Precise Nature of Zeta 1 Zeros

I wish to probe further here as to the precise nature of the Zeta 1 non-trivial zeros.

We will illustrate with a simple example where we start with the Zeta 2 finite expression for the 2 non-trivial roots of the 3 roots of 1, which again is represented as,

1 + s¹ + s² = 0.

Once again we can give values to the 3 terms of the expression by substituting s with the 2 non-trivial roots of s
i.e. – .5 + .866i  and – .5 – .866i respectively.

So when we substitute the first value for s (– .5 + .866i ) , the three terms on the RHS of the equation are

1,  – .5 + .866i  and – .5 – .866i .

Then we sustitute the second value for s (– .5 – .866i ), the three terms on the RHS of the equation are 

 1,  – .5 – .866i  and – .5 + .866i   respectively.


So with t here - representing the 3 roots of 1 - the formula for the total number of terms generated for the 2 non-trivial roots = t * (t - 1) = 3 * 2 = 6.

It is apparent that we are referring here to the natural numbers as representing dimensional values (i.e. exponents or powers) rather than base quantities. In other words, number here corresponds to Type 2 rather than Type 1 interpretation.

Thus for example the t - 1 non-trivial roots (of the t roots of 1) would be represented in Type 2 terms through the Zeta 2 Function as,

1 + s¹ + s² + s³ +.... +  s^{t – 1} = 0 with again the total number of values generated on the RHS of the expression

= t * (t - 1).

If however we concern ourselves with respect to these dimensional natural numbers (i.e. from 1 to t) with roots corresponding to the average spacing between primes (with respect to those natural numbers), we obtain

(log t - 1) non-trivial roots.

Then when we continue with these roots in regular groupings of log t - 1, through all the t - 1 non-trivial roots of 1, the total number of possible values = t * (log t - 1).

These non-trivial roots of t, with respect to the Zeta 1 Function, relate to circular values i.e. drawn on the unit circle in the complex plane (where the sum = 0).

So again with respect to 1 + s¹ + s² = 0, as the expression for the 2 non-trivial roots,

we have for s = – .5 + .866i  

1  – .5 + .866i   – .5 – .866i  = 0 and for s = – .5 + .866i  

1 – .5 + .866i  – .5 + .866i   = 0.

However when we express these values on a linear scale the sum of terms ≠ 0

Now in the unit circle when r = 1, the circumference has length 2π.

Therefore to convert on the linear scale to single units we divide by 2π.

Therefore where we now consider non-trivial roots as the average gap between primes up to t where this average gap = log t/2π = 3, then t/2π = 20.0855 (with t = 126.20116).


Thus on this basis the number of non-trivial zeros up to t

= 20,0855 (3 – 1) = 40.171 (where the actual value is in fact 40).

 

So the heights in this case (with respect to the Zeta 1) actually correspond to linear type measurements with respect to circular Zeta 2 values relating to the circumference of the unit circle in the complex plane. To standardise this value in a linear fashion with respect to single linear units, we thereby divide by 2π.

One final important point relates to the fact that these linear values of t (for Zeta 1 non-trivial zeros) are measured on an imaginary - rather than real - number line.

The reason for this is that we are using a linear (quantitative) way of indirectly representing a circular type relationship (which inherently carries a holistic meaning of a qualitative nature).


And the appropriate way of carrying out this indirect linear translation is with respect to imaginary (rather than real) units.

Indeed when one properly understands the nature of the Zeta 1 non-trivial zeros, by definition, they must all lie on a straight line as the indirect linear representation of number interdependence which is inherently circular (of a holistic qualitative nature).