In yesterday’s blog I simplified the nature of the non-trivial zeros as relating directly to the interdependent aspect of the number system.
Furthermore in line with Zeta 1, Zeta 2 and Zeta 3 approaches, we can provide 3 - relatively - distinct perspectives on what such interdependence entails.
Though its true significance to my mind remains as yet completely unrecognised, it would perhaps be most illuminating to start with the Zeta 2 (non-trivial) zeros.
Once again these zeros arise as the solution to the equation,
s^1 + s^2 + s^3 + s^4 +……+ s^t = 0.
Now one of these solutions is for s = 0.
The others then relate in quantitative terms to the non-trivial roots (excepting the common root 1) of t.
Now if t is a prime number, then all these non-trivial roots will be unique (i.e. cannot occur as the solutions for any other prime value of t).
All this of course is well known in conventional mathematical terms. However it does not convey the true significance of these prime numbered root values.
What is not yet clearly recognised is that associated with all these quantitative values (as relatively independent) is an important complementary meaning of qualitative interdependence in holistic terms.
When seen from this perspective each grouping of prime numbered roots (including the common root 1) provides a unique circle of interdependence from an ordinal qualitative perspective.
For example when t = 3 (where of course t is prime) we have the Zeta 2 equation,
s^1 + s^2 + s^3 = 0.
Once again, s = 0 represents one of these 3 solutions.
The other two solutions (correct to 3 decimal places) relate to the unique (non-trivial) roots of 1,
i.e. – ½ + .866i and – ½ - .866i.
Now in the Type 2 approach (that I have introduced) a dynamic complementary relationship always necessarily exists between such values in both quantitative and qualitative terms.
So if we include the trivial root i.e. t = 1, the three roots of 1 in relative quantitative terms,
are 1, – ½ + .866i and – ½ - .866.
However these now bear a complementary relation to the same 3 values now considered as a circular interdependent group in a qualitative holistic manner.
In other words, the true unrecognised significance of these 3 circular numbers in this context is that they provide a unique ordinal means of identifying a group (containing 3 members).
Again this issue is completely overlooked in conventional mathematical terms where it is - wrongly - assumed that ordinal identification of members can be carried out in an unambiguous fashion.
So therefore from this perspective we can identify unambiguously the 3 members of a group in ordinal terms as the 1st, 2nd and 3rd respectively.
However the key problem is that with the numbers of a group increasing the very meaning of 1st, 2nd and 3rd (given earlier) is no longer relevant in the new changed context.
So for example if the number t now increases to 5, 1st, 2nd and 3rd acquire a new – relatively - distinctive meaning in the context of this larger grouping of ordinal members.
Therefore when one reflects carefully on the matter, ordinal natural number rankings have a merely relative validity depending on the overall size of any group in question.
This raises then the serious issue of providing some means of unambiguously distinguishing such rankings (as the finite size of the grouping changes).
And the key to this is with reference to the new circular number system that is associated with the various roots of 1.
And in this system for any value t, we can unambiguously identify the different ordinal members of the group in a – relatively – quantitative numerical manner, while also allowing for the overall interdependence of the group members in a qualitative holistic fashion.
So once again in the context of 3, we unambiguously identify the 3 members of this group in a quantitative manner through the circular number system (defined with respect to points on the circumference of the unit circle in the complex plane).
So in relative quantitative terms, these 3 members are identified as
1, – ½ + .866i and – ½ - .866i
The qualitative aspect arises through combing these 3 values as a whole.
And as the sum of roots = 0, this means that the qualitative interpretation (with respect to the interdependence of this group) strictly has no meaning from a quantitative perspective.
Now again the key significance of t as representing a prime dimensional number in this context, is that the various internal members of the group (except 1) will be defined in a unique manner.
Now the fact that 1 is not unique (yet a member of the prime group) is necessary so as to provide a link with the complementary linear manner of defining the relationship of primes to natural numbers.
And we have already seen - again in a complementary fashion - that the one value for t for which the sum of roots ≠ 0 is where t = 1!
So we are always interpreting in a relative – rather than absolute – terms.
Thus – in this relative sense – each prime group is defined uniquely in ordinal terms by its natural number constituents (through quantitative numbers in the circular system).
And the corresponding interdependence of such a group - represented as the sum of individual members) strictly has no quantitative meaning! In other word the quantitative value of this sum = 0.
When seen in this light, the true significance of the solutions for t (as the non-trivial zeros) for the Zeta 2 equation is that they provide the means to define number in an interdependent manner.
And as the sum of roots = 0 for all values of t (except 1), group independence with respect to all these numbers is thereby seen to relate to the holistic qualitative nature of number.
And when t becomes very large, dynamic interactivity so increases that the relative independence of each quantitative member becomes inseparable from the relative interdependence collectively of these same members (in qualitative terms).
So therefore any distinction between a collective prime number grouping (in qualitative terms) and its uniquely distinctive ordinal number members (in quantitative terms) thereby ceases. So in this sense the non-trivial zeros of the Zeta 2 (for unlimited t) entail the identity of the prime with the natural numbers.
We will see again that the Zeta 2 provides a directly inverse way of defining the quantitative/qualitative relationship between number to the Zeta 1 approach.
So with the Zeta 2 we have shown how the sum of all the natural numbers to t, as solutions for the base numbers s (in ordinal fashion) of the equation = 0 with t having no upper finite limit!
Then with the Zeta 1 we will show how the sum of all the natural numbers to infinity as solutions for dimensional numbers s (in cardinal fashion) = 0 likewise have no upper limit.
Thus the non-trivial zeros for Zeta 1 and Zeta 2 simply represent two sides of the same coin with respect to demonstrating the interdependent nature of the number system.
Whereas Zeta 2 demonstrates this interdependence with respect to their ordinal nature (in a circular number fashion), Zeta 1 demonstrates such interdependence with respect to their cardinal nature (in an imaginary linear number fashion).