Monday, November 10, 2014

Interesting Connections

We have seen that log n approximates the average gap between prime numbers.

However equally - though less well known - it measures the average amount of natural factors contained by a large number. (In this context natural implies any natural number which is a factor of the number in question!)

In fact Dirichlet proved in 1838 the approximate relationship (for such factors) as

log n + 2γ – 1 = log n +.15443..

So clearly when n is very large, log n provides a very good approximation 

It is also interesting to observe that log n equally approximates the sum of the harmonic series to n (which contains the reciprocals of the natural numbers). The sum (to n) is generally given as log n + γ = log n + .57721...

Once again however when n is very large, log n offers a very good approximation.


Now log log n (Hardy-Ramanujan Theorem) offers an approximation of the average number of (distinct) prime factors contained by a number (when n is very large)

It is tempting therefore to presume that, equally as with log n, a complementary type relationship might exist with respect to the average gap as between natural numbers.

After some consideration, I came up with the notion of spacing as between composite natural numbers composed of non-repeating prime factors. So for example 6 (2 * 3) and 10 (2  *5) would represent appropriate examples. However 4 (2 * 2) and 8 (2 * 2 * 2) and  9 (3 * 3) would be excluded. All single primes however could be included.  

I am offering no proof of this and have only had the time to carry out limited empirical testing of the proposal. However it would represent an interesting hypothesis to test.

It is also once more interesting to observe that just as log n approximates the sum (to n) of the reciprocals of the natural numbers, that equally log log n approximates the sum (to n) of the reciprocals of the prime numbers.

Again the more correct approximation is given as log log n + B (where B is Merten's number = .261497...). However for large n the simpler expression i.e. log log n would offer a good approximation!


It is also fascinating to note that log n/log log n measures the ratio of the average amount of natural number to (distinct) prime factors for a large number.

Equally it measures the ratio of the sum of the reciprocals of natural numbers (to n) to the corresponding sum of reciprocals of the prime numbers.

Also it has been proven that if n is primorial - i.e example 2 * 3 * 5 * ...* n, that

log n/log log n approximates the number of factors in n (for very large n).

Once again log log n measures the average number of (distinct) prime factors in n.

It would therefore be fascinating to obtain a corresponding measurement for the total number of prime factors (allowing for repetition).

So for example 2 and 3 are the (distinct) prime factors of 24. However 2, 2, 2 and 3 represents the prime composition of 24 (allowing for repetition).

Now once again - based on a limited amount of empirical testing - I offer the expression (log log n)2
as the simple appropriate expression in this case.

Finally, log n measures the average amount of natural factors.

However for example if we say that 24 contains 1, 2, 3, 4,, 6, 8, 12, and 24 as factors (There is always some arbitrariness as whether to include 1 and the number itself as factors! However for very large n it does not significantly affect approximation results!)

This would therefore represent 8 factors.

However, these factors could be uniquely expressed as 1 * 24, 2 * 12, 3 * 8 and 4 * 6.

So 8 is now reduced to 4.

So if we represent factors in this latter fashion (as unique combinations of 2) the corresponding expression for the average number would be log n/2.


Addendum 21/3/16

I have since discovered - through excess to detaiiled tables listing the prime fators of all natural nos. to 1015 - that the above contention that the average number of prime factors (including those that repeat) would ~ (log log n)2 is incorrect.

It seemed to me to be be intuitively likely (without initial access to the prime factor tables) that the natural numbers with recurring prime factors would steadily increase, as a proportion of all numbers, as n increased.

However this is is not the case with a constant relationship apparently being maintained throughout the number system.

Therefore the average frequency of all the prime factors of a number (i.e. where factors can recur) can be given as k(log log n) where k is a constant.

Now this constant would appear to work out approximately at just a little in excess of 1.2.

I have speculated in recent blog entries that the value the value of k ~ (1 + √2)/2.

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