Ordinal Nature of Prime Numbers

One of the great limitation of the conventional approach to  the Riemann Hypothesis, is that it attempts to interpret the primes and natural numbers in a merely cardinal (i.e Type 1) manner.

However the primes and natural numbers can be equally given a distinctive ordinal (i.e. Type 2) interpretation.

So the mystery of the relationship of the primes to the natural numbers (and the natural numbers to the primes) can only be properly understood in a dynamic fashion, entailing the two-way interaction of both cardinal and ordinal aspects.

We have already looked at the ordinal aspect of each individual prime number form the Type 2 perspective.

So once again a prime such as 3 is - by definition - composed of three members i.e. 1st, 2nd and 3rd, in a natural number ordinal fashion.

Ist, 2nd and 3rd refer directly to a qualitative rather than quantitative notion of number. This is due to the fact that their respective meanings imply interdependence through a necessary relationship with other group members.

So what is 1st in the context of 3, implies 3 group members and in principle what is 1st could entail any one of these members (depending on context).
What is then 2nd implies two remaining members. However what is 3rd (in the context of 3) then implies only one possible member. So the notion of interdependence here no longer holds.

In an indirect quantitative manner, these 3 ordinal numbers can be expressed in The Type 2 system through raising 1 to 1/3, 2/3 and 3/3 respectively.

So 1^1/3 (i.e. – .5 + .866...i  is the indirect quantitative representation of the notion of 1st (in the context of 3 members).

1^2/3 (i.e.  – .5 – .866...i is then the indirect quantitative representation of  the notion of 2nd (in the context of 3 members).

1^3  i.e. 1, is finally the indirect quantitative representation of the notion of 3rd (in the context of 3 members).
As this final result is always 1 (and indistinguishable from the cardinal notion of 1), it can be considered as a trivial result.

So where t is a prime number, the ordinal notion of the nth member (in the context of t group members) is always trivial in this sense (= 1).

The remaining (t – 1) non-trivial results are then the various solutions to the finite equation,

1 + s¹  + s²  + s³  +….. + s^{t – 1}   = 0.

So therefore in the context of a group of 3 members (where t = 3),

the two non-trivial results are the solutions of the equation,

 1 + s¹  + s²    = 0.

These again are what I refer to as the Zeta 2 non-trivial zeros.

As I have stated before, this finite equation can be extended in an infinite cyclical manner (with each full cycle made up of 3 successive terms).

Fascinatingly the expected value of this infinite equation = 1/2.

So the ordinal nature of each prime (in this Type 2 sense) derives from the fact that each prime is composed by definition of a natural number succession of terms in an ordinal manner).

So again for example, 3 in this sense, is composed of individual 1st, 2nd and 3rd members!

However there is another complementary manner in which the ordinal nature of prime numbers arises.

As is well known every natural number (except 1) represents a unique combination of prime number factors (in cardinal terms).

So therefore 6 for example us uniquely expressed as the product of 2 and 3 (i.e. 2 * 3) in cardinal terms.

However the ordinal nature of primes arises in the related context that the notion of 6th, likewise reflects a unique product combination of 2nd and 3rd (in an ordinal manner).

Now indirectly this qualitative notion of number is expressed through the Type 2 aspect of the number system through using the reciprocals of the dimensional powers involved.

So 1^1/6  = 1^{(1/2) * (1/3)} = .5 – .866...i.


Now with respect to each prime number, a perfect balance is maintained as between its individual members (indirectly expressed in a quantitative manner) and its collective identity (qualitatively expressed through the sum of its members).

So, for example, when we add the three roots of 1 (as quantitative expressions of 1st, 2nd and 3rd in the context of 3) the resulting collective sum = 0.

Therefore the combined interdependence of the 3 members (representing a strictly qualitative notion) thereby has no quantitative value!

So the Zeta 2 non-trivial zeros uniquely reconcile within each prime number group, the relative independence of each individual member, in quantitative terms, with the corresponding relative interdependence of the collective group of members in a qualitative manner.

The Zeta 1 non-trivial zeros as solutions of s to the infinite equation,

1^–s  + 2^–s  + 3^–s  + 4^–s  +…….. = 0,

solve a complementary problem.

So the Zeta 1 non-trivial zeros (that occur in pairs of the form .5 + it and .5 – it respectively) uniquely reconcile in a complementary manner, for the number system as a whole, the corresponding relative independence of each prime number (in an individual quantitative manner) with the overall relative interdependence of the prime numbers as a collective group (in the natural number system).

Once again we are accustomed to look at the prime numbers in a quantitative cardinal manner.

However the qualitative (ordinal) nature of the primes - representing their corresponding collective interdependence with the natural number system - is expressed through the Zeta 1 (i.e. Riemann) non-trivial zeros.

So now in a complementary manner (to the Type 2), the qualitative nature of the zeros is expressed through each individual zero (thereby representing a formless energy state) while their quantitative nature is expressed through the overall collective nature of the zeros.

There is of course a two-way dynamic interdependence as between Zeta 1 and Zeta 2 zeros (in quantitative and qualitative terms).

The reason why the Zeta 2 zeros uniquely express the individual nature of the members of each prime group, is because the Zeta 1 zeros seamlessly preserve the interdependence of the prime numbers with the number system as a whole.

And the reason why the Zeta 1 zeros uniquely express the collective nature of the primes with the natural number system is because the Zeta 2 zeros uniquely achieve such uniqueness for each individual member of a prime (considered as a group).

In fact, this two-way relationship as between the primes and the natural numbers, perfectly expresses in a mathematical fashion the original relationship as between whole and part and part and whole in both quantitative and qualitative terms.