First there is the external aspect where - as we have seen - we can classify numbers with respect to various general configurations with which such factors occur e.g. that which I have identified as Class 2, containing all numbers where 1 or more primes occur at most 2 times.
However there is also an internal aspect where we concentrate now on the combined number of factors belonging to these configurations. So for example, we have again identified that - ultimately - 1/4 of the prime factors of numbers belong to that same configuration i.e. Class 2 (where 1 or more prime factors occur at most 2 times) with all the other major classes defined with respect to the simple terms of the geometric series 1, 1/2, 1/4, 1/8, 1/16, ... which in fact represents the Zeta 2 function,
ζ2(s2) = 1 + s21 + s22 + s23 + s24 +….,
where s2 = 1/2.
So with respect to the external (horizontal) aspect of the number system, the general frequency of occurrence of numbers according to the various prime factor configurations is governed by the Riemann (Zeta 1) function
ζ1(s1) = 1– s1 + 2– s1 + 3– s1 + 4– s1 +… ,
where s1 = a positive real integer.
Then with respect to the internal (vertical) aspect of the number system the general frequency of occurrence of the combined prime factor totals of these numbers is governed by a specific key version of the complementary Zeta 2 function (where s2 = 1/2).
And the truly important thing to recognise that this in fact offers confirmation of the inherently dynamic interactive nature of the number system where - ultimately - both external and internal aspects are determined in a holistic synchronous manner!
However the striking rational number features of the number system by no means end here.
So let us return now to further clarification with respect to our example as illustrated by the Class 2 system.
As we have seen in an external manner, the most general classification of numbers relates to the case where 1 or more primes can occur at most 2 times. Let us for greater clarity refer to this as Class 21!
However - as we have seen - a more restricted classification relates to the case where 2 or more primes can occur at most 2 times. Let us refer to this as Class 22.
Now if we were to count up all the prime factors for those numbers in Class 22 and then express this as a fraction of the combined total of factors for Class 21 the result = 1/8.
Now of course, rather like the tossing of an unbiased coin, where for example with 1000 tosses we would be unlikely to get 500 heads, equally if we counted all the factors belonging to both classes over a range of 1000 numbers, the ratio would be likely to differ somewhat from 1/8.
However what is remarkable is how quickly the correct result tends to emerge. For example in one sampling I attempted over a range of 1000 numbers, the combined sum of factors of Class 22/Class 21 = 1/7.91.
In another sampling over a range of 2000 numbers I obtained the answer = 1/7.99!
There is perhaps an even simpler way of expressing the above result!
If one were to extract a prime factor from Class 2, the probability that it would belong to Class 22 = 1/8. And since the probability that a prime factor will belong to Class 2 = 1/4 (with respect to the number system as a whole), this means that the probability that a factor will belong to Class 22 (where 2 or more prime factors can occur at most 2 times) in relation to all factors = 1/32!
Now, perhaps I should say something here regarding my overall rationale. As should be clear, I am not presenting these findings a proven results. Rather I am deliberately using the holistic aspect - based on the complementary interaction of twin aspect of the number system - to generate striking findings that would not easily emerge through the standard analytic approach.
So through and through my findings have primarily emerged from intuitive insight as to what relationships should be from a holistic persepctive, combined then with often - extensive - verification from the prime factor tables at my disposal. So I only offer results, when experimental evidence is in support of intuitively inspired conjectures (arising from an inherently dynamic appreciation of the number system).
Of course, I have no problem with acceptance of the importance of mathematical proof. Rather the conventional analytic approach often acts as a straight jacket preventing more original insights regarding the nature of the number system from flourishing. And this is especially true regarding the relationship of the primes to the natural number system.
In fact - as I will demonstrate in a simple illustration in a future approach - the holistic approach can itself sometimes suggest novel ways of proving - in an analytically acceptable manner - striking mathematical results.
In fact - even more remarkably - all possible configurations regarding the ratio of factor counts (according to varying prime factor combinations) will result in similar rational number results.
For example if we were now to look at the even more restricted subclass of Class 23, where 3 or more prime factors occur at most 2 times, the fraction of prime factors belonging to this class in relation to Class 222 = 1/16.
And in turn Class 24/Class 23 = 1/32, Class 25/Class 24 = 1/64, and so on.
Now if we turn to Class 3, we can define the most general case as Class 31 (where 1 or more prime factors occur at most 3 times).
Then the more restricted class where 2 or more primes occur at most 3 times = Class 32.
Then with respect to the ratio of the count of prime factors Class 32/Class 31 = 1/32.
For subsequent ratios we then keep increasing by a factor of 4.
So Class 33/Class 32 = 1/128, Class 34/Class 33 = 1/512, and so on.
In fact if we look at the most general classes (where 1 or more primes can occur at most 1, 2, 3, 4,...n times), a remarkable feature is evident.
In other words the resulting probability that a random prime factor will belong to each of these classes in turn can be expressed as the reciprocal of 2 raised in turn to each of the natural integers.
So the probability that a factor will belong to Class 1 (where each prime occurs but once) = 1/21.
The probability that a factor will belong to Class 2 (where 1 or more primes can occur at most 2 times) = 1/22.
The probability that a factor will belong to Class 3 (where 1 or more primes can occur at most 3 times) = 1/23.
And in general terms the probability that a factor will belong to Class n (where 1 or more primes can occur at most n times) = 1/2n.
So conventionally we look at the prime factors as the building blocks of the natural number system in Type 1 terms (where each entry represents the base aspect of number).
However, here we have demonstrated in a different way, how the prime factors represent the building blocks of the natural number system in Type 2 terms (where each entry represents the dimensional aspect of number).
And just as the conventional way of looking at the relationship between the primes and the natural numbers is based on the Zeta 1 (Riemann) zeta function, this alternative way is based on a key special case of the Zeta 2 function.
The crucial point once again is that as both these functions are complementary with each other in a dynamic interactive manner, the ultimate nature of the number system (in the relationship between primes and natural numbers) is holistic and purely relative in a two-way synchronous manner.
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