We are still looking at the holistic mathematical significance of the Euler Zeta Functions for positive integer values.
We have seen that in qualitative terms, each dimensional value of s is associated with a unique structural manner of rational interpretation.
Once again from this perspective the great limitation of conventional mathematical appreciation is that in formal terms it is solely confined to linear type appreciation (i.e. where s = 1).
And the key defining characteristic of this mode of interpretation is the qualitative aspect of understanding is not recognised and in effect is thereby reduced to the quantitative!
However an infinite set of more refined alternative "higher" interpretations exist where both quantitative and qualitative aspects are given distinctive recognition.
This entails that Mathematics is no longer formally defined in mere conscious rational terms but rather in more subtle terms where both conscious and unconscious aspects interact.
In direct terms the unconscious aspect of such understanding relates to holistic intuition. However indirectly this holistic aspect can be indirectly translated in rational terms as imaginary (based on a circular manner of interpretation).
So just as in quantitative terms the structure of 1 (for 3 or more roots) always entails both real and imaginary parts, likewise in corresponding qualitative terms, the holistic structure of 1 (for 3 or more dimensions) always contains both real (conscious) and imaginary (unconscious) aspects of interpretation.
Though all my essential thinking on these matters had been long formulated before I became interested in the Riemann Hypothesis, this Hypothesis does indeed provide an extremely important application of such notions. And in reverse manner understanding the Hypothesis in this way has helped to considerably clarify my own understanding with respect to the higher structures of psychological understanding.
Using spiritual type language, the higher integral dimensions (positive) relate to the transcendent type structures of understanding that progressively unfold through the process of contemplative type development. So through increasing dynamic interaction of both rational and intuitive aspects of understanding, interpretation becomes ever more refined in rational terms.
Once again the even numbered structures relate to integral type understanding at such higher dimensions (where one attempts to define the appropriate manner through which overall holistic interdependent nature of reality can be rationally encoded).
Then the odd numbered dimensions relate to corresponding differentiated understanding at these higher dimensions (where one now tries to equally define the appropriate manner through the relatively independent nature of specific phenomena of form can be likewise encoded).
Now ultimately both the - relatively - continuous nature of the integral aspect and the discrete nature of the differentiated aspect become so refined as to be indistinguishable from each other which would relate to pure contemplative experience of reality. However this represents a limiting state that can only be approximated imperfectly in human terms.
Incidentally this brings us back to the holistic mathematical nature of e (which plays such crucial role with respect to the behaviour of prime numbers).
Just as in conventional mathematical terms both the integral and differential of e^x are indistinguishable from each other, in holistic mathematical terms it is quite similar. So here we are no longer able to distinguish the discrete (differentiated) aspect of phenomenal form from the corresponding continuous (integral) aspect of holistic emptiness.
And again we can identify such a state as one of pure contemplative awareness (in the proper balanced sense of fully harmonising both intuitive and rational aspects of understanding). So this ideal state - insofar as it can be humanly approximated - combines both profound intuitive depth with incredible rational clarity).
Indeed it is in this state that the mystery of the prime numbers is finally resolved (which is the same state any lingering problem with respect to involuntary primitive instincts) is also resolved. So once again the quantitative nature of prime number behaviour and the qualitative nature of primitive instinctive behaviour entail the relationship of both specific conscious and holistic unconscious aspects of understanding (corresponding to both the linear and circular aspects of understanding respectively).
Friday, November 19, 2010
Thursday, November 18, 2010
Holistic Mathematical Connections (2)
We have now looked at the Euler Zeta Function for positive even integers of s (2, 4, 6,...) to find that the resulting value can be always be expressed in terms of Pi.
I have also been at pains to indicate the qualitative significance of this fascinating numerical behaviour.
One might initially think therefore that a similar expresion would exist for corresponding odd integers of s (3, 5, 7,...) but, as is well known, this is not the case. No closed value expressions have yet been found (though several ingenious closely approximating formulae have been derived).
Once again this is where holistic mathematical understanding can prove very illuminating.
In all psychological behaviour we have the two related aspects of differentiation and integration respectively entailing both conscious and unconscious appreciation. Differentiation is (analytically) associated with the linear logical system thus enabling the separation of opposite polarities in experience (as independent).
By contrast integration is (holistically) associated with the circular logical system thus enabling the complementarity - and ultimate identity - of these same polarities (as interdependent).
Now when we look at the higher dimensions of understanding, whereby rational understanding becomes refined in an increasingly intuitive manner, both processes of differentiation and integration are at work.
We have already identified 1-dimensional appreciation as the representative of the linear and 2-dimensional as representative of the pure circular aspect (that exhibits full complementarity) respectively.
So 1-dimensional appreciation is properly geared for the differentiated interpretation of reality (in analytic terms); then 2-dimensional appreciation is properly geared for integrated interpretation (in holistic terms) .
In quantitative terms we see this reflected in the corresponding root structures (of 1).
The 1st root of unity = + 1 is fully independent of its negative and thereby exhibits no complementarity; by contrast the two roots of unity = + 1 and -1 exhibit perfect complementarity (as interdependent).
Now, the higher even numbered dimensions (that are positive) can best be seen as the movement towards ever more refined integral sttructures . So, as we have seen, when we obtain the roots of unity for any positive even integer, we are always able to match each of these roots with a corresponding negative. For example the four roots are + 1, - 1, + i and - i. So here each root (as positive) can be matched with a corresponding root (as negative).
In like manner, the holistic mathematical interpretation of such even dimensions relate to qualitative structures of higher integration (representing ever more refined rational appreciation of interdependence).
However the higher positive odd integral dimensions do not represent such perfect symmetry. In fact with all such root structures we will always have one root that is + 1 separated from the rest (with all other complex roots representing symmetry only with respect to their imaginary parts). Though the sum of the corresponding real parts will indeed sum up to - 1, this represants a form of broken symmetry (with respect to the real part). So - quite literally - in psychological terms, there is an inevitable delay with respect to the unconscious process of negation of phenomenal forms. In this way though experience is indeed of an increasingly refined nature, the very process of differentiating phenomena creates a degree of linear rigidity (by which they are given a degree of independence).
So the holistic mathematical interpretation of such odd dimensions (in qualitative terms) is perfectly reflected in the quantitative structure of its corresponding roots.
The clear implication therefore is that pure circular complementarity (requiring the perfect complementarity of opposites) cannot exist with the positive odd integral dimensions).
Because of a degree of breaking (of integral symmetry) in qualiative terms, such dimensions cannot represent the pure relationship of circular to linear understanding. In like manner the quantitative value of such zeta expressions cannot be conveyed through closed pi expressions.
Now a great deal of debate has continued as to whether such zeta values (for positive odd integer values of s > 1) are rational or irrational.
It has indeed been proved that - at least when s = 3 - the value is irrational.
Now there are good holistic mathematical reasons for believing that not alone is the value of such expressions always irrational (but necessarily of a transcendental nature).
If the value was in fact rational (for s > 3) this would imply a rational value in the prime product formula over the infinite range of terms. This would imply that each term bears a relationship to previous terms that can be expressed in a rational manner. However the very nature of prime numbers is that their individual quantitative nature is uniquely distinct from their overall holistic behaviour with respect to each other. Therefore no such rational formula could exist.
Therefore the convergent sum of terms inevitably entails a relationship involving both linear and circular notions (which directly implies that it is transcendental). And as we have seen what distinguishes the even from the odd dimesnions here, is that with the even we have a pure relationship of circular to linear (whereas with the odd this is not the case).
Once again the key assumption here (regarding the inherent nature of prime numbers) literally transcends conventional mathematical interpretation. Indeed it is the same assumption that ultimately establishes the axiomatic nature of the Riemann Hypothesis!
I have also been at pains to indicate the qualitative significance of this fascinating numerical behaviour.
One might initially think therefore that a similar expresion would exist for corresponding odd integers of s (3, 5, 7,...) but, as is well known, this is not the case. No closed value expressions have yet been found (though several ingenious closely approximating formulae have been derived).
Once again this is where holistic mathematical understanding can prove very illuminating.
In all psychological behaviour we have the two related aspects of differentiation and integration respectively entailing both conscious and unconscious appreciation. Differentiation is (analytically) associated with the linear logical system thus enabling the separation of opposite polarities in experience (as independent).
By contrast integration is (holistically) associated with the circular logical system thus enabling the complementarity - and ultimate identity - of these same polarities (as interdependent).
Now when we look at the higher dimensions of understanding, whereby rational understanding becomes refined in an increasingly intuitive manner, both processes of differentiation and integration are at work.
We have already identified 1-dimensional appreciation as the representative of the linear and 2-dimensional as representative of the pure circular aspect (that exhibits full complementarity) respectively.
So 1-dimensional appreciation is properly geared for the differentiated interpretation of reality (in analytic terms); then 2-dimensional appreciation is properly geared for integrated interpretation (in holistic terms) .
In quantitative terms we see this reflected in the corresponding root structures (of 1).
The 1st root of unity = + 1 is fully independent of its negative and thereby exhibits no complementarity; by contrast the two roots of unity = + 1 and -1 exhibit perfect complementarity (as interdependent).
Now, the higher even numbered dimensions (that are positive) can best be seen as the movement towards ever more refined integral sttructures . So, as we have seen, when we obtain the roots of unity for any positive even integer, we are always able to match each of these roots with a corresponding negative. For example the four roots are + 1, - 1, + i and - i. So here each root (as positive) can be matched with a corresponding root (as negative).
In like manner, the holistic mathematical interpretation of such even dimensions relate to qualitative structures of higher integration (representing ever more refined rational appreciation of interdependence).
However the higher positive odd integral dimensions do not represent such perfect symmetry. In fact with all such root structures we will always have one root that is + 1 separated from the rest (with all other complex roots representing symmetry only with respect to their imaginary parts). Though the sum of the corresponding real parts will indeed sum up to - 1, this represants a form of broken symmetry (with respect to the real part). So - quite literally - in psychological terms, there is an inevitable delay with respect to the unconscious process of negation of phenomenal forms. In this way though experience is indeed of an increasingly refined nature, the very process of differentiating phenomena creates a degree of linear rigidity (by which they are given a degree of independence).
So the holistic mathematical interpretation of such odd dimensions (in qualitative terms) is perfectly reflected in the quantitative structure of its corresponding roots.
The clear implication therefore is that pure circular complementarity (requiring the perfect complementarity of opposites) cannot exist with the positive odd integral dimensions).
Because of a degree of breaking (of integral symmetry) in qualiative terms, such dimensions cannot represent the pure relationship of circular to linear understanding. In like manner the quantitative value of such zeta expressions cannot be conveyed through closed pi expressions.
Now a great deal of debate has continued as to whether such zeta values (for positive odd integer values of s > 1) are rational or irrational.
It has indeed been proved that - at least when s = 3 - the value is irrational.
Now there are good holistic mathematical reasons for believing that not alone is the value of such expressions always irrational (but necessarily of a transcendental nature).
If the value was in fact rational (for s > 3) this would imply a rational value in the prime product formula over the infinite range of terms. This would imply that each term bears a relationship to previous terms that can be expressed in a rational manner. However the very nature of prime numbers is that their individual quantitative nature is uniquely distinct from their overall holistic behaviour with respect to each other. Therefore no such rational formula could exist.
Therefore the convergent sum of terms inevitably entails a relationship involving both linear and circular notions (which directly implies that it is transcendental). And as we have seen what distinguishes the even from the odd dimesnions here, is that with the even we have a pure relationship of circular to linear (whereas with the odd this is not the case).
Once again the key assumption here (regarding the inherent nature of prime numbers) literally transcends conventional mathematical interpretation. Indeed it is the same assumption that ultimately establishes the axiomatic nature of the Riemann Hypothesis!
Monday, November 15, 2010
Holistic Mathematical Connections
As we have seen when s is a positive even integer, the value of the Euler Zeta function can be expressed in terms of an expression comprising (Pi^s) * k (where k represents a rational number).
Now what I am concerned with here is to give a holistic mathematical interpretation of the significance of this result.
To do this it is easiest to concentrate initially on the simplest - and best known - case where s = 2.
As we have seen here the value of the Zeta function = (Pi^2)/6
As stated repeatedly, in holistic mathematical terms, standard conventional interpretation is qualitatively of a linear nature (i.e. 1-dimensional).
What this means in effect is that the polarities of experience are inherently separated giving just one positive (unambiguous) direction.
So for example all understanding - including of course mathematical - entails a dynamic interaction of both external (objective) and internal (subjective) aspects.
However in mathematical interpretation these are clearly separated. So mathematical truth is given just one direction i.e. dimension as objective (with which psychological mental constructs are thereby assumed to statically correspond).
So linear logic is decidedly of the unambiguous either/or variety!
However corresponding to all other dimensions is a unique qualitative mode of interpretation and the simplest of these is 2-dimensional.
So with 2-dimensional interpretation, two polar directions are always given (which dynamically interact in a paradoxical manner).
So for example interpretation is not identified with either the external or internal pole of understanding (as separate) but rather with their mutual interdependence. This then leads to an alternative circular mode of both/and logic that is based on the complementarity (and ultimate identity) of polar opposites.
So the qualitative nature of 2-dimensional interpretation corresponds with the structural nature of the two roots of unity.
However whereas the two roots of 1, i.e. + 1 and - 1 are interpreted using either/or (linear) logic in quantitative terms, qualitatively they are interpreted using circular both/and circular logic.
However the use of 2 initially always requires the corresponding use 1-dimensional interpretation. For if we are not able to identify polar opposites as initially separate in experience, then we cannot hope they see how they are complementary!
Thus a key issue that thereby arises is the reconciliation of both the linear and circular modes of understanding.
Now in quantitative terms the constant pi relates to the relationship as between the (circular) circumference and its (linear) diameter.
Likewise in qualitative terms, for correct 2-dimensional interpretation we must reconcile both linear (either/or) and circular (both/and) interpretation.
Thus the very form of the quantitative result for the Euler Zeta function (where s = 2) has its corresponding qualitative holistic mathematical explanation (i.e. as the relationship of linear and circular understanding).
Though in quantitative terms, the Euler function here does indeed have a direct quantitative numerical result, indirectly it also contains a qualitative significance in the form of the result (entailing pi).
Now we will see later, when s takes on negative integer values, that the numerical results arising have but an indirect quantitative significance. Rather in direct terms, the numerical values that result can only correctly be given a qualitative interpretation!
Briefly the holistic mathematical interpretation for all other even integer values of s operates along the same lines. Just as the 2-dimensional case represents the simplest version of the complementarity of opposites, all higher dimensions point to more intricate arrangements of the same complementarity. This corresponds in quantitative terms with the fact that the s roots of 1 can also be arranged in a complementary manner.
In my own work, I have mainly concentrated on interpretation that corresponds to 2, 4 and 8 dimensions respectively (which I believe are the most important). However in principle the basic requirement regarding matching positive and negative directions (now taken in the complex plane) can be easily extended to all even integer values of s.
And once again this constitutes the holistic mathematical explanation as to why the form of all such zeta results can be expressed in the form of pi.
Now what I am concerned with here is to give a holistic mathematical interpretation of the significance of this result.
To do this it is easiest to concentrate initially on the simplest - and best known - case where s = 2.
As we have seen here the value of the Zeta function = (Pi^2)/6
As stated repeatedly, in holistic mathematical terms, standard conventional interpretation is qualitatively of a linear nature (i.e. 1-dimensional).
What this means in effect is that the polarities of experience are inherently separated giving just one positive (unambiguous) direction.
So for example all understanding - including of course mathematical - entails a dynamic interaction of both external (objective) and internal (subjective) aspects.
However in mathematical interpretation these are clearly separated. So mathematical truth is given just one direction i.e. dimension as objective (with which psychological mental constructs are thereby assumed to statically correspond).
So linear logic is decidedly of the unambiguous either/or variety!
However corresponding to all other dimensions is a unique qualitative mode of interpretation and the simplest of these is 2-dimensional.
So with 2-dimensional interpretation, two polar directions are always given (which dynamically interact in a paradoxical manner).
So for example interpretation is not identified with either the external or internal pole of understanding (as separate) but rather with their mutual interdependence. This then leads to an alternative circular mode of both/and logic that is based on the complementarity (and ultimate identity) of polar opposites.
So the qualitative nature of 2-dimensional interpretation corresponds with the structural nature of the two roots of unity.
However whereas the two roots of 1, i.e. + 1 and - 1 are interpreted using either/or (linear) logic in quantitative terms, qualitatively they are interpreted using circular both/and circular logic.
However the use of 2 initially always requires the corresponding use 1-dimensional interpretation. For if we are not able to identify polar opposites as initially separate in experience, then we cannot hope they see how they are complementary!
Thus a key issue that thereby arises is the reconciliation of both the linear and circular modes of understanding.
Now in quantitative terms the constant pi relates to the relationship as between the (circular) circumference and its (linear) diameter.
Likewise in qualitative terms, for correct 2-dimensional interpretation we must reconcile both linear (either/or) and circular (both/and) interpretation.
Thus the very form of the quantitative result for the Euler Zeta function (where s = 2) has its corresponding qualitative holistic mathematical explanation (i.e. as the relationship of linear and circular understanding).
Though in quantitative terms, the Euler function here does indeed have a direct quantitative numerical result, indirectly it also contains a qualitative significance in the form of the result (entailing pi).
Now we will see later, when s takes on negative integer values, that the numerical results arising have but an indirect quantitative significance. Rather in direct terms, the numerical values that result can only correctly be given a qualitative interpretation!
Briefly the holistic mathematical interpretation for all other even integer values of s operates along the same lines. Just as the 2-dimensional case represents the simplest version of the complementarity of opposites, all higher dimensions point to more intricate arrangements of the same complementarity. This corresponds in quantitative terms with the fact that the s roots of 1 can also be arranged in a complementary manner.
In my own work, I have mainly concentrated on interpretation that corresponds to 2, 4 and 8 dimensions respectively (which I believe are the most important). However in principle the basic requirement regarding matching positive and negative directions (now taken in the complex plane) can be easily extended to all even integer values of s.
And once again this constitutes the holistic mathematical explanation as to why the form of all such zeta results can be expressed in the form of pi.
Sunday, November 14, 2010
Thr Pi Connection
Euler also demonstrated another remarkable connection as between his Zeta function and the value of pi.
So whenever (representing the dimension) in the function is a positive even integer, then the resulting value can be expressed as pi (to the power of s) * by a rational no.
So in the simplest - and best known - case when s = 2,
∑[1/n^2] = ∏{p^2/(p^2 – 1)} = (pi^2)/6
One of the interesting implications of this result is that it provides another means of proving the infinitude of primes (i.e. in the accepted reduced nature of the infinite).
For if the the no. of primes was finite then the product formula (involving the primes) would ensure a rational value.
However because the actual answer is irrational (involving pi), then this implies that the no. of primes must be infinite.
However the deeper qualitative implications of this result are not properly appreciated.
Again no matter how many finite terms are included in the product formula, a rational result will ensue. Therefore the fact that the result is irrational, implies that an additional qualitative aspect of understanding is required.
Now some infinite series in the limit of an infinite process - again accepting the reduced conventional appreciation of the infinite - do result in a rational finite answer.
For example if we sum the series 1 + 1/2 + 1/4 + 1/8 +......,
the actual sum for any finite no. of terms will be rational. However the limiting value for an infinite no. of terms will also be rational (i.e. 2).
However one reason why this infinite series does not lead to an irrational value is the fact that each successive term can itself be expressed as a rational fraction of the previous term. So just as rational number can be expressed in decimal form with a consistent repeating sequence of digits, likewise if successive terms in a series are related in a similar manner (through the application of a consistent rational operation) then a rational limit will result.
However clearly in the case of series entailing the prime numbers, this is not in fact the case. Rather, as I have stated before the actual location of each prime number intimately depends on the overall holistic relationship of the primes to the natural numbers.
And, again the key point is that this holistic relationship is qualitatively of a different nature as it relates to the potential infinite nature of these numbers whereas rational interpretation relates properly to actual finite type considerations.
So once again the key limitation of Conventional Mathematics is that it can only attempt to deal with potential infinite notions - properly relating to circular paradoxical type appreciation - in a grossly reduced manner. Here they are treated as an extension of the finite (so as to become amenable to an unambiguous linear type logic).
As we know the constant pi (which is irrational and transcendent in nature) pertains quantitatively to the relationship of the (circular) circumference to its (linear) diameter.
Likewise in holistic mathematical terms, qualitative understanding (that is irrational and transcendental) pertains to the corresponding relationship as between (pure) circular and linear type interpretation.
This thereby implies that - correctly understood in an appropriate qualitative manner - interpretation of the Zeta function (for even values of s) implies the pure relationship as between linear and qualitative type notions that is expressed in an indirect rational manner.
In other words - when appropriately understood - we then realise that the very nature of prime numbers entails a pure relationship as between actual finite notions (in the precise identity of specific prime numbers)and potential infinite notions (in the general distribution of the primes among the natural number system).
However there is another fascinating connection in these pi expressions and the nature of the primes.
If we confine ourselves to the rational nos. by which the pi expressions must be multiplied, then the denominators of such expressions bear a very important relationship to the primes.
So, when s = 2n (where n = 1, 2, 3, 4,...)
then, where 2n represents an integral power of 2, the denominator of the rational number part will represent a product of all prime numbers (in various combinations) up to 2n + 1 and only these primes.
In all other cases, the denominator of the rational part will represent the product of all prime numbers - in varied combinations - from 3 to 2n + 1(and only these primes).
For example when s = 2n = 4, this represents an integral power of 2. The denominator of the rational part = 90 = 2 * 3^2 * 5 which represents all primes from 2 to 2n + 1(i.e. 5).
Then when s = 2n = 6, this does not represent an integral power of 2.
The denominator of the rational part = 945 = 3^3 * 5 * 7 which represents all primes from 3 to 2n + 1 (i.e. 7).
I will just give one more example to illustrate
Wne s = 2n (i.e. n = 8) = 16, s can again be expressed as 2 (raised to a positive integral power) i.e. 2^4.
The denominator here of of pi^16 = 325641566250.
And 325641566250 = 2 * 3^7 * 5^4 * 7^2 * 11 * 13 * 17. So we can see here how all the prime numbers from 2 to 2n + 1 i.e. 17, are included as factors in the denominator (and only these primes).
So whenever (representing the dimension) in the function is a positive even integer, then the resulting value can be expressed as pi (to the power of s) * by a rational no.
So in the simplest - and best known - case when s = 2,
∑[1/n^2] = ∏{p^2/(p^2 – 1)} = (pi^2)/6
One of the interesting implications of this result is that it provides another means of proving the infinitude of primes (i.e. in the accepted reduced nature of the infinite).
For if the the no. of primes was finite then the product formula (involving the primes) would ensure a rational value.
However because the actual answer is irrational (involving pi), then this implies that the no. of primes must be infinite.
However the deeper qualitative implications of this result are not properly appreciated.
Again no matter how many finite terms are included in the product formula, a rational result will ensue. Therefore the fact that the result is irrational, implies that an additional qualitative aspect of understanding is required.
Now some infinite series in the limit of an infinite process - again accepting the reduced conventional appreciation of the infinite - do result in a rational finite answer.
For example if we sum the series 1 + 1/2 + 1/4 + 1/8 +......,
the actual sum for any finite no. of terms will be rational. However the limiting value for an infinite no. of terms will also be rational (i.e. 2).
However one reason why this infinite series does not lead to an irrational value is the fact that each successive term can itself be expressed as a rational fraction of the previous term. So just as rational number can be expressed in decimal form with a consistent repeating sequence of digits, likewise if successive terms in a series are related in a similar manner (through the application of a consistent rational operation) then a rational limit will result.
However clearly in the case of series entailing the prime numbers, this is not in fact the case. Rather, as I have stated before the actual location of each prime number intimately depends on the overall holistic relationship of the primes to the natural numbers.
And, again the key point is that this holistic relationship is qualitatively of a different nature as it relates to the potential infinite nature of these numbers whereas rational interpretation relates properly to actual finite type considerations.
So once again the key limitation of Conventional Mathematics is that it can only attempt to deal with potential infinite notions - properly relating to circular paradoxical type appreciation - in a grossly reduced manner. Here they are treated as an extension of the finite (so as to become amenable to an unambiguous linear type logic).
As we know the constant pi (which is irrational and transcendent in nature) pertains quantitatively to the relationship of the (circular) circumference to its (linear) diameter.
Likewise in holistic mathematical terms, qualitative understanding (that is irrational and transcendental) pertains to the corresponding relationship as between (pure) circular and linear type interpretation.
This thereby implies that - correctly understood in an appropriate qualitative manner - interpretation of the Zeta function (for even values of s) implies the pure relationship as between linear and qualitative type notions that is expressed in an indirect rational manner.
In other words - when appropriately understood - we then realise that the very nature of prime numbers entails a pure relationship as between actual finite notions (in the precise identity of specific prime numbers)and potential infinite notions (in the general distribution of the primes among the natural number system).
However there is another fascinating connection in these pi expressions and the nature of the primes.
If we confine ourselves to the rational nos. by which the pi expressions must be multiplied, then the denominators of such expressions bear a very important relationship to the primes.
So, when s = 2n (where n = 1, 2, 3, 4,...)
then, where 2n represents an integral power of 2, the denominator of the rational number part will represent a product of all prime numbers (in various combinations) up to 2n + 1 and only these primes.
In all other cases, the denominator of the rational part will represent the product of all prime numbers - in varied combinations - from 3 to 2n + 1(and only these primes).
For example when s = 2n = 4, this represents an integral power of 2. The denominator of the rational part = 90 = 2 * 3^2 * 5 which represents all primes from 2 to 2n + 1(i.e. 5).
Then when s = 2n = 6, this does not represent an integral power of 2.
The denominator of the rational part = 945 = 3^3 * 5 * 7 which represents all primes from 3 to 2n + 1 (i.e. 7).
I will just give one more example to illustrate
Wne s = 2n (i.e. n = 8) = 16, s can again be expressed as 2 (raised to a positive integral power) i.e. 2^4.
The denominator here of of pi^16 = 325641566250.
And 325641566250 = 2 * 3^7 * 5^4 * 7^2 * 11 * 13 * 17. So we can see here how all the prime numbers from 2 to 2n + 1 i.e. 17, are included as factors in the denominator (and only these primes).
Wednesday, November 10, 2010
The Harmonic Series Again
Euler made great advances with respect to better understanding of the prime numbers.
In what must constitute one of the most truly memorable contributions to Mathematics he was able to demonstrate an intimate connection as between the natural numbers on the one hand and the prime numbers on the other.
Now once again the zeta function is defined as ∑1/n^s (for n = 1 → ∞)
The harmonic series results from setting s = 1
Thus ∑1/n = 1 + 1/2 + 1/3 + 1/4 +...... which is divergent.
Now the Euler zeta function is defined for values of s (> 1) with
∑[1/n^s] = ∏{p^s/(p^s – 1)} where again p ranging from 2 → ∞
Therefore when s = 2
1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 +... = {2^2/[(2^2) - 1]}*{3^2/[(3^2) - 1]}*{[5^2/([(5^2) - 1]}
So
1/1 + 1/4 + 1/9 + 1/16 +..... = 4/3 * 9/8 * 25/24 *.....
So we have here an intimate connection as between the natural numbers on the one hand (connected through addition)and the prime numbers (connected through multiplication).
Furthermore a unique dimensional connection exists between the two for all numbers where s > 1.
The question then arises as to whether a unique connection also exists in the vitally important case where s = 1.
Clearly in this case both the natural number sum series and the prime number product series will both diverge.
However if we confine ourselves to a limited finite number of terms, then an interesting connection does indeed exist.
So where the harmonic series is summed to a finite number of terms (n)
∑1/n → ∏{p + 1)/p}
So
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → 3/2 * 4/3 * 6/5 *.......
So as we obtain the sum of the first n terms of the harmonic series on one side, we approximate the corresponding product of all p terms up to n on the other side.
This relationship can also be written in another interesting way!
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → (1 + 1/2) * (1 + 1/3) * (1 + 1/5) *.....
Also when the no. of terms on the LHS = n, the corresponding no. of terms on the RHS approximates n/log n.
Thus on the LHS we have the sum of terms entailing the reciprocals of the natural nos (starting with 1); on the RHS we then have the product of terms entailing the reciprocals of the primes (in each case added to 1).
Now what this formulation clearly demonstrates is the intimate relation as between addition and multiplication in the connection of the primes to the natural number system.
However though - correctly - understood, multiplication involves both a quantitative and qualitative transformation with respect to number interpretation, Conventional Mathematics is based on the merely quantitative aspect. Therefore it inevitably reduces the qualitative aspect to the quantitative.
This quite simply then constitutes not alone the key barrier to solving the Riemann Hypothesis but in fact likewise the key barrier to its proper interpretation.
For ultimately the Riemann Hypothesis relates to the reconciliation of both the quantitative and qualitative aspects of Mathematics.
So again putting it bluntly, not alone can the Riemann Hypothesis not be solved in a conventional mathematical fashion, it is not even capable of being properly understood in this manner!
In what must constitute one of the most truly memorable contributions to Mathematics he was able to demonstrate an intimate connection as between the natural numbers on the one hand and the prime numbers on the other.
Now once again the zeta function is defined as ∑1/n^s (for n = 1 → ∞)
The harmonic series results from setting s = 1
Thus ∑1/n = 1 + 1/2 + 1/3 + 1/4 +...... which is divergent.
Now the Euler zeta function is defined for values of s (> 1) with
∑[1/n^s] = ∏{p^s/(p^s – 1)} where again p ranging from 2 → ∞
Therefore when s = 2
1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 +... = {2^2/[(2^2) - 1]}*{3^2/[(3^2) - 1]}*{[5^2/([(5^2) - 1]}
So
1/1 + 1/4 + 1/9 + 1/16 +..... = 4/3 * 9/8 * 25/24 *.....
So we have here an intimate connection as between the natural numbers on the one hand (connected through addition)and the prime numbers (connected through multiplication).
Furthermore a unique dimensional connection exists between the two for all numbers where s > 1.
The question then arises as to whether a unique connection also exists in the vitally important case where s = 1.
Clearly in this case both the natural number sum series and the prime number product series will both diverge.
However if we confine ourselves to a limited finite number of terms, then an interesting connection does indeed exist.
So where the harmonic series is summed to a finite number of terms (n)
∑1/n → ∏{p + 1)/p}
So
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → 3/2 * 4/3 * 6/5 *.......
So as we obtain the sum of the first n terms of the harmonic series on one side, we approximate the corresponding product of all p terms up to n on the other side.
This relationship can also be written in another interesting way!
1 + 1/2 + 1/3 + 1/4 + 1/5 +.... → (1 + 1/2) * (1 + 1/3) * (1 + 1/5) *.....
Also when the no. of terms on the LHS = n, the corresponding no. of terms on the RHS approximates n/log n.
Thus on the LHS we have the sum of terms entailing the reciprocals of the natural nos (starting with 1); on the RHS we then have the product of terms entailing the reciprocals of the primes (in each case added to 1).
Now what this formulation clearly demonstrates is the intimate relation as between addition and multiplication in the connection of the primes to the natural number system.
However though - correctly - understood, multiplication involves both a quantitative and qualitative transformation with respect to number interpretation, Conventional Mathematics is based on the merely quantitative aspect. Therefore it inevitably reduces the qualitative aspect to the quantitative.
This quite simply then constitutes not alone the key barrier to solving the Riemann Hypothesis but in fact likewise the key barrier to its proper interpretation.
For ultimately the Riemann Hypothesis relates to the reconciliation of both the quantitative and qualitative aspects of Mathematics.
So again putting it bluntly, not alone can the Riemann Hypothesis not be solved in a conventional mathematical fashion, it is not even capable of being properly understood in this manner!
Sunday, November 7, 2010
Zoning in on the Primes
We have seen how important the harmonic series is with respect to the nature of prime numbers.
Once again - as an initial approximation - the sum of the first n terms of the series provides an estimate of the gap as between primes (in the region of n).
Secondly the reciprocal of n provides a good estimate of the change in the gap as between primes in the region of n.
So as we have seen as we move for example from 1,000,000 to 1,000,001 the change in the gap would be close to 1/1,000,000.
Now the extent to which the harmonic series falls short in terms of predicting the actual gap between primes is explained by Euler's Constant (= .5772...).
More accurately the gap between primes (in the region of n) is given by log n.
And as the harmonic series approximates log n + λ (where λ = Euler's Constant) then using the harmonic series as a prediction will overestimate the gap by .5772.
However remarkably the value of Euler's Constant is itself related to all other positive integer values of the Zeta function.
So λ = ζ(2)/2 - ζ(3)/3 + ζ(4)/4 - ζ(5)/5 +.....
Therefore when we confine ourselves to finite values of n, the value of log n (representing the average gap between primes) can be expressed in terms of the positive integer values for s of the Zeta Function
i.e. ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -......
We can likewise modify the initial estimate of the change in average spread between primes (in the region of n).
So now using all of these Zeta values a better approximation is given by
1/n - 1/{2*(n^2)} + 1/{3*(n^3)} - 1/{4*(n^4)} +.....
We could also express the above as the value of log (n + 1) - log n
However as n becomes very large the modifications to the initial estimate of the change in spread (i.e. 1/n) are so small as to be negligible. Therefore 1/n serves as a particularly good estimate!
We can also express log n in another way as the sum of Zeta function values for positive integer values of s > 1.
Therefore the average gap between primes is given as
ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +.....
So n/log n representing the frequency of primes (among the first n natural numbers)
→ n/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
→ n/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
As we have seen n in fact represents the value ζ(0) where the sum of terms is taken over a finite limited range.
Therefore
n/log n → ζ(0)/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
n/log n → ζ(0)/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
Once again - as an initial approximation - the sum of the first n terms of the series provides an estimate of the gap as between primes (in the region of n).
Secondly the reciprocal of n provides a good estimate of the change in the gap as between primes in the region of n.
So as we have seen as we move for example from 1,000,000 to 1,000,001 the change in the gap would be close to 1/1,000,000.
Now the extent to which the harmonic series falls short in terms of predicting the actual gap between primes is explained by Euler's Constant (= .5772...).
More accurately the gap between primes (in the region of n) is given by log n.
And as the harmonic series approximates log n + λ (where λ = Euler's Constant) then using the harmonic series as a prediction will overestimate the gap by .5772.
However remarkably the value of Euler's Constant is itself related to all other positive integer values of the Zeta function.
So λ = ζ(2)/2 - ζ(3)/3 + ζ(4)/4 - ζ(5)/5 +.....
Therefore when we confine ourselves to finite values of n, the value of log n (representing the average gap between primes) can be expressed in terms of the positive integer values for s of the Zeta Function
i.e. ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -......
We can likewise modify the initial estimate of the change in average spread between primes (in the region of n).
So now using all of these Zeta values a better approximation is given by
1/n - 1/{2*(n^2)} + 1/{3*(n^3)} - 1/{4*(n^4)} +.....
We could also express the above as the value of log (n + 1) - log n
However as n becomes very large the modifications to the initial estimate of the change in spread (i.e. 1/n) are so small as to be negligible. Therefore 1/n serves as a particularly good estimate!
We can also express log n in another way as the sum of Zeta function values for positive integer values of s > 1.
Therefore the average gap between primes is given as
ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +.....
So n/log n representing the frequency of primes (among the first n natural numbers)
→ n/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
→ n/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
As we have seen n in fact represents the value ζ(0) where the sum of terms is taken over a finite limited range.
Therefore
n/log n → ζ(0)/{ζ(1)/1 - ζ(2)/2 + ζ(3)/3 - ζ(4)/4 + ζ(5)/5 -…..}
or alternatively,
n/log n → ζ(0)/{ζ(2)/2 + ζ(3)/3 + ζ(4)/4 + ζ(5)/5 +…..}
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