 - is viewed in ordinal terms as representing a unique circle of interdependence in natural number terms.
Thus, if we again take the number 3 this is now viewed in ordinal terms as comprising a 1st, 2nd and 3rd member.
When we reflect on the matter ordinal rankings have a merely relative nature. Thus for example what is 2nd (with respect to a group of 2) clearly has a different meaning than what is 2nd (with respect to a group of 3). So in fact for every ordinal number, an unlimited set of possible interpretations exist (depending on the size of the number group).
Now to give an unambiguous meaning to these ordinal numbers we use the Type 2 number system to obtain the corresponding roots of 1.
So the 3 roots of 1 are 1, - .5 + .866i and - .5 - .866i. 1 here corresponds to the 3rd root of 1 (with exponent 3/3 = 1); - .5 + .866i corresponds to the 1st root with exponent 1/3 and - .5 - .866i corresponds to the 2nd root of 1 with exponent 2/3.
The circular interdependence of these roots is demonstrated by the fact that the sum of the 3 values = 0. In more general terms the circular interdependence of the n root of 1 is demonstrated by the fact that the sum of the n roots = 0 (except where n = 1). Once again this clearly indicates that qualitative notions of interdependence can be given no meaning in a merely linear interpretation (i.e. 1-dimensional). And once again it is such linear rational interpretation that characterises the very nature of Conventional Mathematics!
The uniqueness of the prime numbers with respect to the number of roots of unity is that with the exception of 1 (which is common to all roots) the other members are uniquely defined i.e. cannot occur in any smaller number sequence of roots. I refer to these roots (other than 1) as the non-trivial zeros of the Zeta 2 Function.
So where n = p roots are involved the non-trivial zeros arise as unique solutions for the equation,
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0
The non-trivial zeros for the Zeta 1 Functions arise as solutions for the infinite sequence of terms:
1^(- s) + 2^(- s) + 3^(-s) + 4^(- s) + ...... = 0
The non-trivial zeros for the Zeta 2 Function correspondingly arise for the complementary reverse equation (i.e. where the exponent becomes the base quantity and the base quantity the exponent) with a finite no. of terms.
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0.
Thus from the Type 2 perspective the very nature of the prime numbers is completely inverted. Whereas in Type 1 cardinal terms they are considered as the most independent of numbers (serving as the building blocks of the natural number system), in Type 2 terms they are now seen in complementary fashion as the most interdependent of all numbers (that are defined in circular terms by a unique set of natural numbers now given an ordinal interpretation).
So from the Type 1 perspective we use the primes in a cardinal manner to generate the natural numbers in a unique fashion; from the corresponding Type 2 perspective we use the natural numbers in ordinal terms to define uniquely each prime number!
Now when one provides both the Type 1 and Type 2 perspectives, it becomes readily apparent that the primes and natural numbers are ultimately fully interdependent with each other in an ineffable manner that both transcends all phenomenal attempts at interpretation and yet is already immanent (i.e. inherent) in physical phenomena as soon as they arise in experience.
The Type 2 approach to number is fully complementary with the Type 1 perspective.
Whereas in the Type 1, number is viewed in linear terms as a separate independent entity, by contrast in the type 2 it is viewed in circular terms as a holistic interdependent relationship.
So once again in the Type 1, the number 3 for example is viewed as an independent whole integer on the number line in cardinal terms; by contrast in the Type 2, the number 3 is viewed in holistic terms as entailing the interdependence of its 3 individual ordinal members - expressed as the 3 roots of 1 - in a circular manner.
Of course the sum of these 3 roots = 0 entailing that this qualitative notion of interdependence strictly has no quantitative significance.
However as stated earlier whereas the Type 1 analytic aspect of number has a (hidden) holistic interpretation, equally the Type 2 holistic aspect of number can be given a (hidden) analytic expression.
Thus again, the Type 1 approach in qualitative terms is based on a linear (1-dimensional) method of interpretation. The (hidden) holistic aspect requires reflecting “higher” dimensional interpretations (where numbers representing dimensions are given a corresponding qualitative meaning).
By contrast the Type 2 approach is based directly on such higher dimensional interpretation. The (hidden) analytic aspect requires reducing all numerical values in a linear quantitative manner.
This simply implies with respect to the various roots of 1, treating all values as positive (and imaginary values as real).
We can thereby give a positive real expression to all cos and sin parts of the roots of 1.
I have expressed before that such a treatment leads to a complementary (Type 2) Prime Number Theorem and equally a complementary (Type 2) Riemann Hypothesis.
What we do here is to initially sum up the cos and sin components of roots separately.
For example the sum of the cos parts of the 3 roots of 1 (in this real positive manner) = 1 + ½ + ½ = 2. We then get the average of the sum of these 3 parts = .66666.
We then do the same for the 3 sin parts. The value for one of these parts = 0 with each of the other parts = .866.
Therefore the sum of the 3 roots = 0 + .866 + .866 = 1.732 (approx.)
The average of the sum of 3 parts = .577 (approx.)
Now what I found many years ago is that when the number of n roots increases (initially sticking to prime numbers for n) that the average of the n roots quickly zones in on the constant value 2/π.
What is remarkable is that 2/π = i/ln i.
Though I did not realise the connection immediately, this latter formulation points to the Type 2 (imaginary) equivalent of the Type 1 (real) Prime Number Theorem that the average number of primes up to n approximates n/ln n.
I further began to see an important connection in the ratio of the deviations of the cos part from i/ln i to the corresponding deviation (in absolute terms) of the sin part from this value.
In this example illustrating 3 roots the ratio = . 0300…/.0596… = .5033…
Already in this really example the value is very close to .5 and this value is approximated ever more closely as the value of n increases.
Again, though I did not make the connection initially, I gradually began to see this relationship as in fact offering the complementary (Type 2) formulation to the Riemann Hypothesis.
Once again all the non-trivial solutions (i.e. except 1) arise through solving the equation
1 + s + s^2 + s^3 + ... + s^(n - 1) = 0.
As we have seen this in fact represents the complementary formulation of the Type 1
equation,
1^(- s) + 2^(- s) + 3^(-s) + 4^(- s) + ...... = 0
So in fact when properly viewed from both the Type 1 and Type 2 perspectives, we have two complementary sets of non-trivial zeros arising with two corresponding formulations of the Riemann Hypothesis!
As we know the non-trivial zeros can be used to zone in on the precise location of the prime numbers with respect to the Type 1 formulation.
In other words we can gradually eliminate all unexplained deviations with respect to the general prime number frequency among the natural numbers in this manner.
What is fascinating is that in reverse manner the prime numbers themselves can be used to explain the deviations with respect to the average sin and cos values for a natural number (n) of roots.
Indeed after some investigation I have found this remarkably simple expression to explain to a high level of accuracy such deviations.
This simple expression for the cos part is π/12(n^2) where n is prime.
I had stated before on these blogs that some years ago I manually worked out the average values of the sum of all prime numbered roots up to 127 (calculating also their deviation from 2/π (= i/ln i).
For example the average of the cos part (in this reduced linear quantitative manner) for the 53 roots of 1 = .636712981….
The corresponding deviation from 2/π (= i/ln i) = .000093209698..
Using the expression π/12(n^2) where n = 53, we obtain .000093202…
So we have already predicted the deviation correctly for the first 8 decimal places and this accuracy increases with higher n values!
To get an even more accurate prediction the following expression can be used
π/12(n^2)*{1 + [π/12(n^2)]}.
For n = 53 this yields .0000932087… which now rounded to the 5th significant figure would be correct for the first 9 decimal places.
As the deviation for the average sin value from quickly approaches twice that of the cos value from 2/π (= i/ln i),
The corresponding simple expression to predict this deviation = π/6(n^2)
The actual deviation for n = 53 is .000186411434…
The predicted deviation, i.e. π/6(n^2) = .0001864004…
When we combine cos and sin values the deviation from 4/π (= 2i/ln i)
= π/12(n^2)
So where n is prime the change in the deviation is seen to be based very closely on the square of the prime value!
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