As we have seen, the non-trivial zeros represent the points on an imaginary number scale, where the primes and natural numbers approach mutual identity.
Strictly, this relates in absolute terms to an ineffable reality as the identity of both form (1) and emptiness (0).
So the identity of the primes with the remaining natural numbers, necessarily pertains to a phenomenal reality, where both are identified in a relative approximate manner.
Thus, the non-trivial zeros, from the enlarged perspective that I have adopted, represent the Type 1 presentation of the Type 3 nature of mathematical interpretation.
So this represents an attempt to represent the set of non-trivial zeros as independent points drawn through ½ with respect to the real axis, on an imaginary number line.
Once again, precisely speaking, these points represent the (relative) identity of the interdependence of both prime and natural numbers.
However, there is no way of meaningfully intuiting the true nature of these zeros from this perspective.
Fortunately, the alternative Type 2 presentation does allow for a remarkably simple illustration of the profound nature of this two-way identity of primes and natural numbers.
As we have seen the Type 2 aspect of the number system relates to the circular representation of number arising from the extraction of the n roots of 1 in quantitative terms. These numbers then can be represented as equidistant points on the circle of unit radius in the complex plane.
From a quantitative perspective they are considered as - relatively – independent; however from the complementary perspective, they have a new qualitative meaning as representing the interdependent nature of number.
Thus n has a Type 2 interpretation as a measurement of the (internal) interdependence of its n circular points (which are independent from a quantitative perspective).
This qualitative interdependence can be easily appreciated through combining the circular numbers representing the n roots of 1 (through addition).
For all values of n (except 1) this sum = 0.
For example the 4 roots of 1 are 1, – 1, i and – i respectively with their sum = 0.
This simply implies that the qualitative notion of interdependence has no strict meaning in quantitative terms. Indeed this illustrates the holistic - as opposed to the analytic - interpretation of number (for which the Type 2 aspect is directly suited).
Of course with the standard Type 1 linear aspect (based on 1-dimensional interpretation where n = 1) the one root of unity = 1 in quantitative terms. So the sum of roots also = 1. This again graphically illustrates how the Type 1 aspect - which defines all conventional mathematical interpretation - directly reduces the notion of qualitative interdependence in a quantitative independent manner!
The Type 2 aspect of mathematical interpretation is inherently of a dynamic interactive nature, where the ordinal identity of the internal units of each number, are defined in both linear and circular terms (as the relationship of quantitative to qualitative, and qualitative to quantitative respectively).
So 4 as a (whole) number integer has 4 internal constituents as 1st, 2nd , 3rd and 4th members of 4 respectively.
These 4 number members are then represented as equidistant points on the circumference of the circle of unit radius in the complex plane.
So each number as a point in the complex plane, can be seen to simultaneously lie on a straight line (i.e. trough the radius drawn to each point) while equally lying on the circular circumference.
Likewise we equally give an interpretation that now combines both linear (quantitative) and circular (qualitative) notions.
So we start the Type 2 aspect with 1 – s^t = 0, as the equation for calculating the t roots of 1.
Now 1 – s represents the one trivial solution, for s = 1, which always represents one of the t roots.
Dividing by 1 – s, we obtain
1 + s + s^2 + s^3 + ……+ s^(t – 1) = 0.
Then multiplying by s we obtain,
s^1 + s^2 + s^3 + s^4 +……+ s^t = 0
Then apart from the solution s = 0, the other solutions represent the t – 1 non-trivial solutions of this equation.
I term this equation the Zeta 2 Function, which complements the standard Riemann Zeta Function that is appropriate for interpretation according to the Type 1 aspect, which in this context I term Zeta 1.
So the Zeta 1 Function which extends over an infinite number of terms (in accordance with the reduced nature of the Type 1 aspect) can be written,
1^(– s) + 2^(– s) + 3^ (– s) + 4^ (– s) + ……. = 0
The corresponding Zeta 2 Function which extends over a finite number of terms, involves the reverse complementary positioning of both the base natural numbers and dimension (s) with respect to Zeta 1.
So s now becomes the base number in Zeta 2 and the natural numbers 1, 2, 3, 4,… its corresponding dimensional numbers.
Once again with the Type 1 aspect the primes (as cardinal quantities) are represented as the building blocks of the natural numbers.
Then in the Type 2 aspect this relationship is reversed with the natural numbers (as ordinals with qualitative characteristics) represented as the building blocks of the primes.
So the Zeta 1 is in line with the former Type 1 representation, whereas the Zeta 2 (where base and dimensional numbers with respect to Zeta 1 are reversed) enables the latter Type 2 representation.
So correctly understood we generate two sets of non-trivial zeros.
In the Zeta 1, the non-trivial zeros represent the solutions for the dimensional number s, which are all complex in nature and of the form ½ + it and ½ – it respectively (with t transcendental in nature).
In the Zeta 2, the non-trivial zeros represent the solutions for the base number s, which again are complex in nature of the form a + it (which though generally irrational is algebraic in nature).
Now as we have seen, the non-trivial zeros relating to Zeta 1, represent points on an imaginary number line (through ½) where the prime and natural numbers are identical (in a relative manner).
The very nature of Type 1 interpretation is based on the notion of number representing analytic recognition in terms of independent quantities. However the authentic nature of these non-trivial zeros requires - by contrast - their holistic recognition (in the qualitative recognition of interdependence).
Therefore Type 1 interpretation cannot provide an appropriate appreciation of what such non-trivial zeros represent. Thus - in a quite literal manner - their nature cannot therefore be appropriately intuited using Type 1 interpretation.
However this situation is happily reversed through Type 2 interpretation.
It combines both the linear and circular notions of number (as representing quantitative independence and qualitative interdependence respectively) in a dynamic interactive manner, thereby providing a simple means of demonstrating the nature of the identity of prime with natural number behaviour.
Now doing this requires a fascinating complementary form of reductionism to that which defines Type 1 interpretation!
As we have seen, Type 1 entails reductionist interpretation, with quantitative results all defined qualitatively in 1-dimensional terms.
Now in reverse fashion in Type 2, we can provide a corresponding form of quantitative reductionism, where all results (representing the qualitative nature of interdependence) are defined quantitatively in 1-dimensional terms.
This implies that with respect to the non-trivial solutions of the Zeta 2 for s, we simply ignore negative signs (defining all numbers as positive) and likewise ignore imaginary signs (treating all numbers as real).
In this way we can represent the solutions for s (as points on the unit circle in the complex plane) as positive real numbers.
So with the dimension t as prime, we can represent its natural number ordinal members in positive real number terms.
I have explained elsewhere that when we then obtain the mean average of these numbers that both the cos and sin parts approximate ever more closely to 2/π as t increases in value representing ever larger primes.
We also saw that 2/π = i/log i, giving us a fascinating Zeta 2 Prime Number Theorem (to complement the n/log n of Zeta 1).
Also the ratio of the deviations of the average cos and sin values from 2/π (= i/log i) approaches ever closer to .5 (in absolute terms) as t (representing primes) increases.
Now what is truly fascinating is that if now instead of the all the natural number ordinal members of t as prime (measured as I have explained in a quantitative 1-dimensional manner as real positive numbers) , we concentrate only on the prime number ordinal members, the mean average of these members will likewise approximate to 2/π (= i/log i).
So for example, 127 is a prime number.
So I obtained the 127 roots of 1 (representing the ordinal natural numbers from 1 to 127) converted them in a reduced 1-dimensional quantitative manner and then obtained the mean average for both cos and sin parts which already approximated in both cases very closely to 2/π (= i/log i).
I then obtained the 31 prime numbered roots (from 1 to 127) again calculating their mean average for both cos and sin parts.
Again in both cases the answer approximated quite closely to 2/π (= i/log i).
I also examined behaviour with respect to the natural and prime numbered roots of several other values of t (where t is a prime number) with similar results emerging in both cases.
So I would confidently conjecture that in the limit where t (as prime number) →∞, that both the mean average for both natural and prime number ordinal roots = 2/π (= i/log i).
In strict terms however as we can only meaningfully define the roots of 1 (with respect to finite values of t) this identity of natural and prime numbers will necessarily be of an approximate relative nature.
Thus in this way, we have demonstrated the identity of both the prime and natural numbers (in a manner that relates to the interdependent nature of both their linear and circular behaviour).
Therefore in a direct sense the non-trivial zeros associated with Zeta 2 can demonstrate the identity of both the prime and natural numbers in a holistic ordinal manner.
By contrast the non-trivial zeros associated with Zeta 1 can only demonstrate this identity - now in a quantitative analytic sense - in an indirect manner that does not directly intuit with Type 1 understanding.
So from a Type 3 perspective both the Zeta 1 and Zeta 2 represent two sides of the same coin, both of which - for full comprehension - must closely interact with each other in the mutual two-way identity of the primes and the natural numbers (and the natural numbers and the primes) in both quantitative and qualitative terms.