New Perspective on Prime Number Theorem (5)

In the last two blog entries I showed, how coming from the standard Type 1 perspective, that the Prime Number Theorem can be indefinitely extended within the number system to an unlimited set of relationships involving Higher Order Primes.

So what are commonly referred to as the primes thereby in this context represent the 1st Order Primes. However 2nd Order, 3rd, Order,.....Nth Order Primes can be subsequently defined, all of which are bound by a corresponding Prime Number Theorem equivalent.

However, though more difficult to properly envisage it is also equally possible to view the Prime Number Theorem from the (largely unrecognised) Type 2 perspective, once again resulting in a potentially unlimited set of Prime Number theorem equivalents.

Remember once again that the key formulation here relates to the average frequency of the natural factors of a number (up to n) divided by the corresponding average frequency of its (distinct) prime factors!

We suggested that this was given (for large n) by log n/log(log n).

Therefore by letting t₁= log n, the relationship could then be expressed as t₁/log t₁ .

We also pointed out the fascinating connection here with the well known harmonic series (sum of the reciprocals of the natural numbers) and the corresponding prime series (of the sum of the reciprocals of the prime numbers).

Indeed n/log n could equally be expressed (for large n) as,

1 + 1 + 1 + 1 +....../(1 + 1/2 + 1/3 + 1/4 +.....)

t₁/log t₁ i.e. log n/log(log n) is then expressed as,

1 + 1/2 + 1/3 + 1/4 +......../(1/2 + 1/3 + 1/5 + 1/7 +.......)

So we are using the reciprocals of the 0th Order Primes (i.e. natural numbers) and 1st Order Primes (i.e. the regular primes) in this relationship.

However we could equally defines a new (infinite) series based on the reciprocals of the 2nd Order Primes

i.e. 1/3 + 1/5 + 1/11 + 1/17 + .........

It is fascinating to speculate that the value of this sum would in turn be given by,

log{log(log n)} + C where C represents a constant.

Then for very sufficiently large n, this would approximate to log{log(log n)}.

Then in turn we could define a new (infinite) series based on the reciprocals of the 3rd Order Primes i.e.

1/5 + 1/11 + 1/31 + 1/41 +....

Again it would be interesting to speculate that the value of this sum in turn would be given by:

log[log{log(log n)}]  + D where D represents a constant.

Then for sufficiently large n, again the result would be approximated well as log[log{log(log n)}].

And we could continue on in this manner for 4th Order, 5th Order,....Nth Order Primes.

One interesting and - perhaps - extremely surprising - conclusion from all this is that the (infinite) series of reciprocals for all Higher Order Primes, would ultimately be divergent in value.

Thus again, no matter how often we thin out the original series of prime numbers to derive Higher Order Primes, the (infinite) sum of the reciprocal series (based on these primes) will ultimately diverge (albeit extremely slowly).

Now we could express the original ratio as between the average number of natural factors (up to n) contained by a number and the corresponding average number of prime factors as log n/ log(log n), which is replicated in terms of the ratio of the sum of the reciprocals of the natural numbers and the corresponding sum of reciprocals of the standard (1st Order) primes

i.e. 1 + 1/2 + 1/3 + 1/4 +......../(1/2 + 1/3 + 1/5 + 1/7 +.......).

This therefore suggests that we can extend this indefinitely in a Type 2 fashion throughout the number system.

So for example, if we confine ourselves to (distinct) prime factors that are 2nd Order Primes (i.e. 3, 5, 11, 17, 31,...) and also confine ourselves to the natural factors of numbers based on these primes), then the new ratio of natural to prime factors will be given as log(log n)/ log{log(log n)}.

If we let t₂ = log(log n) then this ratio could then be expressed as t₂/log t₂.

This ratio could then in turn be expressed as the ratio of the sum of reciprocals based on the 1st Order Primes divided by the corresponding sum of reciprocals based on 2nd Order Primes

i.e. 1/2 + 1/3 + 1/5 + 1/7 +....../(1/3 + 1/5 + 1/11 + 1/17 +.....).

We could then go on to get a new ratio of natural to prime factors based on consideration of 3rd Order Primes which would be approximated by log{log(log n)}/log[log{log(log n)}].

Then by letting t₃ =  log{log(log n)}, we could express this ratio as t₃/log t₃.

This result once again could then be expressed as the ratio of the sums of reciprocals of the 2nd Order and 3rd Order Primes,

i.e. 1/3 + 1/5 + 1/11 + 1/17 +..../(1/5 + 1/11 + 1/31 + 1/41 + .....).

And we could proceed indefinitely in this manner to - in the general case - expressing a ratio of natural to prime factors based on consideration of Nth Order Primes which would be expressed in turn as t_N/log t_N .

And this result in turn could then be expressed as the ratio of the sums of reciprocals of the (N – 1)th and Nth Order Primes respectively.