Differing Orders of Primes and Natural Numbers (1)

I have raised before the notion of different primes with respect to the natural number system.

The well-known set of primes 2, 3, 5, 7, 11, … in this context can be referred to as Order 1 primes, with the corresponding natural numbers 1, 2, 3, 4, 5, …, in corresponding fashion referred to as Order 1 natural numbers.

However we can now order the cardinal primes 2, 3, 5, 7, 11, …, with respect to ordinal natural number rankings 1, 2, 3, 4, 5, …

If we now choose to list only those primes with a corresponding ordinal prime ranking we now obtain a new set of cardinal primes i.e. 3, 5, 11, 17, 31, 41, …

I then refer to this new set of cardinal primes as Order 2 primes with the corresponding set of natural numbers (including 1) based on the use of these primes as factors as

1, 3, 5, 9, 11, 15, 17, 25, 27, 31, …

And I refer to this new set of natural numbers as Order 2 natural numbers.

We can now again give the Ordinal 2 cardinal primes an (Order 1) natural number ranking in an ordinal fashion i.e. 1, 2, 3, 4, 5, …

Then if we once again only choose those remaining primes that correspond with prime rankings we derive yet a new set of cardinal primes i.e. 5, 11, 31, 59, 127, …

I then refer to this latest set of primes as Order 3 primes.

And once more we can derive the corresponding Order 3 set of natural numbers (including 1) based on the use of these primes as factors i.e. 1, 5, 11, 25, 31, 55, 59, 125, 127, …

And we can continue on indefinitely in this fashion deriving an ever sparser set of primes and corresponding natural numbers (based on the use of these primes as factors).

For example the Order 4 primes are 11, 31, 127, … and the corresponding Order 4 natural numbers are 1, 11, 31, 121, 127, …

So we keep interchanging in this manner as between the quantitative notion of cardinal primes and the corresponding qualitative notion of ordinal natural number rankings.

Indeed one fascinating insight that derives from this approach is that the full set of natural numbers, can thereby be expressed as Order 0 primes!

So in this way the true independence as between the primes and natural numbers with respect to both analytic (quantitative) and holistic (qualitative) meanings is made readily apparent.

And what is very interesting in this regard is that we can derive an unlimited number of corresponding L-functions based on matching the Order k primes with the corresponding Order k natural numbers.

So for example, we can thereby match the Order 2 primes with the corresponding Order 2 natural numbers.

Therefore in general terms, with convergent answers where s > 1.

1 + 1/3^s  + 1/5^s + 1/9^s  + 1/11^s + …        = 1/(1 – 1/3^s) * 1/(1 – 1/5^s) * 1/(1 – 1/11^s) * …

So for example, where s = 4,

1 + 1/3⁴  + 1/5⁴ + 1/9⁴  + 1/11⁴ + …        = 1/(1 – 1/3⁴) * 1/(1 – 1/5⁴) * 1/(1 – 1/11⁴) * …

The LHS (for listed terms) = 1.01416…, whereas the RHS = 1.01419… 

So already the two results are very similar.

This latest procedure - entailing Order k natural numbers with corresponding Order k primes - is different from previous procedures where we eliminated a finite number of terms with respect to the RHS (product over primes) with then made consequent adjustments for all terms where the primes acted as constituent factors in the LHS (sum over the integers) expression.

In this case, with each successive Order, we eliminate an infinite series of individual prime terms with respect to the RHS with then consequent adjustments to the LHS natural numbers expression.

And the product over primes expression is no longer adjusted for a finite number of rational terms - but rather an infinite number - we can no longer preserve results of the form kπ^s (where k again is a rational number and s a positive even integer).

However with respect to new Order k expressions we can again adjust for a finite number of individual terms (with respect to the RHS prime expression).

So, as we have seen the Order 2 function is,

1 + 1/3^s  + 1/5^s + 1/9^s  + 1/11^s + …        = 1/(1 – 1/3^s) * 1/(1 – 1/5^s) * 1/(1 – 1/11^s) * …

Therefore if we now eliminate the individual term related to 3 as prime in the RHS, then we must correspondingly eliminate all terms which include 3 as factor in the corresponding LHS expression.

So  1 + 1/5^s + 1/11^s + 1/17^s + 1/25^s + …  = 1/(1 – 1/5^s) * 1/(1 – 1/11^s) * 1/(1 – 1/17^s) * …