Return of the Zeta 2 Function (2)

We have been looking now at the finite version of the Zeta 2 function to find that it too is necessarily embedded in the corresponding Zeta 1 (Riemann) function.

The terms in the infinite product over primes expression are given as 1 – 1/pⁿ.

And 1 – 1/pⁿ = (1 – 1/p)(1 + 1/p¹ + 1/p² + ...  + 1 + 1/p^{n – 1}).

So the second expression (inside brackets) corresponds directly with the finite version of the Zeta 2 function i.e. ζ(s₂) =  1 + s₂¹ + s₂² + s₂³ + ... + s₂^{n – 1}, where s₂ = 1/p¹ .

And we also saw how this finite version of the Zeta 2 is fundamental to explaining why the behaviour of the Zeta 1 (Riemann) function with respect to the infinite sum over natural numbers differs for (positive) even and odd integers of s respectively.

So where even integers are concerned 1 – 1/pⁿ = (1 – 1/p^n/2)(1 + 1/p^n/2) with n/2 a positive integer.

However where odd integers are concerned 1 – 1/pⁿ = (1 – 1/p)(1 + 1/p¹ + 1/p² + ...  + 1 + 1/p^{n – 1}) with no complementary relationship possible therefore as between both sets of terms (inside brackets).


When we earlier looked at the (reduced) version of the Zeta 1 (Riemann) function we saw how the elimination of successive terms in the product over primes is related to the corresponding sum over natural numbers expression.

So again, to remind ourselves of this we will take the well known case for the function ζ(s₁), where s = 2!

Thus ζ2) = 1/1² + 1/2² + 1/3² + 1/4² +...  = 4/3 * 9/8 * 25/24 * 49/48 * ...  = π²/6.

Therefore, when we eliminate all the even numbered terms (i.e. those divisible by 2) in the sum over natural numbers expression, we eliminate the corresponding term related to p = 2 i.e. 4/3, in the product over primes expression.

So 1/1² + 1/3² + 1/5² + 1/7² ...  = 9/8 * 25/24 * 49/48 * 121/120 *...

Then when we eliminate additionally all terms (in the original series) divisible by 3 in  the sum over natural numbers, we likewise eliminate the next term in the product over primes expression related to p = 3 i.e. 9/8.

So  1/1² + 1/5² + 1/7²  + 1/11²  +...  = 25/24 * 49/48 * 121/120 * 169/168 *.... 

And finally to illustrate further here when we eliminate additionally all terms (in the original series) divisible by 5 in the sum over natural numbers expression, we likewise eliminate the next term in the product over primes expression related to p = 5 i.e. 25/24.

So 1/1² + 1/7² + 1/11²  + 1/13²  + ...   = 49/48 * 121/120 * 169/168 * 289/288 *....

Thus what is taking place here is a progressive thinning out of the means by which composite  numbers are formed (through the factors of the frequently occurring prime numbers).

Therefore it resembles an orchestra where most of the best known instruments are quickly eliminated from consideration.

Though a certain music can still be created through the relationship of the remaining (scarcely heard) instruments we would necessarily listen to a very sparse sounding symphony.


Now to be clear, we are referring here to the external aspect of the number system, where we concentrate on the generation of natural numbers in cardinal terms from corresponding prime "building blocks" (likewise understood in a cardinal manner).
And the key preoccupation from this perspective is the corresponding frequency in a collective manner - as we accumulate more natural numbers - of (cardinal) primes to (cardinal) natural numbers.

However there is a corresponding internal aspect of the number system, whereby we concentrate on the factor composition of each individual number demonstrating a complementary relationship as between the frequency of (distinct) prime factors among all the natural numbered factors of each individual number.

So we can start out with primes and extend in either of two directions either externally with respect to the collective natural number system or internally (in terms of prime factors) with respect to each individual number.

However there is an extremely important - though very subtle difference - as between the two uses of the primes.

In the first case i.e. external, we view the primes with respect to their independent identity i.e. once again as constituent "building blocks" of the natural number system (as it collectively unfolds).

However in the second case i.e. internal, we properly view the primes with respect to their interdependent identity i.e. in terms of their unique factor relationships with other prime numbers.

So properly understood, whereas the external aspect relates to the cardinal nature of the primes, the internal by contrast relates to the ordinal nature of the primes.

And both of these aspects - external and internal - can only be properly viewed as complementary aspects of prime behaviour in a dynamic interactive manner, where notions of cardinal independence (as quantitative) and ordinal interdependence (as qualitative) respectively, carry a merely relative meaning.

So when we shift reference frames, the external aspect now attains a qualitative meaning entailing interdependence and the internal a quantitative meaning of independence respectively.

Therefore, though again from the external perspective, we might attempt to start out with merely quantitative notions of the prime "building blocks" i.e. 2, 3, 5, 7,...and so on, it gradually becomes apparent - at least it should become apparent - that these primes really can have no meaning without an implicit ordinal identity (that already assumes the natural numbers). So if we were not already able to implicitly recognise 2 as the 1st , 3 as the 2nd, 5 as the 3rd, 7 as the 4th prime and so, we would - literally - be unable to give any coherent order to our understanding of primes.

Likewise though we may from the internal perspective, recognise prime factors with respect to their interdependent identity with other prime factors, again this can have no meaning in the absence of our ability to quantitatively identify these factors.

Therefore for example, though the two prime factors of 6 i.e. 2 * 3 entail a unique interdependent relationship (in the context of 6) of the prime nos. 2 and 3, we can also clearly identify in a quantitative manner that 2 factors are involved here.

So notions of (quantitative) independence and (qualitative) interdependence always necessarily co-exist with each other in a complementary interactive manner in both external and internal terms, with thereby a merely relative identity (depending on context).

When we now return to the Zeta 2 function (in finite terms) we can again show that operates in a complementary fashion to that of the Zeta 1.

As repeatedly stated we start with respect to the Zeta 2 (in this context)  Let us look for example at

ζ(s₂) =  1 + s₂¹ + s₂² + s₂³ + ... + s₂^{t – 1} where t = 3.

So ζ(s₂) =  1 + s₂¹ + s₂².

And for zeros

1 + s₂¹ + s₂² = 0.

So this yield the two (non-trivial) roots of the 3 roots of 1 i.e. – .5 + .866i and – .5 – .866i respectively.

Now let us eliminate the even numbered terms here (i.e. 2nd).

Thus 1 + s₂² = 0

So s₂  = + i and – i respectively.

In other words we have now being able to extend from prime numbered to natural numbered roots.

So  + i and – i represent the two additional roots of 4 besides + 1 and – 1 (which are already the 2 prime roots of 1).

In other words just as in an external cardinal manner, 2¹ * 2¹ enables us to extend from the prime number 2 to the new composite natural number 4, likewise in a complementary internal ordinal manner, 1² * 1² enables us to extend from the 2 roots of 1 to the 4 roots of 1, which indirectly express, in a quantitative manner, the ordinal notions of 1st, 2nd, 3rd and 4th (in the context of 4).

So though the elimination of terms (from the Zeta 1) restricts the external symphony (of the collective relationship of primes to natural numbers) in an inverse manner, the corresponding elimination of terms extends the internal symphony (of the individual relationship of prime factors to natural numbered factors).

And we can continue on in this manner to build up the range of factor connections.

For example

1 + s₂¹ + s₂² + s₂³ + s₂⁴ provide the 4 non-trivial roots (of the 5 roots of 1).

When we eliminate even numbered terms here we get,

1 + s₂²  + s₂⁴  = 0.

Now if we let s₂² = x, then 1 +  x¹ + x² = 0.

So x = – .5 + .866i and – .5 – .866i respectively (i.e. the 2 non-trivial roots of 3).

Therefore s₂ provides the means to extend the 3 roots of 1 to the 6 roots of 1 which in Type 2 terms

= 1^{2 * 3} .

So just as all composite natural numbers in Type 1 (cardinal) terms represent the unique composition of primes as base nos., equally all composite natural numbers in Type 2 (ordinal) terms represent the unique composition of primes as dimensional nos.