A Brief Addendum

As is well known log n or the more accurate log n – 1 provides a good approximation of the average spread or gap as between primes in the region of n.

However what is not properly known is that log n - 1 equally provides a good approximation of the average amount of natural factors of a number (also in the region of n).

And notice the complementarity as between both estimates! In one case we are referring to primes and in the other case to natural numbers; then again in the first case we are referring to the Type 1 notion of number (i.e. defined in a 1-dimensional manner); then in the second case we are referring to the corresponding dimensional notion of number (relating to its factor components).

This complementarity once again points to the truly dynamic interactive nature of the number system.

Thus the behaviour of the primes (with respect to the average gap as between primes in the region of n) is dynamically inseparable from the corresponding behaviour of natural number factors (this time with respect to their combined ratio in relation to n).

Put another way the behaviour in both cases is ultimately determined in a holistic synchronous manner.

Whereas great attention has been placed on one aspect of this behaviour (i.e. with respect to prime number frequency), precious little attention has been devoted to the complementary form of behaviour with respect to the frequency of the natural factors of numbers.

So in principle, just as it is is possible through use of the Riemann (Zeta) 1 zeros to eventually determine the exact frequency of the primes to any number, equally it should be possible to determine the exact cumulative frequency of natural number factors of the composites. through corresponding adjustments using the Zeta 2 zeros.

Another interesting fact!

n/log n – 1  approximates the frequency of primes to n.

The connection then as between the primes and natural numbers is of an additive nature,

n/log n - 1 equally can be used to measure the ratio as between n and the corresponding average number of factors of n.

However the connection here is of a multiplicative (rather than additive) nature.

Thus the two uses of the same formula illustrate the complementary relation as between addition and multiplication (which represent the Type 1 and Type 2 aspects of number with respect to each other).


Also it should be be observed that the average frequency of prime factors of a number is given approximately as log log n.

Therefore the ratio of natural number to prime factors is given as log n/ log log n.

If we now let log n = n₁, then this ratio can be given as n₁/log n₁.

Or if we wish to use the more accurate expression already employed, this would be given as n₁/log n₁ – 1 .

So now we can see clearly how a derivation of the formula for calculation of frequency of primes (with respect to the natural numbers) can be used for calculation of the ratio of natural number to prime factors.

And once again whereas the connection as between primes and natural numbers in the first (Type 1) is of an additive nature, in the second (Type 2) case it is of a complementary multiplicative nature.


Just one more observation!

log log n (or alternatively log log n – 1) in Type 2 terms can be used to approximate the average number of prime factors of n.

Therefore log log n equally can be given a Type 1 meaning.

Thus whereas log n measures the average gap between primes at n, log log n measures the average gap as between primes whose gap is log n..

So for example the average gap as between primes at n =1,000,000 = 13.815...
Therefore log log n measures the average gap of primes to 14 (approx).