Return of the Zeta 2 Function (1)

We have so far concentrated on the infinite expression of the Zeta 2 function, i.e.

ζ(s₂) =  1 + s₂¹ + s₂²  + s₂³ + ..., giving attention to real values of s_2, that lead to finite answers for the corresponding sum of terms.

And in this regard, we have been at pains to show the truly complementary role that this Zeta 2 function has with the corresponding Zeta 1 (Riemann) function.

So therefore with respect to real positive integer values of the Zeta 1 (Riemann) function, an exactly matching formulation can be provided through the corresponding Zeta 2 function.

And this applies with respect to both the infinite sum over the all natural numbers and the corresponding infinite product over all primes expressions.

So the (horizontal) expression for the infinite sum over the base natural numbers of the Zeta 1 (Riemann) function, with dimensional number (s) fixed is matched by the corresponding (vertical) expression for the infinite sum over the (dimensional) natural numbers of the Zeta 2 function, with base number fixed.

And we also saw how each term in the infinite product over all primes expression of the Zeta 1 (Riemann) function, with dimensional number (s) fixed, can be expressed through a corresponding infinite sum over the dimensional natural numbers of the Zeta 2 function with base number fixed.


However the finite version of the Zeta 2 function plays an equally important role, which can likewise be shown to be embedded in the corresponding Zeta 1 (Riemann) function.

Now the zeros of the Zeta 2 function arise directly in the context of its finite expression.

Thus in general terms

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

And in previous entries we defined it - initially - with respect to prime values of t.

Therefore the zeros of the function arise in this context for prime values, where,

ζ(s₂) =  1 + s₂¹ + s₂² + s₂³ + ... + s₂^{t – 1}   = 0.

The simplest possible case then arises where t = 2 so that,

ζ(s₂) =  1 + s₂¹, with  s₂¹  = – 1.

And this represents the non-trivial root of the equation 1 – s₂² = 0.
So, by definition, 1 – s₂¹ is always a default (i.e. trivial) root of the equation 1 – s₂^t = 0, with
1 + s₂¹ + s₂² + s₂³ + ... + s₂^{t – 1} = 0, thereby representing the remaining non-trivial roots.

So therefore in the case of t = 2, when we combine the one non-trivial root i.e. – 1 with the default trivial root + 1, we thereby are provided with the (indirect) means of expressing the ordinal notions of 1st and 2nd (in the context of two members).

Now once again, the 2-dimensional expression,

1 – s₂² = 0 , when correctly interpreted in holistic mathematical terms, provides the direct intuitive  
recognition of 1st and 2nd in this context as fully interdependent i.e. where 1st and 2nd are potentially interchangeable with each other.

The reduced 1-dimensional expressions i.e. where  s₂¹ = 1 and  s₂^1/2 =  – 1 respectively, provide the fixed interpretations of 1st and 2nd (in any actual context).

As stated often before, in the interpretation of the directions at a crossroads, both forms illustrated above, naturally arise.

Thus if we view the approach to the crossroads from both N and S directions, left and right turns (1st and 2nd) are fully interchangeable, depending on relative context. However if we view the approach to the crossroads as from either N or S directions (taken separately) then left and right will in this context have a fixed unambiguous meaning. So left i.e. the 1st turn (+ 1) thereby excludes right i.e. the 2nd turn  (– 1) and vice versa.

However a big question that then arises relates to the issue of extending from the prime numbered non-trivial solutions (which are - by definition - unique) to corresponding solutions for composite values of t (where some non-unique values necessarily arise).

So the 2-valued solution for 1st and 2nd arises where,

1 – s₂²  = 0.

Quite simply the 4 valued solution arises where 1 – s₂⁴  = 0 i.e. (1 – s₂²)(1 + s₂²) = 0.

This could then be expressed as,

1 – x²  = 0, where x = s₂².


Thus the two new solutions are given by 1 + s₂² = 0 i.e. + i and – i respectively.

So just as  in standard cardinal terms (through the Type 1 aspect of the number system)

4 = 2 * 2, i.e. 4¹ = 2¹ * 2¹ (as unique factors) we now have the means to indirectly express in corresponding unique terms the ordinal notions of 4 members (in the context of 4) as

1⁴ = 1² * 1², i.e.  strictly 1⁴ = 1² *( – 1)².

Thus there are two related processes taking place in the number system (which are fully complementary).

From the quantitative perspective, we have the process by which the cardinal primes (in base number terms) are uniquely combined with each other to form new composite natural nos. (in base number terms).

However, from the corresponding qualitative perspective we have the process by which the ordinal primes (in dimensional number terms) are uniquely combined with each other to form new composite natural numbers (in dimensional number terms).

So, for example, in cardinal terms, we can show (in the simplest case) that 4 represents the unique combination of 2 * 2 (as constituent factors).

However in corresponding ordinal terms, we can equally show (in this case) that the ordinal relationship of 1st, 2nd, 3rd and 4th (as the 4 individual members of a group of 4) represents the unique extension of 1st and 2nd to a higher dimensional context.

So what is vital to appreciate is that the cardinal notion of 4 (as representing unique constituent numbers) has no strict meaning in the absence of the corresponding ordinal notions of 1st, 2nd, 3rd and 4th. Likewise, in reverse fashion, the ordinal notions of 1st, 2nd, 3rd and 4th have no strict meaning in the absence of the cardinal notion of 4.

Thus we cannot conceive of the quantitative notion of "4" (as representing independent units) in the absence of the qualitative notion of 4 (as representing interdependent units).

Likewise we cannot conceive of the qualitative notion of "4" i.e. "fourness" (as representing interdependent units) in the absence of the quantitative notion of 4 (as representing independent units).

And when one properly grasps this simple fact, one must then accept that the conventional interpretation of the number system (in an absolute quantitative type manner) is no longer fit for purpose.

In fact the true nature of the number system is dynamic and relative, with complementary quantitative and qualitative aspects (which synchronistically arise in very process of understanding).