Emergence of Zeta 1 and Zeta 2 Functions

We started with the two basic approaches to the natural number system representing the Type 1 and Type 2 aspects of mathematical understanding respectively.

Once again Type 1 is defined in terms of the natural numbers defined as quantities (expressed with respect to the invariant default number dimension of 1).

So here we have

1^1, 2^1, 3^1, 4^1,........

The Type 2 is then defined in complementary terms with respect to the natural numbers - relatively - expressing qualitative dimensions (expressed with respect to the invariant default base number quantity of 1).

So here by contrast we have

1^1, 1^2, 1^3, 1^4,......

When considered in relative isolation from each other the Type 1 can be directly associated with appreciation of the (specific) quantitative aspect of number relationships, with Type 2 then associated - by contrast - with the (holistic) qualitative aspect.


However in Type 3 understanding - where both quantitative and qualitative aspects progressively interact in understanding - the quantitative is seen to have a qualitative aspect and the qualitative a quantitative aspect respectively.


Bearing comparison with Quantum Mechanics, Type 1 provides the particle aspect of number understanding and Type 2 the wave aspect (in isolation).


However with Type 3 understanding the particle likewise has a wave aspect (and the wave aspect a particle aspect) respectively.

Now the findings of Riemann with respect to his Zeta Function that a harmonic wave aspect underlines the accepted particle approach to number is really a manifestation of what properly belongs to Type 3 understanding.

However because of the lack of any explicit Type 2 interpretation, Conventional Mathematics lacks the means to properly explain the true nature of this wave system.


Now the importance of the (accepted) Zeta Function is that it can be seen as a natural extension of the Type 1 system where the various natural number terms are expressed with respect to the dimensional number s (which can be any complex number) and then summed to infinity (in accordance with conventional reduced notions of the infinite)and set = 0.

So we now have 1^s + 2^s + 3^s + 4^s + ........ = 0

Now s is conventionally expressed (when the function is written in this form) with respect to a negative value.

i.e. 1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) + ..... = 0


I refer to this as the Zeta 1 Function (as it reflects simply the Type 1 approach).

Now the truly remarkable fact about this distribution is that the only value for which it remains undefined is where s = 1.

And as the the conventional Type 1 approach is defined in terms of the default dimension of 1, this actually implies that the Riemann Zeta Function cannot truly be understood from the conventional mathematical perspective.

And the simple reason for this is that the Riemann Zeta Function (when properly decoded) establishes the dynamic relationship as between the quantitative and qualitative aspects of the number system. So this clearly cannot be achieved within a merely quantitative interpretation of number!


However in line with the Type 2 approach is a corresponding Zeta 2 Function (that is complementary with Zeta 1).

So we start with the expression for obtaining the roots of 1,

i.e. 1 - s^n = 0

However 1 - s is a factor so that 1 - s^n = (1 - s){1 + s^2 + s^3 + ... + s^(n - 1)}.

Therefore,

(1 - s){1 + s^2 + s^3 + ... + s^(n - 1)} = 0

So dividing by 1 - s, we obtain

1 + s^2 + s^3 + ... + s^(n - 1) = 0.

Now once again this implies that 1 - s = 0 is redundant as a solution.

What this means in effect is that when we take 1 as the root of unity, it is not unique.


So for example, 1 represents one of the 3 roots of 1; however it equally represents one of the 5 roots of 1. Therefore though 3 and 5 are prime numbers the common root = 1 is not unique.

So again remarkably from the Zeta 2 perspective the one value for which the Function remains undefined is for s = 1.

So with respect to Zeta 1, The Function is undefined when s = 1 (with 1 here representing a qualitative dimensional value).

Then with respect to Zeta 2, the Function remains undefined when s = 1 (with 1 here - in complementary relative terms - representing a base quantitative value)!


So just as we cannot consider quantitative meaning (in isolation from qualitative (as with Zeta 1) we likewise cannot consider qualitative meaning in isolation from quantitative (as with Zeta 2).



So in the appropriate dynamic interpretation of the Zeta Function, both Zeta 1 and Zeta 2 aspects interact. So what is hidden from the perspective - say - of Zeta 1, can be explained with respect to Zeta 2 (and vice versa).

Indeed we could accurately say in Jungian terms, that when one aspect is made conscious and thereby known the other aspect remains hidden (and unconscious). Then when the latter aspect is made known the first aspect remains hidden.


Now just a couple of more points with respect to Zeta 2!

Strictly, to obtain direct comparability (in complementary terms) with Zeta 1 we must multiply by s. So s = 0 is also a solution (which really implies the quantitative answer in an approach that is intrinsically of a qualitative nature).


Secondly the Zeta 2 is necessarily of a finite - rather - than infinite nature (which again is complementary with Zeta 1).

Therefore with this modification

s^1 + s^2 + s^3 + ... + s^(n - 1) + s^n = 0

And when n is a prime number, the solutions to this equation (as roots) will produce unique values which then likewise have unique qualitative interpretations as the structure of their corresponding dimensional values.

To put this more simply, the solutions to the Zeta 2 function, establish the qualitative uniqueness of the natural numbers (excepting 1) among the primes.

So Zeta 1 relates to the quantitative uniqueness of the primes (among the natural numbers). Zeta 2 - in complementary fashion - relates to the qualitative uniqueness of the natural numbers (among the primes).

Thus properly understood - in dynamic interactive terms - the primes and natural numbers (and natural numbers and primes) are ultimately totally interdependent in an ineffable manner!

And this is the great central mystery that underlies the number system, the proper comprehension of which will change for ever the very nature of mathematical enquiry!