Wednesday, December 21, 2011

Sum of Reciprocals of Primes

As is well known the sum of the terms in the harmonic series

1 + 1/2 + 1/3 + 1/4 +.... ~ ln n + γ (where γ = the Euler-Mascheroni constant =.5772..)

It is fascinating therefore that the sum of reciprocals of primes

1/2 + 1/3 + 1/5 + 1/7 + ..... ~ ln ln n + B (where B = Merten's constant = .261497..)

This would of course suggest that the sum of this series diverges for large n!

However just as the harmonic series can be used to calculate the spread as between cardinal prime numbers, likewise this latter series can be used to calculate the spread as between ordinal prime numbers.

In other words all prime numbers can be linked with the ordinal set of natural numbers.

So 2 is the 1st, 3 the 2nd, 5 the 3rd, 7 the 4th prime respectively.

So if we now order these primes in an ordinal prime fashion, then both 3 and 5 are prime (i.e. as the 2nd and 3rd primes).

We could then reorder these surviving primes in an natural number ordinal fashion before selecting once again those surviving numbers that are prime in an ordinal fashion.

Let us take all the primes up to 31 to illustrate,

(1) 2, (2) 3, (3) 5, (4) 7, (5) 11, (6) 13, (7) 17, (8) 19, (9) 23, (10) 29, (11) 31.

These these starting cardinal primes are listed in natural number ordinal fashion (in brackets). We will refer to these as Order 1 Primes. So all prime numbers are Order 1 primes.

Then if we extract the primes (whose ordinal numbers are also prime), we are left with 3, 5 11, 17 and 31.
If we now again rank these ordinally in natural number fashion we have
(1) 3, (2) 5, (3) 11, (4) 17 and (5) 31.
We can refer to this smaller group as Order 2 Primes.

Once again we can then extract only those primes that have a prime number ordinal ranking i.e. 5, 11 and 31.

Then shifting to ordinal natural number ranking we have a new - even smaller - set of surviving primes,

(1) 5, (2) 11 and (3) 31.

We can refer to these then as Order 3 Primes.

Then once more extracting those remaining with an ordinal prime ranking we are left with 11 and 31

So giving natural number ordinal rankings we have,

(1) 11 and (2) 31

These are Order 4 Primes.

Finally extracting the one remaining prime with an ordinal prime ranking we are left with 31.

Finally ranking this as (1) 31, this qualifies as an Order 5 Prime.

It is no accident that 31 corresponds to the Mersenne prime 2^n - 1 (where n = 5).

Indeed if we were to continue up to 127 for example, 127 would then qualify as the one remaining Order 7 Prime. And 127 is the Mersenne prime (where n = 7).

I have suggested at various times that perhaps we could guarantee the generation of Mersenne primes by starting with 2 and then proceeding through switching in an orderly fashion as between quantitative (base) and qualitative (dimensional) use of prime numbers.

So 2^2 - 1 = 3 (as base number).

Then substituting 3 as dimensional number we have,

2^3 - 1 = 7 (which is prime).

Once again substituting 7 as dimension we have,

2^7 - 1 = 127 (which is prime).

Then substituting 127 as dimension we have,

2^127 - 1 = 170141183460469231731687303715884105727 (which is prime).

It is tempting to argue that by using this number as exponent of 2 that we can generate a new Mersenne prime that is incomparably larger than any yet discovered!

Now going back to the harmonic series and sum of the reciprocals of primes.

Once again the sum of the harmonic series for large n approximates to ln n.

The sum of the reciprocals of primes for large n approximates to ln ln n.

Now the prime numbers used as denominators in this series are Order 1 Primes.

It is possible therefore to extend this result for Order 2, Order 3, Order 4 primes etc.

For example ultimately the sum of reciprocals of Order 2 Primes should approximate (for sufficiently large n) to Ln Ln Ln n, Order 3 to Ln Ln Ln Ln n, Order 4 to Ln Ln Ln Ln Ln n etc.
This would suggest that no matter how high the Order of Primes involved that the sum of the series of its reciprocal terms would diverge (for sufficiently large n).

Also by this reckoning the harmonic series could be interpreted as the sum of Reciprocals of Order 0 Primes.

So the natural numbers are prime numbers of Order 0!
What simply this means in effect is that the ordinal ranking of the complete set of primes (i.e. Order 1 Primes) is given by the natural numbers!

As for the spread as between primes, once again for Order 1 Primes the answer approximates to log n.

Clearly the spread will grow for higher Order Primes.

So we can postulate that the number of Order 1 Primes up to 1,000,000 = n/ln n = n1 = 72,283 (approx).

Therefore the number of Order 2 Primes up to 1,000,000 approximates n1/ln n1 = 6469 (approx).

Thus the average gap as between Order 2 Primes approximates n/{n1/ln n1} = 1,000,000/6469.
Therefore in the region of 1,000,000 we would expect the average gap as between Order 2 Primes to approximate 154.6.
Now because n is still of a relatively small magnitude, the actual number of Order 2 Primes would differ significantly from this estimate. However the approximation of estimated to actual would continue to improve (in relative terms) as n increases.

The upshot of what we are doing here is that the prime and natural numbers are in fact completely interdependent with each other.
From one perspective the (individual) natural numbers are derived from the primes; however equally from the complementary perspective, the (general) distribution of the primes is derived from the natural numbers.

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