In my last blog entry, I defined the simple Functional Equation for the Zeta 2 Function i.e.
ζ2(1/s) = 1 – ζ2(s)
This implies that ζ2(1/s) = ζ2(s) = 1/2 (when both are equal).
Of course a similar type of condition applies to the Zeta 1 Function.
In this case ζ1(s) = ζ1(1 – s) = 0, when real part of s = 1/2.
Now by appreciating the precise significance of this equality in the context of the Zeta 2 Function, we can learn a great deal more regarding the true significance of the Riemann Hypothesis (where all non-trivial zeros are postulated to lie on the imaginary line through 1/2).
Indeed the well-known Eta 1 Function (as the alternating counterpart to the Zeta 1 Function where s = 0 ) can be defined where s = 0 as,
1 – 1 + 1 – 1 + ....... .
Now this in turn corresponds to the Zeta 2 Function
i.e. ζ2(s) = 1 + s1 + s2 + s3 +... = 1/(1 – s) where s = – 1
So ζ2(– 1) = 1 – 1 + 1 – 1 + ....... = 1/2.
The key significance of this value for s (i.e. – 1) is that it represents the simplest of all the Zeta 2 non-trivial zeros.
These non-trivial zeros again correspond to the n – 1 roots of 1, where n is prime (i.e. all roots other than the default root = 1).
The n – 1 roots of 1 correspond to the n – 1 solutions of the finite equation,
1 + s1 + s2 + s3 +.... + sn – 1 = 0.
Then as explained in the previous blog entry, when we keep repeating (with regular cycles of n terms) these n – 1 roots will likewise act as solutions to the infinite Zeta 2 equation
i.e. ζ2(s) = = 1 + s1 + s2 + s3 +... = 0.
However when this is given a linear expression i.e. where the series can be increased by 1 one term at a time (rather than a regular cycle of terms) it will acquire a finite value.
We can illustrate this most easily with reference to the first of the non- trivial zeros i.e. – 1.
This arises as the 2nd root of 1 and represents the solution to the finite equation
1 + s = 0.
Then when we repeat with regular cycles of 2, – 1 is equally the solution to the infinite Zeta 2 expression,
1 + s1 + s2 + s3 +... = 0;
However from the formula, the value of 1 + s1 + s2 + s3 +... = 1/(1 – s).
Thus when s = – 1, 1/(1 - s) = 1/2!
Now this can be easily explained as an average of circular and linear type interpretations of the series.
Once again for s = – 1 ,
ζ2(– 1) = 1 – 1 + 1 – 1 + .......
If we take these terms in regular cycles of 2 (with complementary pairing of positive and negative values for 1),
ζ2(– 1) = 0.
However if we allow linear progression (with the series increasing by 1 term at a time) the value will keep alternating equally as between 0 and 1.
Thus the value of 1/2 in the formula relates to this linear interpretation of the alternating infinite series whose average value = 1/2.
So once again for the circular grouping of complementary even number of terms
ζ2(– 1) = 0.
However then in linear terms for an odd number of terms ζ2(– 1) = 1.
Thus the significance of 1/2 in this context is that it represents the mid-point in quantitative (analytic) terms as between circular ( = 0) and corresponding linear ( = 1) interpretation.
More crucially, from a corresponding holistic perspective it represents a perfect balance as between quantitative (cardinal) and qualitative (ordinal) type interpretation with respect to the number system.
And once again this is the significance of 1/2 in the context of the Riemann Hypothesis (with respect to the Zeta 1 Function).
It simply represents the condition for the ultimate identity of both the cardinal and ordinal aspects of the number system!
However this can only be properly understood in the context of an inherently dynamic mathematical approach (that gives equal emphasis to its Type 1 (cardinal) and Type 2 (ordinal) aspects.
Conventional Mathematics is quite unsuited for appreciation of this central issue as - by its very nature - it reduces qualitative to mere quantitative interpretation!
So we have illustrated the significance of 1/2 with respect to the Zeta 2 Function, in the simplest situation where the finite equation,
1 + s1 + s2 + s3 +.... + sn – 1 = 0, has just one non-trivial solution (as the 2nd root of 1). However all other situations can then be seen as extensions of this simplest case in a crucial respect i.e. that the sum of the n – 1 roots of 1 (excepting the default root of 1) = – 1.
Thus the circular interdependence of these n – 1 roots is demonstrated through their group sum = – 1.
So in this manner, 1/2 remains the value that ensures the equal emphasis - in both quantitative and qualitative terms - as between number (both with respect to linear interdependence and circular interdependence).
It must be remembered that this simplest case of the Zeta 2 series where s = – 1,
i.e. 1 – 1 + 1 – 1 + ....... provides an easy means for obtaining a numerical value for s = 0 (with respect to the Zeta 1 Function).
Indeed in an earlier blog entry, I showed in this manner how to obtain the corresponding Zeta 1 values for ζ1(0), ζ1(– 1), ζ1(– 2) and ζ1(– 3).
Just one more important point at this stage!
As we know with respect to the Zeta 1,
ζ1(0) = 1 + 1 + 1 + 1 +..... = – 1/2.
Then with respect to the Zeta 2 when s = 1,
ζ2(1) = 1 + 1 + 1 + 1 +.....
Though this from a merely quantitative perspective might seem to represent exactly the same expression as ζ1(0), it remains undefined with respect to the Zeta 2 Function!
Now the key explanation for this seeming anomaly is that the dimensional value to which 1 is raised = 0, with respect to the Zeta 1, whereas the base value in the case of the Zeta 2 = 1.
So we now can see that the crucial condition for both the Zeta 1 and Zeta 2 Functions remaining undefined is that the dimensional and base values in both cases respectively = 1.
And as the linear type interpretation that characterises Conventional Mathematics is precisely defined in qualitative terms by the dimensional number of 1, and the circular type interpretation that characterises Holistic Mathematics by the base number of 1, this means that neither the Zeta 1 nor Zeta 2 Functions can be properly appreciated in this manner. In other words proper interpretation entails the dynamic relationship of both types of meanings!
The crucial point is that - properly understood - all mathematical understanding is inherently dynamic (with two-way interaction as between both its quantitative and qualitative aspects).
When seen in this light the existing problem with what we misleadingly identify as Mathematics, could not be more fundamental (with the true nature of the Riemann Hypothesis remaining undefined from this limited perspective).
Friday, May 24, 2013
Wednesday, May 22, 2013
Functional Equation for Zeta 2 Expression
As is well known, Riemann defined an important equation with respect to the Riemann Zeta Function whereby for any value of s with respect to ζ (s) the corresponding value of ζ (1 – s) can be obtained from the Functional Equation.
So for example when s = 2, ζ (2) = π2/6, then through the Functional Equation
ζ ( – 1) = – 1/12.
As we have seen, there are in fact two complementary aspects to the Zeta Function, with the Riemann Zeta Function relating to Zeta 1, which - in the notation that I adopt - is ζ1(s).
Therefore in this context, Riemann's Functional Equation establishes a relationship as between ζ1(s) and ζ1(1 – s).
Now for s > 1, the infinite sum of terms of ζ1(s) results in a finite value value that is meaningful in terms of linear type interpretation.
ζ1(s) = 1/1s + 1/2s + 1/3s + 1/4s + ....
So for example,
ζ1(2) = 1 + 1/4 + 1/9 + 1/16 + .... = π2/6 = 1.644934...
However clearly the corresponding value through the Functional Equation for
ζ ( – 1) = – 1/12, does not conform to linear type interpretation.
Thus from this linear perspective,
1/1– 1 + 1/2– 1 + 1/3– 1+ 1/4– 1 + .... = 1 + 2 + 3 + 4 + .... which diverges to infinity.
However when we interpret this expression from a circular - rather than linear - perspective it can then be demonstrated how this series obtains a meaningful finite value (i.e. – 1/12).
Now switching from the quantitative (cardinal) to the qualitative (ordinal) interpretation of number, implies a corresponding switch from linear to circular type interpretation i.e. from the Type 1 to the Type 2 aspect of the number system.
So when properly interpreted in dynamic interactive terms, the Riemann Functional Equation establishes the appropriate relationship (with respect to the Zeta 1 Function) as between the Type 1 (cardinal) and Type 2 (ordinal) aspects of the number system.
Then when ζ1(s) = ζ1(1 – s) = 0, the real part of s = .5!
Thus the requirement that all the non-trivial zeros of the Zeta 1 Function lie on a line through .5, points to the key fact that it is at this point (on the real axis) and only this point that both the Type 1 (cardinal) and Type 2 (ordinal) aspects of the number system are identical.
Thus the famous Riemann Hypothesis can be seen thereby as the necessary condition for the ultimate identification of both the quantitative (Type 1) and qualitative (Type 2) aspects of the number system.
However we can equally approach this mutual identification of the quantitative and qualitative aspects of the number system from the perspective of the Zeta 2 Function (which in some important respects is easier to understand than Zeta 1).
So once again the Zeta 2 Function is initially defined with respect to a finite series.
So ζ2(s) = 1 + s1 + s2 + .....+ sn – 1
However this can then be extended to an infinite sequence of terms in two ways.
(i) in the accepted linear manner where we keep adding on single additional terms in a quantitative manner.
(ii) in the corresponding circular manner where we keep extending the series through taking regular groups of n terms while using a modular (clock) arithmetic as the appropriate means of interpreting the value of the series.
Now when s > 1, the infinite series for ζ2(s) will converge to a finite value according to the standard linear method of interpretation.
For example when s = 1/2,
ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ..... = 2.
However in the case of Zeta 2, if the series converges for s, then it will diverge for 1/s in linear quantitative terms.
Therefore when now s = 2,
ζ2(2) = 1 + 2 + 4 + 8 + .... which diverges to infinity in linear terms.
However if we now view this relationship in a circular manner, the series will indeed converge.
So for example 2 in this context would be represented 2 by modular or clock arithmetic (where 2 would be represented as 0).
So in terms of one cycle,
ζ2(2) = 1 + 2 = 1 (in modular arithmetic terms).
Thus when we extend this to an infinite sequence of terms (through orderly groups of 2) the value of the series will remain unchanged as 1!
Now interestingly,
1/(1 – s) = 1 + s1 + s2 + s3 + ......
Now, when – 1 < s < 1, this takes on a finite value in the normal linear quantitative) interpretation of a series.
However when s > 1 or < – 1, the infinite series will diverge from this same linear perspective.
However when we now apply the alternative circular perspective (implying modular arithmetic) the infinite series acquires a finite result for these values.
For example when s = 2, the LHS of the equation = 1/(1 – 2) = – 1.
Interestingly the value obtained using clock arithmetic = 1.
However whereas the LHS value is defined in linear terms, the corresponding RHS value is defined in a complementary circular manner.
Now, in the very dynamics of understanding one moves from linear (quantitative) to circular (qualitative) understanding through the negation of what is linear.
Therefore to express the circular result (on the RHS) in an appropriate linear manner we must - literally - negate this value to obtain – 1.
Then when s = 3, in modular (clock terms) again the sum of the infinite series (this time for groupings of 3 terms) = 1.
However we have now two non-trivial values (in the opening finite sequence of terms i.e. 1 + s1 + s2 . So the average value = 1/2 which when expressed in appropriate linear terms = - 1/2.
And again the LHS = 1/(1 – 3) = – 1/2.
We can now define the simple - though remarkable - Functional Equation for the Zeta 2 expression as follows:
ζ2(1/s) = 1 – ζ2(s) which holds for all values of s except where s = 1.
Here we have an important similarity with the Zeta 1 Function which likewise is defined for all values on the complex plane except s = 1.
The reason for this is again striking. Quite obviously when s = 1, no distinction can be made as between ζ2(s) and ζ2(1/s). So we lack the means therefore of establishing a distinctive relationship as between the linear (quantitative) and circular (qualitative) explanations of values.
So once again - when appropriately interpreted in dynamic interactive terms, the Zeta 2 Functional Equation is revealed as an expression that establishes intimate connections as between both the quantitative and qualitative type interpretation of number.
There is another striking aspect worth noting.
Clearly in multiplicative terms s * (1/s) = 1.
Well, remarkably the Zeta 2 Functional Equation expresses the corresponding combination of Functions (based on s) in terms of addition.
Thus ζ2(s) + ζ2(1/s) = 1.
So for example when s = 2, ζ (2) = π2/6, then through the Functional Equation
ζ ( – 1) = – 1/12.
As we have seen, there are in fact two complementary aspects to the Zeta Function, with the Riemann Zeta Function relating to Zeta 1, which - in the notation that I adopt - is ζ1(s).
Therefore in this context, Riemann's Functional Equation establishes a relationship as between ζ1(s) and ζ1(1 – s).
Now for s > 1, the infinite sum of terms of ζ1(s) results in a finite value value that is meaningful in terms of linear type interpretation.
ζ1(s) = 1/1s + 1/2s + 1/3s + 1/4s + ....
So for example,
ζ1(2) = 1 + 1/4 + 1/9 + 1/16 + .... = π2/6 = 1.644934...
However clearly the corresponding value through the Functional Equation for
ζ ( – 1) = – 1/12, does not conform to linear type interpretation.
Thus from this linear perspective,
1/1– 1 + 1/2– 1 + 1/3– 1+ 1/4– 1 + .... = 1 + 2 + 3 + 4 + .... which diverges to infinity.
However when we interpret this expression from a circular - rather than linear - perspective it can then be demonstrated how this series obtains a meaningful finite value (i.e. – 1/12).
Now switching from the quantitative (cardinal) to the qualitative (ordinal) interpretation of number, implies a corresponding switch from linear to circular type interpretation i.e. from the Type 1 to the Type 2 aspect of the number system.
So when properly interpreted in dynamic interactive terms, the Riemann Functional Equation establishes the appropriate relationship (with respect to the Zeta 1 Function) as between the Type 1 (cardinal) and Type 2 (ordinal) aspects of the number system.
Then when ζ1(s) = ζ1(1 – s) = 0, the real part of s = .5!
Thus the requirement that all the non-trivial zeros of the Zeta 1 Function lie on a line through .5, points to the key fact that it is at this point (on the real axis) and only this point that both the Type 1 (cardinal) and Type 2 (ordinal) aspects of the number system are identical.
Thus the famous Riemann Hypothesis can be seen thereby as the necessary condition for the ultimate identification of both the quantitative (Type 1) and qualitative (Type 2) aspects of the number system.
However we can equally approach this mutual identification of the quantitative and qualitative aspects of the number system from the perspective of the Zeta 2 Function (which in some important respects is easier to understand than Zeta 1).
So once again the Zeta 2 Function is initially defined with respect to a finite series.
So ζ2(s) = 1 + s1 + s2 + .....+ sn – 1
However this can then be extended to an infinite sequence of terms in two ways.
(i) in the accepted linear manner where we keep adding on single additional terms in a quantitative manner.
(ii) in the corresponding circular manner where we keep extending the series through taking regular groups of n terms while using a modular (clock) arithmetic as the appropriate means of interpreting the value of the series.
Now when s > 1, the infinite series for ζ2(s) will converge to a finite value according to the standard linear method of interpretation.
For example when s = 1/2,
ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ..... = 2.
However in the case of Zeta 2, if the series converges for s, then it will diverge for 1/s in linear quantitative terms.
Therefore when now s = 2,
ζ2(2) = 1 + 2 + 4 + 8 + .... which diverges to infinity in linear terms.
However if we now view this relationship in a circular manner, the series will indeed converge.
So for example 2 in this context would be represented 2 by modular or clock arithmetic (where 2 would be represented as 0).
So in terms of one cycle,
ζ2(2) = 1 + 2 = 1 (in modular arithmetic terms).
Thus when we extend this to an infinite sequence of terms (through orderly groups of 2) the value of the series will remain unchanged as 1!
Now interestingly,
1/(1 – s) = 1 + s1 + s2 + s3 + ......
Now, when – 1 < s < 1, this takes on a finite value in the normal linear quantitative) interpretation of a series.
However when s > 1 or < – 1, the infinite series will diverge from this same linear perspective.
However when we now apply the alternative circular perspective (implying modular arithmetic) the infinite series acquires a finite result for these values.
For example when s = 2, the LHS of the equation = 1/(1 – 2) = – 1.
Interestingly the value obtained using clock arithmetic = 1.
However whereas the LHS value is defined in linear terms, the corresponding RHS value is defined in a complementary circular manner.
Now, in the very dynamics of understanding one moves from linear (quantitative) to circular (qualitative) understanding through the negation of what is linear.
Therefore to express the circular result (on the RHS) in an appropriate linear manner we must - literally - negate this value to obtain – 1.
Then when s = 3, in modular (clock terms) again the sum of the infinite series (this time for groupings of 3 terms) = 1.
However we have now two non-trivial values (in the opening finite sequence of terms i.e. 1 + s1 + s2 . So the average value = 1/2 which when expressed in appropriate linear terms = - 1/2.
And again the LHS = 1/(1 – 3) = – 1/2.
We can now define the simple - though remarkable - Functional Equation for the Zeta 2 expression as follows:
ζ2(1/s) = 1 – ζ2(s) which holds for all values of s except where s = 1.
Here we have an important similarity with the Zeta 1 Function which likewise is defined for all values on the complex plane except s = 1.
The reason for this is again striking. Quite obviously when s = 1, no distinction can be made as between ζ2(s) and ζ2(1/s). So we lack the means therefore of establishing a distinctive relationship as between the linear (quantitative) and circular (qualitative) explanations of values.
So once again - when appropriately interpreted in dynamic interactive terms, the Zeta 2 Functional Equation is revealed as an expression that establishes intimate connections as between both the quantitative and qualitative type interpretation of number.
There is another striking aspect worth noting.
Clearly in multiplicative terms s * (1/s) = 1.
Well, remarkably the Zeta 2 Functional Equation expresses the corresponding combination of Functions (based on s) in terms of addition.
Thus ζ2(s) + ζ2(1/s) = 1.
Thursday, May 16, 2013
No Longer Fit for Purpose!
The most fundamental distinction as between quantitative and qualitative notions in Mathematics pertains to the relationship as between finite and infinite!
Once again, to properly understand this relationship, finite and infinite must be defined in dynamic interactive terms.
So when the finite aspect such as a number is interpreted in a quantitative manner, then the infinite aspect is thereby of a – relatively – qualitative nature. In our understanding of number both inevitably interact, with the finite aspect directly conveyed through rational and the infinite aspect through intuition respectively. The finite is of an actual localised where the infinite – by contrast – is of a potential holistic nature.
Thus to form knowledge of the number “2” for example, we psychologically form an individual perception of this number (as finite) while equally relating it to the universal concept of number which – strictly – is of a potential nature i.e. as potentially applying to all specific numbers.
So both finite and infinite notions are necessarily involved in all number experience from a dynamic interactive perspective.
However in a reverse complementary fashion, the finite likewise has an infinite aspect and the infinite a finite aspect respectively.
This in fact is what distinguishes the ordinal from the cardinal notion of number!
So we start by giving a finite number such as “2” a cardinal meaning as a distinct individual entity with an unambiguous localised existence on the number scale.
However when we switch to the corresponding ordinal notion of 2, as 2nd, this number has no strict meaning in the absence of a more general holistic context (involving its relationship with other numbers).
So when we have just two members of a group, the notion of 2nd can be given an unambiguous identity in this restricted context. However its meaning is merely relative as 2nd in the context of a group of 3 is distinct from 2nd in the context of 2!. And as we can define 2 in relation to any sized group, the very meaning of 2nd is therefore potentially unlimited (as we can never exhaust the number system in finite terms).
Thus we started by attempting to interpret the number “2” as absolutely independent in a cardinal manner.
However when we switch to the ordinal notion of “2” we come to realise that its very meaning implies interdependence with other numbers in a merely relative fashion (which is potentially of an infinite nature).
Because cardinal and ordinal notions are necessarily related, this implies that properly speaking the cardinal notion of “2” is independent in a – merely – relative sense.
So in this dynamic context, the cardinal notion of 2 enjoys an independent and the ordinal notion of 2 an interdependent identity in relative terms!
Therefore we see here how each individual number possesses aspects that are quantitative (independent) and qualitative (interdependent) with respect to each other.
However this necessarily applies likewise to the number system as a whole.
Once again we are accustomed to think of the number system in an absolute independent fashion.
So from this conventional perspective, the prime numbers are the building blocks of this system, with every natural number (except 1) representing a unique combination of prime factors.
However the very notion of combining primes necessarily implies a relationship of interdependence. So for example the number 30 can be uniquely expressed as the product of 2 * 3 * 5.
Therefore, proper understanding of this unique combination entails a qualitative - as well as quantitative - aspect of interpretation!
Thus, once again when correctly understood, the number system as a whole possesses aspects that are relatively independent and interdependent with each other.
So from one perspective, the prime numbers are relatively independent in this sense (i.e. distinct from other composite natural numbers).
However they equally are relatively interdependent in the nature of their overall behaviour with respect to the number system.
Thus, once again when correctly understood, the number system as a whole possesses aspects that are relatively independent and interdependent with each other.
So from one perspective, the prime numbers are relatively independent in this sense (i.e. distinct from other composite natural numbers).
However they equally are relatively interdependent in the nature of their overall behaviour with respect to the number system.
So just as each individual number has both quantitative (cardinal) and qualitative (ordinal) aspects that are relatively independent and interdependent with respect to each other, the number system as a whole likewise has both quantitative and qualitative aspects that are relatively independent and interdependent with respect to each other.
Thus the prime numbers, as relatively separate entities, enjoy in cardinal terms an independent quantitative existence (as the building blocks of the number system), However, equally the prime numbers, in their unique relationship with each other (as composite natural numbers) enjoy in ordinal terms an interdependent qualitative existence (where each individual is inseparable from the holistic set of factors comprising these numbers).
Therefore, once again in dynamic terms, number independence implies a localised quantitative existence (without reference to an overall holistic context); number interdependence - by contrast - implies a holistic qualitative existence (where the interpretation of each individual number is inseparable from the overall holistic context to which its is related).
And this relationship as between quantitative and qualitative (and independence and interdependence) is mediated in a complementary bi-directional manner as between the prime and natural numbers.
Again from the Type 1 perspective, the prime numbers reveal themselves as the independent building blocks of the natural number system in cardinal (quantitative) terms.
However from the Type 2 perspective each prime number in turn is revealed as internally composed of an interdependent group of natural numbers in ordinal (qualitative) terms. Thus the prime number group of 3, for example, is necessarily composed of a 1st, 2nd and 3rd member!
So when both perspectives are combined, the direction of causation with respect to each of these partial perspectives is revealed as pardoxical. This clearly implies that ultimately both the quantitative and qualitative aspects are identical with each other in an ineffable manner (where likewise both prime and natural numbers are identical).
Thus the prime numbers, as relatively separate entities, enjoy in cardinal terms an independent quantitative existence (as the building blocks of the number system), However, equally the prime numbers, in their unique relationship with each other (as composite natural numbers) enjoy in ordinal terms an interdependent qualitative existence (where each individual is inseparable from the holistic set of factors comprising these numbers).
Therefore, once again in dynamic terms, number independence implies a localised quantitative existence (without reference to an overall holistic context); number interdependence - by contrast - implies a holistic qualitative existence (where the interpretation of each individual number is inseparable from the overall holistic context to which its is related).
And this relationship as between quantitative and qualitative (and independence and interdependence) is mediated in a complementary bi-directional manner as between the prime and natural numbers.
Again from the Type 1 perspective, the prime numbers reveal themselves as the independent building blocks of the natural number system in cardinal (quantitative) terms.
However from the Type 2 perspective each prime number in turn is revealed as internally composed of an interdependent group of natural numbers in ordinal (qualitative) terms. Thus the prime number group of 3, for example, is necessarily composed of a 1st, 2nd and 3rd member!
So when both perspectives are combined, the direction of causation with respect to each of these partial perspectives is revealed as pardoxical. This clearly implies that ultimately both the quantitative and qualitative aspects are identical with each other in an ineffable manner (where likewise both prime and natural numbers are identical).
The significance of the non-trivial zeros - both for the Zeta 1 and Zeta 2 Functions - arise clearly from this dynamic interactive context.
It thereby can be given quantitative expression as a point, – 1, on the circle of unit radius. Then 2nd (in the context of 3) is given by the 2nd of the three roots of 3 i.e. 12/3 and so on. Thus we can give in this manner a potentially unlimited number of quantitative interpretations to the ordinal notion of 2nd.
However these roots strictly have no meaning in the absence of the overall circular group of members to which they are related. Holistic interdependence then arises through summing the n prime roots of 1. So in the context of 2 roots, this implies + 1 – 1 = 0.
And in similar fashion we can give quantitative expression to all other ordinal numbers such as 3rd, 4th etc. together with corresponding holistic expressions of the qualitative interdependence of the sum of roots (where the number of individual members of a prime group = n and the sum of its n roots = 0).
Thus in dynamic interactive terms, both the quantitative analytic independence of each individual member ultimately approaches full coincidence with the qualitative holistic interdependence of the overall (circular) group 0f members.
And this is what dynamically is meant by the notion of interdependence (which must necessarily be of an approximate relative nature in phenomenal number terms).
Then the Zeta 1 zeros provide - in an imaginary linear number fashion - a ready means to give a corresponding qualitative ordinal meaning to the cardinal number system (as a whole). Now more correctly in a dynamic interactive context, the notion of number interdependence here entails no remaining distinction as between both quantitative and qualitative interpretation!
So quantitative independence implies complete separation from its qualitative aspect (which however can only be approximated in a dynamic relative context). Then quantitative/qualitative interdependence implies that both meanings simultaneously co-exist (though again necessarily in an approximate relative manner from a phenomenal number perspective).
As we have seen multiplication gives rise to a qualitative aspect of number transformation (though this is formally ignored in conventional mathematical terms).
Thus the quantitative independence of the prime numbers (with respect to the natural number system) entails that these can be expressed without need for prior multiplication of other numbers).
So the Zeta 1 non-trivial zeros (in this dynamic interactive context) express the other extreme where the independence of each zero in isolation is (relatively) inseparable from the combined distribution of all these zeros on the imaginary number line.
And in a recent blog entry "Stunning Accuracy", I illustrated this remarkable feature of interdependence with respect to the (Zeta 1) non-trivial zeros, in the manner in which their absolute frequency can be predicted to an incredible degree of accuracy by a very simple general formula!
We saw how in the circular context of the Zeta 2 zeros, the quantitative independence of each individual nth root must be seen in conjunction with the overall sum of n roots (which displays their combined qualitative interdependence).
Then in the corresponding linear imaginary context of the Zeta 1 zeros, the quantitative independence of each zero (as a distinct point on the imaginary number line through 1/2), must be seen in conjunction with the combined qualitative interdependence of all zeros on the line.
Once again, this interdependence is demonstrated through the consistent predictive accuracy in absolute terms with respect to their overall distribution and then the consequent manner in which they can be used to smooth out deviations with respect to the corresponding general prediction of frequency of the primes.
So ultimately both sets of zeros are interdependent in a complementary manner (as likewise are the primes and natural numbers).
In fact the Zeta 1 and Zeta 2 zeros simply express the quantitative/qualitative interdependence of the primes and the natural numbers from two opposite directions.
In the context of the Zeta 1, this interdependence, in the unlimited set of non-trivial zeros on the imaginary number line through 1/2, is expressed with respect to the overall number system (where every natural number represents a unique composition of primes in cardinal terms).
In the context of Zeta 2, this interdependence, in a second unlimited set of non-trivial zeros on the circle of unit radius in the complex plane, is expressed with respect to each prime number (representing a unique composition of natural numbers in ordinal terms).
Perhaps the biggest lesson we must learn is that proper interpretation of both sets of zeros cannot take place in the absence of a truly dynamic interactive approach to numerical relationships (which entails giving equal emphasis to both its quantitative and qualitative aspects).
In this context the conventional mathematical approach with sole emphasis on its quantitative aspect (in an absolute manner) is clearly no longer fit for purpose!
Indeed it never was properly fit for purpose. However, considerable success with respect to the specialised - and necessarily limited quantitative aspect of mathematical development - has blinded us to this fact now for several milennia.
Tuesday, May 14, 2013
Addition and Multiplication as Complementary Facets of Same Phenomenon
In the last blog entry I demonstrated how the Euler Product Formula can be expressed with respect to either the Zeta 1 or Zeta 2 Functions indicating in fact that these represent complementary expressions.
Once again The Euler Product Formula beautifully demonstrates the relationship as between addition (with respect to the natural numbers) and multiplication with respect to the primes.
From a deeper level of appreciation it expresses the complementary relationship as between the quantitative and qualitative aspects of the number system, which are dynamically mediated through the interaction of the primes with the natural numbers (and the natural numbers with the primes) respectively.
Each prime number can be uniquely represented in two ways.
(i) through addition as the sum of its individual members (in quantitative terms). Thus from this perspective the prime number 3 = 1 + 1 + 1. This represents the Type 1 definition of 3 which more fully is represented as 31.
(ii) through multiplication as the product of its individual members (in qualitative terms). Thus from this perspective the prime number 3 = 1 * 1 * 1. This represents the Type 2 definition of 3 which is more fully represented (in an inverse manner) as 13.
Now it is important to point out in this context that what represents multiplication from a Type 1 perspective represents addition from the Type 2 (and vice versa).
So 1 * 1 * 1 = 13 = 11 + 1 + 1
So the Type 1 and Type 2 aspects of the number system again dynamically represent - in the interaction of both cardinal and ordinal aspects - how the notion of number is always quantitative as to qualitative (and qualitative as to quantitative) respectively.
Thus both Type 1 and Type 2 aspects of the number system are defined in their pure form with respect to the default number 1. In the case of the Type 1 aspect, 1 represents the (default) dimensional value, to which the varying base or ground value is raised; with the Type 2, 1 represents the (default) base or ground value which is raised to a dimensional value that varies.
In this way we can clearly distinguish the quantitative aspect (with respect to the Type 1) and the qualitative aspect (with respect to the Type 2) respectively.
Of course when we remove this restriction with respect to base and dimensional values remaining fixed respectively as 1, both quantitative and qualitative transformation is involved with respect to any number expression.
So 32 and 23, which again are the inverse of each other (with respect to both base and dimensional values) from a dynamic interactive perspective, involve both quantitative and qualitative transformation.
And as both the Zeta 1 and Zeta 2 Functions involve number expressions of this kind clearly both quantitative and qualitative aspects (from complementary perspectives) are involved with respect to both Functions! And this is why the Euler Product Formula - entailing the relationship as between addition and multiplication - as a corresponding relationship as between the natural and prime numbers - can be defined in the context of both the Zeta 1 and Zeta 2 expressions!
Now we can attempt to convert the pure (linear) qualitative Type 2 notion of number in an indirect (circular) quantitative manner.
So for example the Type 2 dimensional notion of 3 represents in ordinal terms its 1st, 2nd and 3rd members which inversely are associated in quantitative terms with the corresponding 3 roots of 11, 12 and 13 respectively.
So each of these roots now possesses an individual quantitative identity (on the circle of unit radius in the complex plane). However their holistic qualitative identity is now expressed through the addition of the 3 root values = 0.
Thus whereas from the linear number perspective, addition of the individual units of 3 relates to the cardinal (quantitative) aspect of number, from the circular perspective, addition of the 3 roots of 1, which indirectly represents the ordinal identity of these three individual members, relates to the holistic (qualitative) aspect of number.
Therefore to represent the ordinal aspect of number in cardinal terms we must switch from a linear to a circular (Type 2) representation of number; likewise to represents the cardinal in ordinal terms we must switch from a linear to a circular (Type 1) representation of number!
The same inversion applies to the multiplication aspect. From the Type 1 perspective, pure multiplication such as 11 * 11 * 11 represents the qualitative aspect of number transformation.
However when we multiply the three roots of 1 (as the corresponding circular expression) the answer is always 1 (with 1 the very symbol of its linear counterpart). So multiplication = 1 (for prime roots except where 2 is a factor) now expresses a quantitative holistic relationship in circular terms, whereas it was qualitative in linear terms.
In like manner the addition of roots = 0, expressed a qualitative holistic relationship in circular terms (where it was quantitative from a linear perspective).
Thus when one recognisies both Type 1 and Type 2 aspects of the number system (with their related Zeta 1 and Zeta 2 Functions) both addition and multiplication can be expressed in a complementary fashion, where addition in the context of Type 2 represents multiplication in the context of Type 1 (and vice versa).
Likewise what represents multiplication in the context of Type 2 represents exponentiation in the context of Type 1 (and vice versa).
Once again the relationship between addition (with respect to the natural numbers) and multiplication (with respect to the primes) as embodied in the Euler Product Formula, can be expressed in either Type 1 of Type 2 terms through the Zeta 1 and Zeta 2 Functions respectively.
Now both of these, like a left turn at a crossroads, enjoy a - relatively - unambiguous interpretation within their respective reference frames (as the direction from which the crossroads is approached).
However when we attempt to properly combine both interpretations as interdependent in Type 3 terms through the Zeta 3 Function (representing their combined dynamic interaction) deep paradox results, as what is addition from one context in now multiplication from the opposite; and what is multiplication from one context is likewise addition from the opposite.
This again is exactly analogous to the situation at a crossroads where - simultaneously applying both reference frames - a left turn is equally a right; and a right turn is equally a left.
So in Type 3 terms, addition and multiplication are understood as but complementary facets of the same phenomenon and as ultimately identical in an ineffable manner.
We could of course equally say that the quantitative (cardinal) and qualitative (ordinal) aspects of number in dynamic terms represent complementary facets of the same phenomenon which again are also identical in an ineffable manner.
And here lies the true nature of number that resides ultimately in total mystery! So from this perspective, the phenomenal forms that become manifest in space and time - through the relative separation of both quantitative and qualitative aspects - can be seen as but the alluring veils though which number seeks to hide its most intimate secrets.
Once again The Euler Product Formula beautifully demonstrates the relationship as between addition (with respect to the natural numbers) and multiplication with respect to the primes.
From a deeper level of appreciation it expresses the complementary relationship as between the quantitative and qualitative aspects of the number system, which are dynamically mediated through the interaction of the primes with the natural numbers (and the natural numbers with the primes) respectively.
Each prime number can be uniquely represented in two ways.
(i) through addition as the sum of its individual members (in quantitative terms). Thus from this perspective the prime number 3 = 1 + 1 + 1. This represents the Type 1 definition of 3 which more fully is represented as 31.
Now it is important to point out in this context that what represents multiplication from a Type 1 perspective represents addition from the Type 2 (and vice versa).
So 1 * 1 * 1 = 13 = 11 + 1 + 1
So the Type 1 and Type 2 aspects of the number system again dynamically represent - in the interaction of both cardinal and ordinal aspects - how the notion of number is always quantitative as to qualitative (and qualitative as to quantitative) respectively.
Thus both Type 1 and Type 2 aspects of the number system are defined in their pure form with respect to the default number 1. In the case of the Type 1 aspect, 1 represents the (default) dimensional value, to which the varying base or ground value is raised; with the Type 2, 1 represents the (default) base or ground value which is raised to a dimensional value that varies.
In this way we can clearly distinguish the quantitative aspect (with respect to the Type 1) and the qualitative aspect (with respect to the Type 2) respectively.
Of course when we remove this restriction with respect to base and dimensional values remaining fixed respectively as 1, both quantitative and qualitative transformation is involved with respect to any number expression.
So 32 and 23, which again are the inverse of each other (with respect to both base and dimensional values) from a dynamic interactive perspective, involve both quantitative and qualitative transformation.
And as both the Zeta 1 and Zeta 2 Functions involve number expressions of this kind clearly both quantitative and qualitative aspects (from complementary perspectives) are involved with respect to both Functions! And this is why the Euler Product Formula - entailing the relationship as between addition and multiplication - as a corresponding relationship as between the natural and prime numbers - can be defined in the context of both the Zeta 1 and Zeta 2 expressions!
Now we can attempt to convert the pure (linear) qualitative Type 2 notion of number in an indirect (circular) quantitative manner.
So for example the Type 2 dimensional notion of 3 represents in ordinal terms its 1st, 2nd and 3rd members which inversely are associated in quantitative terms with the corresponding 3 roots of 11, 12 and 13 respectively.
So each of these roots now possesses an individual quantitative identity (on the circle of unit radius in the complex plane). However their holistic qualitative identity is now expressed through the addition of the 3 root values = 0.
Thus whereas from the linear number perspective, addition of the individual units of 3 relates to the cardinal (quantitative) aspect of number, from the circular perspective, addition of the 3 roots of 1, which indirectly represents the ordinal identity of these three individual members, relates to the holistic (qualitative) aspect of number.
Therefore to represent the ordinal aspect of number in cardinal terms we must switch from a linear to a circular (Type 2) representation of number; likewise to represents the cardinal in ordinal terms we must switch from a linear to a circular (Type 1) representation of number!
The same inversion applies to the multiplication aspect. From the Type 1 perspective, pure multiplication such as 11 * 11 * 11 represents the qualitative aspect of number transformation.
However when we multiply the three roots of 1 (as the corresponding circular expression) the answer is always 1 (with 1 the very symbol of its linear counterpart). So multiplication = 1 (for prime roots except where 2 is a factor) now expresses a quantitative holistic relationship in circular terms, whereas it was qualitative in linear terms.
In like manner the addition of roots = 0, expressed a qualitative holistic relationship in circular terms (where it was quantitative from a linear perspective).
Thus when one recognisies both Type 1 and Type 2 aspects of the number system (with their related Zeta 1 and Zeta 2 Functions) both addition and multiplication can be expressed in a complementary fashion, where addition in the context of Type 2 represents multiplication in the context of Type 1 (and vice versa).
Likewise what represents multiplication in the context of Type 2 represents exponentiation in the context of Type 1 (and vice versa).
Once again the relationship between addition (with respect to the natural numbers) and multiplication (with respect to the primes) as embodied in the Euler Product Formula, can be expressed in either Type 1 of Type 2 terms through the Zeta 1 and Zeta 2 Functions respectively.
Now both of these, like a left turn at a crossroads, enjoy a - relatively - unambiguous interpretation within their respective reference frames (as the direction from which the crossroads is approached).
However when we attempt to properly combine both interpretations as interdependent in Type 3 terms through the Zeta 3 Function (representing their combined dynamic interaction) deep paradox results, as what is addition from one context in now multiplication from the opposite; and what is multiplication from one context is likewise addition from the opposite.
This again is exactly analogous to the situation at a crossroads where - simultaneously applying both reference frames - a left turn is equally a right; and a right turn is equally a left.
So in Type 3 terms, addition and multiplication are understood as but complementary facets of the same phenomenon and as ultimately identical in an ineffable manner.
We could of course equally say that the quantitative (cardinal) and qualitative (ordinal) aspects of number in dynamic terms represent complementary facets of the same phenomenon which again are also identical in an ineffable manner.
And here lies the true nature of number that resides ultimately in total mystery! So from this perspective, the phenomenal forms that become manifest in space and time - through the relative separation of both quantitative and qualitative aspects - can be seen as but the alluring veils though which number seeks to hide its most intimate secrets.
Friday, May 10, 2013
Zeta 2 Formulation of Euler Product Formula
In yesterday's blog entry I commented on the fact that the Zeta 1 and Zeta 2 (non-trivial) zeros represent the holistic counterparts to the Type 1 and Type 2 aspects of the number system respectively.
Putting it more precisely, the Zeta 1 zeros represent the (qualitative) holistic counterpart to the cardinal aspect of the number system, whereas the Zeta 2 zeros represent the corresponding (quantitative) holistic counterpart to the ordinal aspect of the number system.
It must be clearly remembered in this context that both the cardinal and ordinal aspects of the number system enjoy a relative identity which are - in direct terms - quantitative and qualitative with respect to each other.
We saw how the ordinal identity implies qualitative interdependence in the ordered relationship of numbers with each other.
The Zeta 2 zeros then consisted of giving a new quantitative numerical identity (in circular terms) to the corresponding qualitative notions of 1st, 2nd, 3rd etc. And these new numbers (with respect to the n roots of 1) have then a holistic identity with respect to their combined sum = o.
Now again, one of the great blind spots of Conventional Mathematics resides in the failure to recognise that ordinal aspect of numbers clearly implies a qualitative - rather than quantitative - identity. This failure leads to the consistent reduction of qualitative to quantitative meaning with respect to the interpretation of number!
By contrast, the cardinal identity of number implies quantitative independence in the individual nature of each number. So the prime numbers 2, 3, 5 etc. are independent in this locally confined sense. All the natural numbers (except 1) therefore depend for their identity on the prime numbers (as the independent building blocks of the number system).
The Zeta 1 zeros thereby consist of giving a new qualitative numerical identity (in linear imaginary terms) to the corresponding shared identity of the primes with the natural number system.
It is important in this context to recognise that the imaginary number line in fact is the appropriate way of expressing meaning, which by nature is inherently qualitative and holistic, in an independent local manner. So in this sense each of the Zeta 1 zeros has an individual local identity on the imaginary number line!
Therefore, if we are then to properly decode the collective nature of the Zeta 1 zeros , we must interpret them in a (holistic) qualitative - rather than (analytic) quantitative - manner.
So the holistic nature of the zeros resides in the combined manner in which they can be used to ultimately demonstrate the perfect (interdependent) order of the primes with respect to the natural number system. And this relationship, which is intimately connected with the distribution of non-trivial zeros, is strictly of a qualitative - rather than quantitative - nature!
I mentioned in my last blog that the Zeta 1 and Zeta 2 zeros are ultimately totally complementary with each other. Put another way, the full appreciation of the cardinal aspect of the number system implies corresponding full appreciation of the ordinal; from the other direction, full appreciation of the ordinal aspect likewise implies full appreciation of the cardinal.
So the Zeta 1 ultimately imply the Zeta 2; and the Zeta 2 in turn ultimately imply the Zeta 1 zeros. And in turn the Zeta 1 Function therefore implies the Zeta 2 and the Zeta 2 the Zeta 1 Function respectively.
With this in mind, I sought recently to demonstrate that the Euler Product Formula (that beautifully relates the additive properties of the natural numbers to the multiplicative properties of the primes) can be equally fully expressed in terms of the Zeta 2 Function.
At present it is exclusively associated with the Zeta 1 Function!
However this equally can be expressed in terms of the Zeta 2 Function!
= 1/(1 - 1/n) = n/(n – 1)
ζ2(1/4) = 1 + 1/4 + 1/16 + 1/64 + ... 1 = 1/(1 –1/4) = 4/3
∞
Putting it more precisely, the Zeta 1 zeros represent the (qualitative) holistic counterpart to the cardinal aspect of the number system, whereas the Zeta 2 zeros represent the corresponding (quantitative) holistic counterpart to the ordinal aspect of the number system.
It must be clearly remembered in this context that both the cardinal and ordinal aspects of the number system enjoy a relative identity which are - in direct terms - quantitative and qualitative with respect to each other.
We saw how the ordinal identity implies qualitative interdependence in the ordered relationship of numbers with each other.
The Zeta 2 zeros then consisted of giving a new quantitative numerical identity (in circular terms) to the corresponding qualitative notions of 1st, 2nd, 3rd etc. And these new numbers (with respect to the n roots of 1) have then a holistic identity with respect to their combined sum = o.
Now again, one of the great blind spots of Conventional Mathematics resides in the failure to recognise that ordinal aspect of numbers clearly implies a qualitative - rather than quantitative - identity. This failure leads to the consistent reduction of qualitative to quantitative meaning with respect to the interpretation of number!
By contrast, the cardinal identity of number implies quantitative independence in the individual nature of each number. So the prime numbers 2, 3, 5 etc. are independent in this locally confined sense. All the natural numbers (except 1) therefore depend for their identity on the prime numbers (as the independent building blocks of the number system).
The Zeta 1 zeros thereby consist of giving a new qualitative numerical identity (in linear imaginary terms) to the corresponding shared identity of the primes with the natural number system.
It is important in this context to recognise that the imaginary number line in fact is the appropriate way of expressing meaning, which by nature is inherently qualitative and holistic, in an independent local manner. So in this sense each of the Zeta 1 zeros has an individual local identity on the imaginary number line!
Therefore, if we are then to properly decode the collective nature of the Zeta 1 zeros , we must interpret them in a (holistic) qualitative - rather than (analytic) quantitative - manner.
So the holistic nature of the zeros resides in the combined manner in which they can be used to ultimately demonstrate the perfect (interdependent) order of the primes with respect to the natural number system. And this relationship, which is intimately connected with the distribution of non-trivial zeros, is strictly of a qualitative - rather than quantitative - nature!
I mentioned in my last blog that the Zeta 1 and Zeta 2 zeros are ultimately totally complementary with each other. Put another way, the full appreciation of the cardinal aspect of the number system implies corresponding full appreciation of the ordinal; from the other direction, full appreciation of the ordinal aspect likewise implies full appreciation of the cardinal.
So the Zeta 1 ultimately imply the Zeta 2; and the Zeta 2 in turn ultimately imply the Zeta 1 zeros. And in turn the Zeta 1 Function therefore implies the Zeta 2 and the Zeta 2 the Zeta 1 Function respectively.
With this in mind, I sought recently to demonstrate that the Euler Product Formula (that beautifully relates the additive properties of the natural numbers to the multiplicative properties of the primes) can be equally fully expressed in terms of the Zeta 2 Function.
At present it is exclusively associated with the Zeta 1 Function!
So in the traditional manner of expression for example Euler’s Product Formula for the Zeta 1 Function is given as,
∞
∑ 1/ns = ∏ 1/(1 – p–s) = ζ1(s)
n = 1 p
Thus for example when s = 2 in the Zeta 1 expression,
ζ1(2) = 1 + 1/4 + 1/9 + 1/16 +….. = 4/3 * 9/8 * 25/24 * 49/48 * …. = π2/6
However this equally can be expressed in terms of the Zeta 2 Function!
Once again,
Thus when s = 1/n,
ζ2(s) = 1 + s1 + s2 + s3 + s4…… = 1/(1 – s).
ζ2(1/n) = 1 + (1/n)1 + (1/n)2 + (1/n)3 + (1/n)4……
= 1/(1 - 1/n) = n/(n – 1)
Therefore when n = 2,
ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + …. = 2 /(2 – 1) = 2
Thus,
ζ2(1/2) – 1 = 1
Then when n = 3
ζ2(1/3) = 1 + 1/3 + 1/9 + 1/27 + …. = 3/(3 – 1) = 3/2
Thus,
ζ2(1/3) – 1 = 1/2
Then when n = 4
ζ2(1/4) = 1 + 1/4 + 1/16 + 1/64 +…… = 4/(4 – 1) = 4/3
Thus,
ζ2(1/4) – 1 = 1/3
Thus summing over the natural numbers i.e. n = 2, 3, 4,….
∞
∑{ζ2(1/n) – 1} = 1 + 1/2 + 1/3 + ………
n = 2
Therefore,
∞
∑{ζ2(1/n) – 1}s = 1s + ( 1/2)s + ( 1/3)s + ………
n = 2
So when s = 2,
∞
∑{ζ2(1/n) – 1}2
n = 2
= 12 + ( 1/2)2 + ( 1/3)2 + ……… = π2/6
∑{ζ2(1/n) – 1}2
n = 2
= 12 + ( 1/2)2 + ( 1/3)2 + ……… = π2/6
Likewise when p = 2
ζ2(1/p) = ζ2(1/2)
Then when s = 2
ζ2(1/2)2 = ζ2(1/4)
ζ2(1/4) = 1 + 1/4 + 1/16 + 1/64 + ... 1 = 1/(1 –1/4) = 4/3
Then when p = 3
ζ2(1/p) = ζ2(1/3)
So again with s = 2,
ζ2(1/3)2 = ζ2(1/9)
ζ2(1/9) = 1 + 1/9 + 1/81 + 1/729 + ... = 1/(1 – 1/9) = 9/8
Finally to illustrate with p = 5 and s = 2
ζ2(1/5)2 = ζ2(1/25) = 1 + 1/25 + 1/625 + …. = 1/(1 – 1/25) = 25/24
Therefore multiplying over all the primes when s = 2,
∏{ζ2(1/p)2} = 4/3 * 9/8 * 25/24 * …… = π2/6
p
So we have now demonstrated the famous Euler Product Formula solely in terms of the Zeta 2 expression i.e.,
∑{ζ2(1/n) – 1}s = ∏{ζ2(1/p)s}
n = 2 p
n = 2 p
Therefore in terms of both Zeta 1 and Zeta 2 Functions, the Euler Product Formula can be expressed:
∑ 1/ns = ∏ 1/(1 – p–s) = ζ1(s) = ∑{ζ2(1/n) – 1}s = ∏{ζ2(1/p)s}
n = 1 p n = 2 p
n = 1 p n = 2 p
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