I also sought to highlight the dynamic complementary nature of these findings. The composite numbers represent the interdependent aspect of number (i.e. as being composed of constituent factors). The primes by contrast represent the independent aspect (i.e. in containing no constituent factors other than themselves and 1).
So the non-trivial zeros in being directly related to the factors of the composite numbers thereby represent the complementary shadow of the primes.
There is a direct link here also with psychological experience where both conscious and unconscious serve as dynamic complements of each other. So in Jungian terms the personal shadow projected by the unconscious, represents the unrecognised unconscious aspect (which properly complements conscious type understanding).
Thus we cannot hope to properly understand the relationship between the primes and the non-trivial zeros without equal recognition of the need to balance both conscious (analytic) and unconscious (holistic) type appreciation in mathematical understanding.
And as I have repeatedly stated in these blog entries, Conventional Mathematics - being based formally on merely conscious analytic type notions - remains in total denial of its unconscious shadow.
Thus the attempt is made to understand both the primes and the zeros in a merely absolute quantitative type manner, when in effect the relationship between them is of a dynamic relative nature being analytic (quantitative) and holistic (qualitative) in relationship to each other.
Thus when we view the primes from an analytic perspective in terms of their individual (quantitative) identity, the non-trivial zeros then collectively represent the holistic complement to the primes in qualitative terms.
Equally from the alternative perspective, when we view each non-trivial zeros from an individual (quantitative) perspective, the prime numbers as a collective group thereby represent the holistic complement to the zeros in qualitative
Therefore depending on perspective, which in dynamic interactive terms is always with respect to complementary (opposite) aspects of understanding, both the primes and the non-trivial zeros can be given both an analytic (quantitative) and holistic (qualitative) interpretation.
However these crucial relationships are rendered strictly meaningless when we attempt to view all number relationships in a merely analytic (quantitative) manner!
Now again in measuring the frequency of the non-trivial zeros, I was at pains to measure the number of factors contained by the composite numbers. And the rationale behind this approach was to include the number itself (when composite) as a factor while excluding the number 1.
So once again, from this perspective, the number 12 for example would contain 5 constituent factors (i.e. 2, 3, 4, 6 and 12).
However having completed these two blog entries, I then started to consider the related problem of finding an estimate for the accumulated sum of factors of the composite numbers.
So to illustrate this more clearly, I will demonstrate how this sum is arrived at with respect to the first 10 numbers.
Now four of these 2, 3, 4 and 6 are prime (which we can ignore) as indeed the number 1..
The factors of 4 are 2 and 4 with the sum = 6.
The factors of 6 are 2, 3 and 6 with the sum = 11.
The factors of 8 are 2, 4 and 8 with the sum = 14.
The factors of 9 are 3 and 9 with the sum = 12.
Finally, the factors of 10 are 2, 5 and 10 with the sum = 17.
So the accumulated sum of factors up to 10 = 6 + 11 + 14 + 12 + 17 = 60.
Now I continued on calculating these actual accumulated sums of factors up to 110.
I then came up with a simple formula 2n(n + 1)/π, that attempts to achieve consistency with respect to a close estimate of the actual values.
Up to n
|
Acc. Factor Total
|
Est. Total
|
% Accuracy
|
10
|
60
|
70
|
85.71 (over est.)
|
20
|
242
|
267
|
90.64 (over est.)
|
30
|
609
|
592
|
97.21 (under est.)
|
40
|
1111
|
1044
|
93.97 (under est.)
|
50
|
1706
|
1623
|
95.13 (under est.)
|
60
|
2618
|
2330
|
89.00 (under est.)
|
70
|
3428
|
3164
|
92.30 (under est.)
|
80
|
4439
|
4125
|
92.93 (under est.)
|
90
|
5653
|
5214
|
92.24 (under est.)
|
100
|
7112
|
6430
|
90.41 (under est.)
|
110
|
8382
|
7773
|
92.73 (under est.)
|
The accuracy here is far from stunning. However as we ascend the number scale it consistently seems to be predicting at over 90%. So we are still at a very low point on the number scale. Unfortunately it becomes progressively more difficult to manually calculate the sums of factors of the composite numbers as these increase. Also we would expect considerable local variations as the sum of factors of even just one highly composite number can make a considerable contribution to the the overall total.
However there is a distinct rationale as to how this formula was arrived at!
n(n + 1)/2 is the formula for the sum of natural numbers from 1 to n.
So for example the sum of 1 to 100 = 100(101)/2 = 5050.
Now if we multiply n(n + 1)/2 by 4/π , we get 2n(n + 1)/π, which is the formula I have used to estimate the accumulated sum of factors (of composite numbers).
So again the estimate for this accumulated sum up to 100 is 6430 (as against the actual total of 7112).
And 4/π, which is the link between the two formulae has a special significance from a holistic mathematical perspective.
Consider the following simple geometrical diagram!
Now if we measure the perimeter of the square and then divide by the circumference of the circle the answer = 4/π.
Alternatively, if we take the area of the square and divide by the area of the circle, again the answer = 4/π.
Now, from a qualitative holistic perspective, this relationship is deeply symbolic of the intersection of linear (quantitative) with circular (qualitative) understanding i.e. where notions of (individual) independence and (collective) interdependence coincide.
We will explore the deeper relevance of this connection in the next blog entry.
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