Yesterday I mentioned how deviations could be eliminated with respect to the Zeta 2 Function. (We will outline the precise nature of this Function in a moment).

Now once again, this Function provides a method of measuring the distribution of natural numbers among the primes (as the reverse of the Zeta 1 where the measurement is with respect to the distribution of the primes among the natural numbers).

So associated in this second Function with each prime number is a set of roots which ranges over the natural numbers in ordinal terms from 1 up to and including the prime number in question.

These complex valued roots can for convenience be broken into cos and sin parts with measurements taken in an absolute fashion.

This enables us therefore for any prime number p, to calculate the mean value of the sum of both cos and sin parts.

This value (for both sin and cos) approaches 2/π (i/log i) quite quickly. Also both cos and sin values line up on either side of this value in a manner where the ratio of cos to sin - again in absolute terms - quickly approaches .5.

We then saw how deviations could be eliminated in a manner that involves the square of the prime numbers.

In fact what this demonstrates - in a manner that is the counterpart to the non-trivial zeros for Zeta 1 - is that every prime number (as an individual member) potentially makes a contribution to the elimination of this deviation. So once again we cannot divorce quantitative from qualitative considerations.

Again in the Zeta 1 case the non-trivial zeros embody in a quantitative manner, the holistic collective nature of the primes. They thereby eliminate deviations with respect to their general occurrence (among the natural numbers) and the precise individual identity of the primes.

Here we have the reverse situation where the individual primes eliminate the deviations with respect to the collective behaviour of the natural numbers (as ordinal prime roots of 1) which directly is of a qualitative nature.

However I only considered the cos aspect in yesterday's contribution. Though this real part of the roots does indeed make the more decisive contribution, the sin aspect is also involved. So just like in atomic physics particles also have a wave aspect, likewise, the cos aspect relating to individual primes also has a collective wave aspect that influence the deviation.

So in principle the complete explanation of the deviations involves consideration of both aspects!

We now will look more closely at the form of this Zeta 2 Function.

We start with the simple equation expression and in the manner of Zeta 1 we will use s (though this time representing the base rather than the dimensional number!)

So s^n = 1.

Therefore 1 - s^n = 0

Now (1 - s^n)/(1 - s) = 1 + s + s^2 + s^3 + s^4 +.....s^(n + 1) = 0.

Therefore - expressed in the standard reduced manner n → ∞, then,

(1- s^n)/(1 - s) = 1 + s + s^2 + s^3 + s^4 +..... = 0.

or alternatively,

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

1 - s = 0 represents the 1st. where s = 1, and is always a solution.

In this sense it is trivial. Therefore if we divide by 1 - s,

We now have

1 + s + s^2 + s^3 + s^4 + ..... = 0. This expression contains all other non-trivial root solutions (i.e. except 1)

So this is Zeta 2. It is of course a well-known Function, but its hidden circular type properties hidden within Zeta 1 have been greatly overlooked!

Now the Zeta 1 Function by contrast is

1/1^s + 1/2^s + 1/3^s + 1/4^s +.......

If we invert each term of this Function we then get

1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) +......

Now there is an obvious complementary significance with respect to both Zeta 1 and Zeta 2 (which makes them identical in this sense)!

When s is used as base number in Zeta 2, it is used by contrast as a dimensional power in Zeta 1

Where the natural numbers are used as dimensional powers in Zeta 2, they are by contrast used as base numbers in Zeta 1!

Finally in their present form what is positive for the dimensional power in Zeta 2, is negative in Zeta 1 (and vice versa).

This is of great significance when we come to interpret numerical values on the LHS for s < 0 in the Zeta 1 (where they can be given no coherent quantitative interpretation)! In fact the reason is that values now actually confirm to the hidden Zeta 2 Function (which is indeed quantitatively defined for s < 0)!

Now you might remember that the very basis of my alternative (Type 2) number system requires such switching of base and dimensional numbers with respect to Type 1!

And this is precisely what happens here with these two formulations of the Zeta Function.

So, we can see that this alternative system, I have been dealing with in the past few days with a direct view to qualitative interpretation is simply the Type 2 counterpart of the Type 1 Riemann Function.

Thus in a sense it has the power - as we have seen - to turn everything on its head.

From a Type 1 perspective, we are accustomed to understanding one-way in sole consideration of the distribution of the individual (cardinal) primes among the natural numbers!

However, from the Type 2 perspective, we are led to see their hidden complementary aspect, where we now have the distribution of the individual (ordinal) natural numbers among the primes (as representing the prime roots of 1).

And this clearly brings out the key fact that there is a qualitative as well as quantitative aspect to the primes (and indeed also the natural numbers).

Now, when we then bring both Type 1 and Type 2 approaches together, we begin to appreciate that the prime and natural numbers are in fact complementary mirrors to each other and indeed are ultimately identical!

Again we can perhaps bring this out in a striking manner using our crossroads analogy.

Using - realtively - isolated frames of reference we can deal with Type 1 and Type 2 in seemingly unambiguous terms.

Thus from the Type 1 perspective, natural numbers are derived from the primes! From the Type 2 perspective, the primes are derived from the natural numbers (again as comprising the collection of roots of each prime number).

So from either perspective in isolation, unambiguous type connections seemingly can be made just as a person can unambiguously identify left or right turns using one isolated direction of movement.

However when we simultaneously bring together both aspects (as interdependent) all such unambiguous distinctions break down. So now depending on direction, what is left can be right and what is right can be left.

It is likewise exacty similar as regards the relationship as between the primes and natural numbers. When, as in a Type 3 approach, both aspects of interpretation (Type 1 and Type 2) are combined, the prime and natural numbers can no longer be clearly distinguished. So what we distinguish as prime and natural numbers are in fact just two aspects of the same mutual identity. What is prime from Type 1, is natural from Type 2; and what is natural from Type 2, is prime from Type 1.

And as Type 3 combines both aspects, the prime and natural numbers are understood as ultimately identical (in both quantitative and qualitative terms).

I mentioned before how in the Type 1 approach, each prime number (as cardinal) could be given a natural number ranking (as ordinal).

Then again in the Type 2 (now reversed) each natural number (as cardinal within this qualitative system) likewise can be given a prime number ranking (as ordinal).

So really what is prime or natural is purely dependent on arbitrary reference frames. So the two aspects, which in truth are ultimately identical as perfect mirrors to each other, only appear distinct as prime or natural numbers respectively, when we impose arbitrary reference frames for their interpretation!

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