In recent blog entries, I have shown how the Zeta 1 function is best used (externally) to interpret the general behaviour of the primes; however the Zeta 2 is then best used (internally) to interpret the specific behaviour of each prime. So I have been at pains to demonstrate, the truly complementary nature of both functions, which requires an inherently dynamic appreciation of the relationship between the primes and natural numbers.
In fact there are strong parallels here with physics. As is well known Einstein's Theory of General Relativity is especially useful in understanding the general operation of the macro universe; however Quantum mechanics is then required to properly understand the micro behaviour of particles at the sub-atomic level.
And the key issue that remains is the coherent reconciliation of both approaches.
So it is exactly similar at an even more fundamental level in Mathematics. Indeed - ultimately I strongly believe that the dynamic interactive behaviour of the number system deeply underlies the manifest physical behaviour of the world in the first instance. Therefore the ultimate key to reconciliation of macro and micro physical behaviour at the physical level is the coherent reconciliation of number with respect to its twin macro and micro aspects or - as I customarily refer to as - its external and internal (horizontal and vertical) features.
And this issue can only be properly understood in a dynamic manner, entailing the two-way interaction of primes and natural numbers (in both quantitative and qualitative terms).
And let me be very blunt here! This cannot be achieved within the present accepted paradigm of Mathematics, which is based on an absolute rigid type interpretation that reduces the qualitative to the quantitative aspect (in every circumstance)!
So I would like to be very clear regarding my true intentions which far exceed specific concern with the Riemann Hypothesis.
Quite simply, a completely revised approach to Mathematics is now required - of an inherently dynamic relative nature - to come to terms with the fundamental nature of the number system.
This will require painful recognition that the the highly important holistic (qualitative) appreciation of mathematical relationships, which is designed to complement the analytic aspect - has remained suppressed to an extraordinary degree in our culture.
However the ultimate task - which can then herald in a true golden age of Mathematics - will only arrive when both existing analytic and holistic appreciation of all mathematical relationships are coherently integrated with each other in a truly balanced manner.
And with this everything else in Mathematics and the Sciences and indeed our understanding of social, economic and political life will be utterly transformed in a manner that is presently unimaginable.
So we will look now at the zeros of the Zeta 2 function i.e. those values for which the function = 0.
Again, ζ2(s2) = 1 + s21 + s22 + s23 + s24 +….
As defined this is an infinite series. However the function only has zero solutions in certain limited circumstances.
If we look on the series in finite - rather than infinite - terms, we can express it as
1 + s21 + s22 + .......+ s2t – 1 = 0
And the solutions for the function represent the (t – 1) roots of 1 (other than for s2 = 1).
Note that (including the 1st term = 1), the finite function here contains t terms!
If then further terms are added in regular cycles of t, the solutions for these (t – 1) roots, would equally constitute the zero solutions for the infinite Zeta 2 function.
However it would be instructive here to first concentrate on the finite version, where t is prime.
For example t = 3 is prime.
Therefore the associated Zeta 2 finite equation is given as,
1 + s21 + s22 = 0
Thus the two solutions for s2 here are s2 = – 1/2 + .866i and – 1/2 – .866i respectively i.e. the 2 of the 3 roots of 1 (other than s2 = 1).
Now, all of this is indeed well known! However the true significance of the results (where t is prime) is not at all appreciated!
And as the key to later appreciation of the corresponding significance of the Zeta 1 (Riemann) zeros is intimately associated with this finding, it is important here to explain it in some detail.
When we try - in the standard manner - to subdivide a prime number such as 3 in this illustration, we get
3 = 1 + 1 + 1.
In other words, 3 is defined in a merely reduced quantitative manner where - literally - its individual component units are viewed as independent in strictly homogeneous terms, thereby lacking any qualitative distinction.
This represents the cardinal appreciation of 3 (i.e. as a merely quantitative entity) which can be represented as a point on the real number line.
However is 3 enjoyed a merely independent identity (as quantitative), no means would exist for establishing its relationship with other numbers. Likewise no means would exist for establishing a relationship as between its individual units!
However we can demonstrate the alternative qualitative nature of these units in an ordinal manner, where the meaning of each term is derived from its relationship with the other terms.
Thus the notions of 1st, 2nd and 3rd only have meaning in this context, through their interdependent relationship with each other (which is directly of a qualitative nature).
Whereas the notion of quantitative independence is best illustrated In Type 1 terms, through the number line (that is - of literally 1-dimensional) , the corresponding notion of qualitative interdependence (with respect to number) is best illustrated in Type 2 terms through the unit circle.
So 1st, 2nd and 3rd can be represented as 3 equidistant points on the unit circle (in the complex plane). Now what is fascinating here is that it does not strictly matter which starting point we choose. Therefore the key to the appreciation that underlies true qualitative appreciation of number is holistic understanding of how 1st, 2nd and 3rd can potentially interchange as between the 3 positions available.
Whereas in actual cardinal terms, a number such as 3 has just one fixed invariant identity, in ordinal terms, its individual units can potentially change as between 1st, 2nd and 3rd respectively. In this way each ordinal number is potentially identified with all (ordinal) positions available.
And this is far from fanciful but in fact implicitly underlies our approach to many problems.
For example, let's imagine that one is ranking 3 cars according to a set of different criteria e.g. age, size, price, safety features etc.
Implicit is this approach is the holistic recognition that what is 1st according to one criterion could be equally 2nd or 3rd according to another and vice versa.
So for example what is ranked 1st (in terms of the age) could equally be ranked 2nd in terms of size and 3rd in terms of price and so on according to a variety of possible classification criteria.
Once a particular criterion is applied for classification the actual ranking - according to this criterion - will be unambiguously fixed in actual terms as 1st, 2nd and 3rd. However underlying this must necessarily also exist the holistic appreciation that this order can potentially change (where multiple criteria are applied).
What happens in the standard treatment of ordinal notions is that their potential (qualitative) meaning are always reduced in an actual quantitative manner.
So let's imagine that instead of using cardinal, one now uses ordinal units to define the nature of 3!
Then 3 = 1st + 2nd + 3rd.
In conventional mathematical terms, this causes no problem for 1st, 2nd and 3rd are immediately reduced in a cardinal manner to 1. So we recognise the 1st unit = 1, then the 2nd unit = 1 and finally the 3rd unit = 1.
So from this perspective,
3 = 1st + 2nd + 3rd reduces directly to 3 = 1 + 1 + 1.
Having contemplated this issue for many years, I gradually realised that the true holistic meaning of ordinal notions are in fact intimately related to the Type 2 nature of the number system.
Recognition of number interdependence (which defines true qualitative holistic appreciation) occurs directly in an intuitive manner.
However indirectly one can then attempt to "convert" this in a rational fashion, which takes place in this case through recognition of the 3 roots of 1.
The "trivial" case where one of the roots = 1, represents reduced actual interpretation (in the standard analytic manner).
In linear terms, this always implies identifying the ordinal notion with the last unit of a group involved. So 1st represents the last of 1, 2nd the last of 2 units, 3rd the last of 3 units, 4th the last of 4 units, and so on.
And this is exactly how we would identify these units on the number line!
However the remaining roots represent - when properly understood - the true relative nature of the ordinal notion.
So for example we could have the 1st of 2, or the 2nd of 3 or the 5th of 13 units and so on where 1st, 2nd and 5th take on unique meanings depending on precise context.
Now the big insight that eventually dawned was the realisation that all these "non-trivial" roots (other than 1) in fact represent a unique means of "converting" Type 2 qualitative notions of number indirectly in a Type 1 quantitative manner.
The great significance of the primes in this context is that - with the exception of the "trivial" case of 1 - all the other roots are unique for each prime.
So when the Zeta 2 function is defined for t = a prime number, all its solutions are thereby unique providing an appropriate indirect means of expressing the ordinal notions of 1st, 2nd, 3rd, ......(t – 1)th, of t members, respectively, in an analytic manner.
Now let's conclude this entry by no looking once more on the alternative Type 2 definition of 3 i.e.
3 = 1st + 2nd + 3rd
Now the last entry here i.e. 3rd of 3 = 1. And the other two i.e. the 1st of 3 and the 2nd of 3 are given by the other two roots as – 1/2 + .866i and – 1/2 – .866i respectively.
So now, when the holistic (qualitative) notion of 3 is indirectly converted in an analytic manner,
3 = – 1/2 + .866i – 1/2 – .866i + 1 = 0.
What this means is that the true qualitative notion of ordinal numbers has no strict meaning in quantitative terms. Because such ordinal rankings necessarily relate to the qualitative nature (i.e. interdependence) of numbers, they thereby lack an independent quantitative identity.
In fact this complements our earlier (Type 1) finding i.e. 3 = 1 + 1 + 1, where the separate units (as homogeneous) strictly thereby have no qualitative identity.
Of course the truth is that both quantitative independence (in cardinal terms) and qualitative interdependence (in ordinal terms) are merely relative notions that can only be properly understood in a dynamic interactive manner.