Wednesday, May 7, 2014

Important Connections!

We initially used the accumulated number of factors contained in the composite numbers to give a very close estimate of the frequency of the Riemann (Zeta 1) zeros.

This was based on the simple formula n(log n – 1). Again a delightful complementarity was in evidence here with the inverse version of this formula i.e. n/(log n – 1) giving a surprisingly accurate measurement of the corresponding frequency of primes (up to n).

Once again true appreciation of the complementary dynamics involved here requires appropriate holistic mathematical appreciation.

In earlier blogs in highlighted a remarkable fact in relation to the reciprocal of a number.

For example 4 in cardinal (Type 1) terms is more fully represented as 4 1.

What this implies is that the conventional (analytic) quantitative interpretation of a number entails understanding with respect to the default 1st dimension (where experiential polar opposites such as objective and subjective are absolutely separated from each other).

The reciprocal of 4 (i.e. 1/4) can be represented as 4 – 1.

This implies in holistic terms the direct negation of linear type conscious rational understanding in the generation of unconscious intuition.

Thus the important point to grasp is that the very dynamics by which one is enabled to move from whole to part (and part to whole) in experience requires holistic intuition.

Now once again the switch is made and we obtain 1/4 this is then quickly interpreted in the standard 1-dimensional terms (i.e. in a linear rational fashion).

So 1/4 is now more fully represented as (1/4) 1.

Then again in a reverse manner, to switch from this part to the corresponding whole notion of number, we obtain (1/4) – 1.
Once more this dynamic switch in experience entails the negation of linear type understanding in the generation of holistic intuition.

However when the switch is made the result is quickly reduced in a linear rational manner as 4 i.e. 4 1.

So the important point that is made that the very means by which one is enabled to switch from whole to part and in reverse manner part to whole notions in experience implies the generation of (unconscious) intuition.

And this can be generalised with respect to switching as between the fundamental polarities that necessarily condition all experience (including of course mathematical).

So rather that Mathematics being absolute in rational terms), properly understood, mathematical understanding is of a dynamic relative nature, entailing both quantitative (analytic) and qualitative (holistic) type appreciation. This equally implies that all mathematical understanding properly entails the dynamic interaction of both conscious (rational) and unconscious (intuitive) modes of appreciation.

Now, I have consistently remarked how the primes and Riemann (Zeta 1) zeros are of a complementary nature i.e. analytic and holistic with respect to each other.

Thus when the primes are understood in individual  terms as quantitative, the Riemann zeros then should be appropriately understood in collective terms as qualitative (i.e. as expressing the interdependent nature of the primes).
Likewise, in reverse manner, when the Riemann zeros are understood in individual terms as quantitative, the primes should then be appropriately understood in collective terms as qualitative (i.e. in their overall relationship with the natural numbers).

So this fact is beautifully demonstrated by the very formulae used here to estimate the frequency of primes and Riemann (non-trivial) zeros respectively.

Thus in moving from the frequency of Riemann zeros i.e. n(log n – 1)  to the corresponding frequency of primes we simply use the reciprocal of (log n – 1).
And then once again, to move from the frequency of primes to the corresponding frequency of the Riemann zeros, we simply revert back to the original number by once again taking the reciprocal!

Of course besides the Zeta 1 zeros we also have the (unrecognised) Zeta 2 zeros.

These arise simply as the finite solutions to the equation,

 ζ2(s)  = 1 + s1 + s2 + s3 + …. + s t – 1  = 0.

Put another way, it provides the t – 1 non-trivial roots of the t roots of 1.

As 1 is always a root of unity, in this sense it is a trivial root. However the other (non-trivial) roots are unique for all prime numbers.

So for example if we wish to calculate the 2 non-trivial roots (representing the Zeta 2 zeros) of the 3 roots of 1, we solve for,

1 + s+ s  = 0.

These two roots (correct to 3 decimal places) are – .5 + .866i and – .5 – .866i respectively.

Now the demonstration of the holistic interdependence associated with these zeros requires their incorporation with the default (trivial) root of 1.

So the sum of 1, – .5 + .866i and – .5 – .866i = 0 and indeed this result universally applies with respect to the sum of the t roots of 1!

However it was in the attempt to give a reduced quantitative expression to such (qualitative) interdependence that I considered an alternative method of measurement.

In this reduced form of measurement all magnitudes are considered in a positive real manner (with both negative and imaginary signs ignored).

Therefore in this modified approach, the sum of the 3 roots of 1 (i.e. the sum of the 2 non-trivial and 1 trivial Zeta 2 zeros)

= 1 + .5 + .866 + .5 + .866 = 3.732.

The average of the 3 roots = 1.244.

I found that the average value of all roots quickly converged towards 4/π (i.e. 1.273...) with both cos and sin parts converging towards 2/π  respectively.

So  we could then consider adding up all the roots for each number and then summing up the total sum of roots for each number up to n!
So for example if n = 100, this would thereby entail multiplying 4/π by (1 + 2 + 3 + ....100) i.e. by n(n + 1)/2.

Thus 4/π  * {n(n + 1)/2} = 2n(n + 1)/π.

This however is exactly the same formula we used to estimate the cumulative sum of factors up to n!

Thus there seems to be an important connection between the two approaches which we will explore in the next blog entry.

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