New Perspective on Prime Number Theorem (1)

The notion of the factors of a number can be defined in different ways.

For example id we take the number 12 it can be uniquely expressed in terms of its prime constituents as 2 * 2 * 3.

However one of these factors (i.e. 2)  repeats. Therefore in terms of distinct primes, 12 entails just two building blocks (i.e. 2 and 3).

However 12 has several more factors when we consider all those numbers with which it can be evenly divided.

Now clearly, 12 can be divided by 1. However as all numbers by definition can be divided by 1, we will exclude this from consideration as a trivial factor.

12 can also be divided by 2, 3, 4, 6 and 12. Now again we could query the inclusion of 12 as an integer - by definition - will include itself as a factor. However 12 can perhaps be validly considered as somewhat more unique than 1 (as all integers are not divisible by 12).

So in this sense, in which I define it, 12 has 5 factors.

Finally where primes are concerned we simply ignore any factors. Therefore for example, though 7 is a factor of 7, because the number is prime, we do not consider it.

So we are now left with two ways of defining factors.

Once again we have the prime factors, relating to the distinct prime building blocks of a number.

Then we have - what I define as - the natural factors which are defined with respect to the composite numbers to include all factors (other than 1).

The important question then arises as to the relationship as between the average number of natural and prime factors contained by a number.

There is a well known theorem i.e. the Hardy-Ramanujan Theorem, which postulates that on average the number of prime factors contained by a number can be given as log(log n) with the accuracy of this prediction increasing for large n.

Then recently in my own researches, I came up with an estimate that on average the number of natural factors contained by a number can be given as log n – 1.

However as for very large n,  log n – 1 would closely approximate log n, we can give this result more simply as log n.

So if we let log n = t, we can now say that on average, for large n, the number of natural factors contained by a number = t.

We can also then say that on average, therefore the number of prime factors contained by the same number = log t.

So the frequency of natural to prime factors is given by t/log t.

So for example in the region of the number system, where each number on average contains 1000 natural factors, we would expect it (again on on average) to contain nearly 7 (6.9) prime factors. Thus we would expect over 140 times more natural than prime factors per number in this region of the number system.

Therefore what we have concluded is that not alone does the prime number theorem apply to the distribution of the primes among the natural numbers, but equally it applies to the distribution of the (distinct) prime factors among all the natural factors of a number.

So once again we have distinguished two complementary aspects to the distribution of primes (among the natural numbers).

Again in Type 1 terms we have the well-recognised distribution of the primes among the natural numbers (in cardinal terms).

However In Type 2 terms, we have the largely unrecognised distribution of the prime among the natural factors of a number.

Once again the Type 1 approach is - literally 1-dimensional in nature, where all numbers are considered in reduced fashion as lying on the same number line.

So here, strictly speaking we view relationships in a merely quantitative manner (as befits notions of number independence).

However, the Type 2 approach is multidimensional in nature where numbers represent differing dimensions against a fixed quantitative base  (as befits notions of number interdependence).

But in dynamic interactive terms, the Type 1 and Type 2 aspects are fully complementary with each other.
Thus we cannot properly view quantitative (Type 1) notions of number independence in the absence of corresponding qualitative (Type 2) notions of number interdependence; likewise we cannot properly view qualitative (Type 2) notions of number interdependence in the absence of corresponding quantitative (Type 1) notions of number independence.

Thus the distribution of primes among the natural numbers for both numbers and factors occur simultaneously in the coincidence of quantitative and qualitative aspects.

Thus the ultimate relationship between the primes and natural numbers (and natural numbers and primes) in both Type 1 and Type 2 terms  is determined in a purely synchronous ineffable manner.