ζ

_{2}(s) = 1 + s

^{1 }+ s

^{2 }+ s

^{3 }+….. + s

^{t – 1 }= 0, for the primes (i.e. where t is prime).

Again the simplest solution is for t = 2 where thereby 1 + s

^{1 }= 0, (i.e. s = – 1).

So this represents therefore the first of - what I refer to as - the Zeta 2 zeros.

The various solutions of s, represent the t – 1 roots of 1 (excluding the common root of 1) which are unique for all prime numbers.

Therefore, when we identify the common root of 1 with the actual fixed notion of position (as t

^{th}) then the Zeta 2 zeros provide the unique means of defining all other ordinal positions i.e. 1st, 2nd, 3rd,....(t – 1)

^{th}.

Where prime roots are concerned, we can never have the repetition of any root values!

So in ordinal terms, we thereby have the seeming paradox that each prime is expressed by the individual uniqueness of its constituent natural number members (except 1) as "converted" in an indirect quantitative fashion.

This thereby complements - in a dynamic interactive fashion - the corresponding finding in cardinal terms, that each natural number is expressed by the unique combination of its constituent prime members (except 1).

This implies that a total circularity defines the relationship as between the primes and natural numbers, which thereby implies that ultimately both perfectly mirror each other in an ineffable manner.

We can then extend the Zeta 2 (finite) function - initially defined for primes - to all natural numbers.

However this implies its interaction with the corresponding Zeta 1 (Riemann) function, which is directly concerned with the relationship between the primes and natural numbers (in a cardinal manner).

Likewise, we can validly maintain that the Zeta 1 (infinite) intimately implies its corresponding interaction with the Zeta 2 function.

This is due to the fact that the unique manner by which each (composite) natural number is defined as the product of primes in cardinal terms, equally implies, in complementary fashion, the corresponding unique manner by which each individual prime is defined by its natural number members in an ordinal manner!

So ultimately when correctly understood - again in a dynamic interactive manner - both the Zeta 1 and the Zeta 2 functions are themselves complementary in two-way fashion with each other!

Again, we can perhaps see the huge limitation of the conventional approach with respect to the Riemann zeta function (i.e. Zeta 1 function) in that it attempts to view the relationship between the primes and natural numbers in a merely reduced one-way cardinal manner (as quantitative).

However, properly understood in dynamic interactive terms, the relationship between the primes and natural numbers (and natural numbers and primes) must be conceived in a bi-directional manner entailing both cardinal and ordinal aspects (in quantitative and qualitative terms).

So this thereby implies the corresponding interaction of both the Zeta 1 and Zeta 2 functions.

However, there is another fascinating way of showing this link as between the Zeta 1 and Zeta 2 functions!

As we have seen, we initially defined the Zeta 2 function in finite terms for primes. We then allowed this to be extended to the natural numbers (in finite terms).

Now finally we can extend the function in an infinite manner.

This can be best illustrated with respect to the simplest finite solution (i.e. where t = 2),

1 + s

^{1 }= 0. So s

^{1 }, i.e. s, = 1.

^{ }

^{}If we now extend this equation in an infinite manner, we get,

1 + s

^{1 }+ s

^{2 }+ s

^{3 }+….. = 0

So when s = 1, this implies

1 – 1 + 1 – 1 + ....

Therefore for an even number of terms the sum of this series = 0 and for an odd number = 1.

Therefore if we attach the same probability to the series having an even or odd number of terms, then the expected value = 1/2.

Though much more complicated to demonstrate, the expected value = 1/2 for any number of Zeta 2 zeros (as initially defined for the finite case) when the series is then extended in an infinite manner.

Thus the infinite Zeta 2 function,

1 + s

^{1 }+ s

^{2 }+ s

^{3 }+….. = 1/2. (However once again this represents the probabilistic notion of an expected value).

Therefore the fact that in the Zeta 1 (Riemann) function, all of the non-trivial zeros are postulated to lie on the imaginary line through 1/2 directly concurs with the value of the infinite Zeta 2 function!.

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