We have seen how the average frequency of the natural factors of a number expressed relative to the corresponding average frequency of the prime factors (up to n) can be approximated as log n/log(log n).
Now it is well known that the sum of the terms of the harmonic series i.e. the sum of the reciprocals of the natural numbers (up to n) = log n + γ, where γ = the Euler-Mascheroni constant (= .5772156649...).
Thus, 1/1 + 1/2 + 1/3 + 1/4 +........+ 1/n = log n + γ.
However for sufficiently large n as γ is constant, log n + γ approximates more simply to log n.
It is also known that the sum of the corresponding series entailing the sum of the reciprocals of the primes (up to n) = log(log n) + B1 ,where B1, = Mertens constant (= .2614972148...).
So, 1/2 + 1/3 + 1/5 + 1/7 +....... + 1/n = log(log n) + B1.
Again however, for sufficiently large n, as B1 is constant, log(log n) + B1 approximates to log(log n).
Therefore we have the delightful connection that the average frequency of the natural factors of a number expressed relative to the corresponding average frequency of its prime factors (up to n) is replicated as the ratio of the sum of the reciprocals of the natural numbers to the corresponding sum of the reciprocals of the primes (up to n).
This would also provide a fascinating new way of expressing the prime number theorem.
So, once again, if we let t = log n, then log n/log(log n) = t/log t.
Therefore t/log t can be expressed as the ratio of the two series (up to n) entailing the sum of the reciprocals of the natural numbers and primes respectively.
This result can be extended indefinitely.
I have talked before of 1st Order, 2nd Order, 3rd Order,....Nth Order Primes.
The 1st Order Primes are - what we directly recognise as - the prime numbers, which (up to 100) are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 and 97.
To obtain the 2nd Order Primes we give natural number rankings to these primes and then choose those relating to the cardinal primes from those natural number ordinal rankings.
So for example with respect to the 1st 5 primes, we rank these in the following manner (with cardinal primes in 1st row and ordinal rankings in 2nd).
2, 3, 5, 7, 11
1, 2, 3, 4, 5
We now choose the numbers that refer to the prime rankings (i.e. 2, 3 and 5)
So 3, 5 and 11 are now the three primes, from the original 5, belonging to the 2nd Order Primes.
Now if we list 2nd Order Primes to 100 we get 3, 5, 11, 17, 31, 41, 59, 67 and 83.
Now again, we can give these 2nd Order primes (in ascending order) natural number ordinal rankings as indicated below:
3, 5, 11, 17, 31, 41, 59, 67, 83
1, 2, 3, 4, 5, 6, 7, 8, 9
Once more to obtain 3rd Order Primes, we choose the numbers corresponding to the the prime rankings
i.e. 5, 11, 31 and 59
We then give these 3rd Order Primes a natural number ordinal ranking before picking those corresponding to (ordinal) prime rankings as our new set of 4th Order Primes, i.e.
5, 11, 31, 59
1, 2, 3, 4
So the 4th Order Primes (up to n = 100) are 11 and 31.
Again we rank these in natural number ordinal terms as 1 and 2 respectively
Then by choosing the cardinal number corresponding to the (ordinal) prime number ranking 2, we obtain the only 5th Order Prime (up to 100) i.e. 31.
However where the value of n is unlimited in finite terms, the number of different Orders of Primes is likewise finitely unlimited.
Thus the point I am making is that the order among the natural numbers and primes i.e. natural numbers and First Order Primes) is continually replicated through the corresponding order throughout the number system, as between 1st Order and 2nd Order Primes, 2nd Order and 3rd Order Primes, 3rd Order and 4th Order,...,Nth – 1 Order and Nth Order Primes (where N ultimately has no limit in finite terms).
The frequency of 1st Order primes is approximated as we have seen by n/log n.
Now if we let n1 =n/log n, then this latter relationship (of the frequency of 2nd Order among the Ist Order Primes) can be expressed as n1/log n1.
In turn, if we now let n2 = n1/log n1, then the frequency of 3rd Order among the 2nd Order Primes can be expressed as n2/log n2.
Then if we let n3 = n2/log n2, then the frequency of 4rd Order among the 3rd Order Primes can be expressed in turn as n3/log n3.
Ultimately when nN = nN – 1/log nN –
1, then the frequency of Nth Order among the (Nth – 1) Order Primes can be expressed in turn as nN/log nN.
In fact, in terms of the above approach, the natural numbers can be expressed as 0th Order Primes!
So rather than the Prime Number Theorem being reserved to just one relationship i.e. the frequency of 1st Order among the 0th Order Primes (primes among the natural numbers), we have in fact potentially an unlimited number of Prime Number Theorems continually reiterated throughout the number system i.e. for the frequency of 2nd Order among the Ist Order, the frequency of 3rd Order among the 2nd Order,....., to finally the frequency of Nth Order among the (Nth – 1) Order Primes.