Thursday, February 23, 2012

Deep Prime Connections

I mentioned in a previous blog that whereas the quantitative aspect of primes relates to their individual aspect (as distinct prime numbers) their qualitative aspect relates, by contrast, directly to the natural numbers i.e. in the manner in which each natural number can be expressed in terms of a unique combination of prime factors. (And of course the key observation I was making in this context was that the multiplication of distinct primes necessarily entails dimensional change of a qualitative nature!)

So the quantitative aspect relates to the individual independence of the primes (with respect to the natural numbers), whereas the qualitative aspect relates to their capacity of relatedness through which they can uniquely generate the natural numbers as prime factors.

Now the simplest manner in which the quantitative and qualitative aspect of number manifests itself is through cardinal and ordinal interpretation respectively.

So if we write down the natural numbers as representing the ordinal ranking of the primes, then

1, 2, 3, 4, 5,.... thereby represent the 1st, 2nd, 3rd, 4th, 5th,... primes respectively which correspond in cardinal terms to 2, 3, 5, 7, 11,....

So in this important respect, the ordinal ranking of the natural numbers (as qualitative) corresponds directly with the cardinal ranking of the primes (as quantitative).

We could carry this process further by now seeking in turn to obtain an ordinal ranking of the primes.

So this ordinal ranking of 2, 3, 5, 7, 11,... would thereby correspond with the 2nd, 3rd, 5th, 7th, 11th ... primes, corresponding to 3, 5, 11, 17, 31, ...etc. And we could continue this process indefinitely in a progressive thinning out of the primes by once again labelling our latest set of primes 1, 2, 3, 4, 5,... and then once again obtaining the new prime set corresponding to ordinal prime rankings with respect to these numbers, then relabelling them in natural number fashion before proceeding to look at again at the numbers corresponding to the new ordinal prime rankings!

However it is enough for our purposes here to confine ourselves to the initial ordinal ranking of primes.

As is well known the sum of the reciprocals of the natural numbers i.e. the harmonic series

1 + 1/2 + 1/3 + 1/4 +.....approximates log n + γ (where γ = Euler-Mascheroni constant = .5772..).

And log n, as we know approximates the average gap or spread between cardinal primes (which continually improves as n becomes progressively larger).

And as the contribution of the Euler-Masheroni constant becomes increasingly less important for very large n, then the sum of the reciprocals of the natural numbers approximates well the average gap as between the prime numbers!

What is equally fascinating is that the sum of reciprocals of the prime numbers (cardinal)

i.e. 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + .... is approximated by log log n + B1 (where B1 = Mertens Constant i.e. .261497...)

So just as the sum of the reciprocals of the natural numbers approximates the average gap as between cardinal primes (as independent numbers) for very large n, in like manner the corresponding sum of the reciprocals of the prime numbers should approximate the relational capacity with respect to the ordinal primes for n (i.e. as the number of unique prime factors for n)

This would suggest that the number of prime factors of very large - and I mean very large - natural numbers, would thereby approximate well with log log n.

Now, Hardy and Ramanujan proposed this some time ago (though I am not aware of the precise details of what they had in mind).

However the very point of this exercise is to show how the same conclusion can be reached through holistic recognition of the qualitative aspect of the primes!

Indeed it would be tempting to conjecture that perhaps a more precise measurement could be given in relation to the number of prime factors of a very large number, by using the original approximation for the sum of the reciprocals of primes,

i.e. log log n + .261497...

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