Enhanced Zeta 2 Functions (2)

In the previous blog entry, I demonstrated the complementary nature of the Zeta 1 (Riemann) and Zeta 2 functions.

Thus when leave out the ordinal terms, divisible by p (representing a prime) in the Zeta 1 sum over natural numbers, the corresponding term (involving p) in the product over primes expression is likewise omitted.

So again if we omit all the even terms (divisible by 2) in the sum over natural numbers, we likewise omit the 1st term (related to 2) in the product over primes.

And when s for example = 2, this implies that the 1st term to be omitted in the product over primes expression = (1 – 1/2²) = 4/3 with the (reduced) sum over natural numbers = the product over the remaining primes.

So 1 + 1/3² + 1/5²  + ...    = 9/8 * 25/24 * 49/48 * .... 

However, a complementary relationship exists with respect to the Zeta 2 function, in that the individual term omitted in the product over primes expression is likewise equal to an alternative sum over natural numbers expression (where in this case the natural numbers represent dimensional powers - rather than - base numbers).

Thus 4/3 = 1 + s₁¹ + s₂² + s₂³ + .... where s₁ = 1/2²).

So, 4/3 = 1 + 1/4 + 1/16 + .....

Therefore though the product over all the primes in the Zeta 1 formulation = a corresponding sum over the natural numbers in a collective manner, each individual term in the product over primes expression = a corresponding sum over the natural dimensional numbers with respect to the Zeta 2 formulation.

And there is an inescapable circularity as between the notion of the primes and natural numbers, in both cardinal and ordinal terms.

So we can see above that the 1st term in the product over primes expression relates to the prime number 2; likewise the 2nd term relates to the prime number 3, the 3rd term to the prime number 5 and so on.

In other words the very notion of the natural numbers in an ordinal manner - by definition - is implicit in our identification of the prime numbers (in cardinal terms).

Without the implicit ability to identify 2 as the 1st, 3 as the 2nd, 5 as the 3rd prime and so on, there would be no way in which we could bring order to our understanding of the primes.

So from this perspective the natural numbers (in ordinal terms) are already implicit in the very understanding of the prime numbers (in cardinal terms).

And it also works in reverse whereby the prime numbers (in ordinal terms) are already implicit in the very understanding of the natural numbers (in a cardinal manner).

For example 6 is a natural number which is uniquely expressed as the product of two primes i.e. 2 * 3.

However, what is not properly realised in conventional terms is that the very nature of primes (as independent "building blocks") is transformed through a multiplicative relationship with each other in the context of composite natural numbers. So though in isolation, we may wish to view the primes as independent, whenever we use them in conjunction with other primes (as the unique factors of composite natural numbers), their identity is transformed in a holistic qualitative manner. And this points back to the key distinction of multiplication from addition i.e. that multiplication entails a (dimensional) transformation in the identity of number from a quantitative to qualitative status!

So in truth, the primes have both a relative independent identity (as constituent "building blocks") and a relative interdependent identity (through their relationship with other primes as the unique factors of the natural numbers). And to be properly meaningful, these aspects must be understood in a dynamic interactive manner entailing both quantitative (analytic) and qualitative (holistic) aspects.


So once again, we demonstrated with respect to the Zeta 1 (Riemann) function how the successive elimination of every 2nd term, every 3rd term, every 5th term, every 7th term and so on in the sum over natural numbers (defined for any integer value of s ≥ 1), leads to the corresponding elimination of the 1st term, the 2nd term, the 3rd term, the 4th term and so on in the product over primes expression.


However, once again a fascinating complementary form of behaviour applies to the corresponding Zeta 2 function.

So again, we define the Zeta 2 function as,

ζ(s₂) =  1 + s₂¹ + s₂² + s₂³ + ....

In the simplest case we let s₂ = 1/p, where p = 2. Then,

ζ(s₂) = 1 + 1/2 + 1/4 + 1/8 +.....   = 2 (i.e. 2/1).

This then concurs with the 1st term in the product over primes expression for the corresponding Zeta 1 function i.e. ζ(s₁) =  1/1¹ + 1/2¹ + 1/3¹ + 1/4¹ + .... (where s₁ = 1).

Though this function (the harmonic series) diverges, a product over primes expression can be given as

2/1 * 3/2 * 5/4 * 7/6 *...

So we can see that the 1st term in this (Zeta 1) product over primes expression, coincides with the corresponding value in the (Zeta 2) sum over the natural numbers (with the natural numbers now representing dimensional exponents).

Now with reference to the Zeta 2 function, while maintaining the value of p = 2, we will now consider the effect of eliminating every 2nd term (in the sum over natural numbers expression).

So we now obtain for this (reduced) function, 1 + 1/4 + 1/16 + ... = 4/3.

This value is now equal to the corresponding 1st term in the product over primes (Zeta 1) series (where s₁ = 2).

If we now additionally eliminate every 3rd term (with respect to the original Zeta 2 series) with again p = 2, we obtain the new reduced sum of 1 + 1/8 + 1/64 + .... = 8/7.

This value is then equal to the corresponding 1st term in the product over primes (Zeta 1) series (where s₁ = 3).  

If we now truncate the original Zeta 2 series further (with again p = 2) by now eliminating every 2nd, 3rd and 4th term, we obtain the new - even more truncated - sum of 1 + 1/16 + 1/256 + ...  = 16/15.

This value is then equal to corresponding 1st term in the product over primes (Zeta 1) series (where s₁ = 4).  

And of course we could continue indefinitely in this manner. 

So in general (with p = 2) in the Zeta 2 sum over natural dimensional numbers series, when we eliminate every 2nd, 3rd, 4th, ....nth term, the resulting value = the corresponding 1st term in the product over primes (Zeta 1) series where s₁ = n.


We then can go on to consider the effect of eliminating corresponding terms in the original Zeta 2 series, when p = 3.

Now for the original series (with p = 3), ζ(s₂) = 1 + 1/3 + 1/9 + 1/27 +.....   = 3/2

Now this this corresponds with the 2nd term in the Zeta 1 product over primes expression (where s₁ = 1).

Then when in the same manner as before (with now p = 3) we eliminate every 2nd term in the Zeta 2 series, we obtain  1 + 1/9 + 1/81 + ....  = 9/8.

This then corresponds with the 2nd term in the Zeta 1 product over primes expression (where s₁ = 2).

Then when we eliminate (with p = 3) every 2nd and 3rd term with reference to the Zeta 2 sum over natural (dimensional) numbers, we obtain 1 + 1/27 + 1/729  +.... = 27/26.

This then corresponds with the 2nd term in the Zeta 1 product over primes expression (where s₁ = 3).

So again in more general terms when we eliminate (for p = 3 i.e. the 2nd prime)  every 2nd, 3rd, ...nth term in the original Zeta 2 sum over natural (dimensional) numbers series, the result = the 2nd term in the corresponding Zeta 1 product over primes expression (where s₁ = n).

And in even more general terms, when we eliminate - where p represents the kth prime - every 2nd, 3rd,...nth term in the original Zeta 2 sum over (dimensional) numbers series, the result = the kth term in the corresponding Zeta 1 product over primes expression (where s₁ = n).