On the one hand, we can start with the natural numbers 2, 3, 4, 5, 6, ....(as base quantities in Type 1 terms) with the unique prime factor configurations shown for each number.
However we can then ultimately show these same numbers 2, 3, 4, 5, 6, ....(as dimensional exponents in Type 2 terms) in every case of the number 1/2, representing the proportion of the combined sum of prime factors (in relation to the number system as a whole) corresponding in each case to where 1 or more primes occur at most 2, 3, 4, 5, 6, .... times!
So notice the complementarity! In the former case where numbers represent base quantities - once again for the number expression an, a represents the base and n the dimensional aspects of number respectively - 2, 3, 4, 5, 6, ... represent a "part" meaning as individual independent natural numbers; however in the latter case, where numbers now represent the dimensional aspect, 2, 3, 5, 6, ... have a "whole" meaning through representing the characteristics associated with a collective group of numbers.
And of course the true significance of this is that the very nature of number is inherently dynamic with complementary aspects that interact in a merely relative fashion, i.e. as relatively independent (in quantitative terms) and relatively interdependent (in qualitative terms) respectively.
Also it seems to me that this represents but an equivalent way of expressing the Riemann Hypothesis which also entails 1/2, where - assuming it is true - all the non-trivial zeros lie on the straight line through 1/2. In this case, in somewhat complementary fashion, 1/2 in our example is raised to all of the natural numbers (excluding 1).
So putting it another way! Assuming the truth of the Riemann Hypothesis, then it follows that the probability that a prime number factor chosen at random belongs to Class 1 (where all numbers are composed of combinations of single i.e. non-repeating prime factors) = 1/2.
It then also follows that the probability that such a factor chosen at random belongs to Class 2 (where 1 or more primes occur at most 2 times) = 1/4, that the probability that such a factor belongs to Class 3 (where 1 or more primes occur at most 3 times) = 1/8, and in general terms the probability that such a factor belongs to Class n (where 1 or more primes occur n times) = 1/2n.
Another interesting relationship can be mentioned.
As we have seen, the probability that a random prime factor will belong to Class 22 (where 2 or more primes can occur at most 2 times) = 1/8 (i.e. 1/23) of the probability that a random factor will belong to Class 21 (where 1 or more primes can occur at most 2 times).
Then the probability that a random prime factor will belong to Class 32 (where 2 or more primes can occur at most 3 times) = 1/32 (i.e. 1/25) of the probability that a factor will belong to Class 31 (where 1 or more primes can occur at most 3 times).
And in like manner, the probability that a random prime factor will belong to Class 42 (where 2 or more primes can occur at most 4 times) = 1/128 (i.e. 1/27) of the probability that a factor will belong to Class 41 (where 1 or more primes can occur at most 4 times).
Thus in general terms, the probability that a random prime factor will belong to Class n2 (where 2 or more primes can occur at most n times) = 1/22n – 1 of the probability that a factor will belong to Class n1 (where 1 or more primes can occur at most n times).
So here this particular ratio involves 1/2 raised to each of the odd natural number dimensions (other than 1).
Of course, as we have seen before, hybrid cases can arise.
For example when 1 or more prime factors occur at most 3 times, other factors can thereby occur at most 2 times.
So one might pose the question! What is the probability that a random prime factor chosen from Class 31 (where 1 or more primes can occur at most 3 times) will belong to that subgroup that is associated with 1 or more other primes occurring at most 2 times?
Now as we know the probability of Class 2 (that 1 or more primes will occur at most 2 times) = 1/4.
Therefore this is simply the answer that we are looking for here. In other words 1/4 of all prime factors belonging to Class 3 (where 1 or more primes can occur at most 3 times, will also have Class 2 characteristics (where 1 or more remaining primes occur at most 2 times).
In fact to test this, I looked at the prime factor composition of the 1000 numbers from 400000000001- 400000001000 and the fraction that I obtained was 1/3.985. So even with a limited amount of sampling, remarkably close approximations emerge quite quickly!
Now in order to move from the initial results regarding the frequency of numbers sharing certain prime factor configurations (all based on the Riemann zeta function defined for the natural number integers), to these simple fractions (all based on powers of 2) a reverse use of the Riemann function is necessarily involved (where irrational results cancel out).
For example let us look at that relatively simple case, where 1 or more primes can occur at most 2 times.
Now, as we have seen, the probability that a number chosen at random will conform to this prime factor configuration is given by the Riemann zeta function as 1/ζ(3) – 1/ζ(2) = .832 – .608 = .224 (approx).
Now this probability (.25) is slightly higher than (.224). This therefore implies that the average frequency of factors for this particular number configuration is thereby greater than with the number system as a whole.
So this average frequency (relative to the number system as a whole) = 1/4 divided by 224
= {ζ(2) * ζ(3)}/4{ζ(2) – ζ(3)} = 1.116
Therefore the average frequency of occurrence with respect to the number of prime factors in any given prime factor configuration is likewise governed by the Riemann zeta function (for the positive integers). However it occurs in a "miraculous" complementary fashion to the corresponding frequency of occurrence of the numbers conforming to the various factor configurations.
In other words there is a horizontal/vertical type complementarity at work both externally with respect to the behaviour of the primes with respect to the number system as a whole and internally with respect to the corresponding behaviour of prime factors within each number.
And though these in isolation are determined by the irrational number behaviour (with respect to both aspects) of the Riemann zeta function, yet remarkably they then cancel out through bi-directional dynamic interaction with each other to leave the rational features we associate with the primes and other integers.
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