We concluded the last contribution with the remarkable finding that

i^i = e^(- pi/2), which is a real number!

Now if we take natural logs of each side

then i(log i) = - pi/2,

therefore 1/i(log i) = - 2/pi.

So, - i/log i = - 2/pi

Thus i/log i = 2/pi.

As we know the prime number theorem relating to the general frequency of the primes among the natural numbers is most simply expressed as n/log n (with the proportionate frequency increasing as n becomes larger).

So by allowing n to become progressively larger we have the linear quantitative attempt to reach the infinite (in an actual manner).

Now properly understood i represents the corresponding holistic notion of the infinite where one attempts to appropriate it (in a potential manner).

We can see this in the common psychological appreciation of the imaginary as something that emanates from the holistic unconscious to be embodied in an actual (conscious) manner.

Therefore understood in this light i/log i represents the qualitative correspondent to the prime number theorem!

Now in quantitative terms, we attempt to understand prime numbers from a Type 1 perspective as base quantities i.e.

2^1, 3^1, 5^1, 7^1,.......

However the prime numbers have in Type 2 terms a corresponding qualitative interpretation as dimensions i.e.

1^2, 1^3, 1^5, 1^7,......

In an inverse quantitative manner we can obtain the circular structure of these dimensions through obtaining the reciprocal roots.

So therefore we can attempt to find the 2 roots, 3 roots, 5 roots, 7 roots of unity and so on for each of the prime numbers.

Now in this approach we consider all roots which will have both a real and imaginary component.

With respect to both parts we take the values in an absolute manner (ignoring negative signs). Then we sum up both parts (both real and imaginary taken separately) and then obtain the average.

We can demonstrate simply here for p = 3.

There are 3 corresponding roots of 1 involved i.e. 1, - .5 +.866i and - .5 - .866i

Now ignoring negative signs the sum of the real part here = 1 + .5 + .5 = 2.

Therefore the mean average = 2/3 = .6666.. .

Then taking the magnitude of the imaginary part (ignoring the i) the sum = .866 + .866 = 1.732

Therefore the mean average = 1.7321/3 = .57735...

Now the remarkable finding here as the value of p increases is that the mean value of the absolute quantitative value for both the real and imaginary parts converges on 2/pi = .636619772...

We can readily find all these values through use of the Euler Identity,

e^(2*i^pi) = cos (2*pi) + i sin (2*pi).

So the 3 roots of 1 - where the dimensional numbers are 1/3, 2/3 and 3/3 respectively are calculated in this manner as

cos {(2/3)*pi} + i {sin(2/3)*pi},

cos {(4/3)*pi} + i {sin(42/3)*pi}, and

cos {(6/3)*pi} + i {sin(6/3)*pi}.

As p becomes ever larger the mean value (for both parts) approximates ever closer to i/log i.

So we seem here in fact to have a circular number equivalent to the prime number theorem (that is couched in a linear quantitative manner).

However it does not end here!

We can see from our example above that the mean value for the real part = .6666.. and the imaginary part = .57735... respectively.

Therefore the mean value for the real part exceeds 2/pi and the corresponding value for the imaginary part is less than 2/pi respectively.

In fact looking at the absolute differences the value is .6666... - .636619772..

= .03005 (approx)... for the real part

and .636619772... - .57735... = .05927(approx)

Now the ratio of this difference real/imaginary = .03005/.05927 = .507 (approx)

This already seems very close to .5 (sound familiar!)

In fact as the value of p increases the ratio of this difference does indeed tend ever more closely to .5!

So once again what we are stating is this!

As p becomes larger, the absolute mean value of both real and imaginary prime roots of 1 converges ever closer to 2/pi (i.e. i/log i).

Insofar as a difference remains the ratio of absolute deviation of real/imaginary value converges ever closer to .5.

Just as Riemann came up with improvements to prediction of the general frequency of the primes, I experimented with my own improvements.

Now let's say that we wish to calculate the deviation of the absolute mean value of the real part from 2/pi for a larger value of p (say 127).

What we do here is to multiply the deviation (for p = 3) by (p/p1)^2 where p = 3 and p1 = 127.

So this gives us .03005 * (3/127)^2 = .000016768 (approx)

Considering that we are using such an early prime number = 3, this compares extremely well with the true deviation = .000016232 (approx).

In fact this and any other calculation can be significantly improved by then dividing the result by (1 + d) where again in this case d = .03005.

So in in this case we can then approximate the true deviation as .000016277..

Thus our answer is already correct to 3 significant figures!

Predictions can be greatly improved through using the deviations of later prime numbers.

For example if we use the deviation associated with p = 61, we can calculate the corresponding deviation associated with p = 127 correct to 6 significant figures!

Variations of this approach can be used likewise to predict corresponding deviations associated with the imaginary part!

Now in principle just as the non-trivial zeros of the Riemann Zeta function can be used to correct the deviations from the actual with respect to the general distribution of the primes, a corresponding method should exist enabling - ultimately - an exact mean of absolute prime root values for both real and imaginary parts.

So just as the Riemann Hypothesis is used to accurately calculate the average number of primes (in a linear context), this latter approach is used to calculate the average value of these primes in a circular context.

And in each case .5 plays a key role. In fact the circular version provides a key indication of what the .5 actually represents.

As we have seen in the circular context .5 represents ratio of (real) cos to (imaginary) sin values which indicates in turn a quantitative (analytic) to qualitative (holistic) connection.

And this is really what The Riemann Hypothesis is all about i.e. in establishing the condition necessary for full reconciliation of both quantitative and qualitative aspects of interpretation!

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