Where s = 0 (2)

We have seen how in dynamic interactive terms, appropriate interpretation of ζ(1) and ζ(0) lie at the two opposite extremes of understanding i.e. analytic and holistic respectively.

Interpretation of ζ(1) represents the absolute  extreme (defined with respect to the default dimensional value of 1). Here the attempt is made to explicitly abstract, in absolute rational manner, the quantitative from the corresponding qualitative aspect of number. Strictly however, as the qualitative must necessarily be implicitly involved, one can only dynamically approach absolute type understanding.

Interpretation of  ζ(0) then represents the relative extreme (defined with respect to the default base value of 1). Here the attempt is made to intimately understand, in a purely intuitive manner, the interdependent relationship as between quantitative and qualitative aspects (which is ultimately ineffable). Again in strict terms, where phenomenal recognition exists, one can likewise in dynamic experiential terms, only approach purely relative type understanding! So purely absolute (rational) and purely relative (intuitive) understanding exist as two limiting extreme positions, within which all actual understanding dynamically takes place.  

Indirectly this latter intuitive approach can be conveyed in a circular (paradoxical) rational manner.

However, in dynamic interactive terms, such circular understanding always entails initial linear understanding. So, for example, again to appreciate the truly relative (circular) nature of left and right turns at a crossroads (when approached simultaneously from the two directions, both N and S) , we must initially interpret each turn in an absolute (linear) manner (as approached from just one direction, either N or S).

So 0 (as representing the dimensional number 0), is best appreciated as representing the point at the centre of the unit circle (in the complex plane), which equally represents the centre of the circle and the centre of its line diameter. Here both linear and circular notions of understanding are ultimately reconciled in a purely intuitive relative manner.

Thus, once again, to explain the appropriate relationship (i.e. in dynamic interactive terms) with respect to the Riemann zeta function as between as between ζ(s) and ζ(1 – s) for s = 1, we must bridge the two opposite extremes of interpretation i.e. rational (in an absolute quantitative sense where the quantitative aspect is understood as fully independent of the qualitative) and intuitive (in a purely relative manner, where both quantitative and qualitative aspects are understood as ultimately fully interdependent with each other). 

And in the manner that I employ these terms, I customarily refer to the former rational extreme (where quantitative and qualitative aspects are explicitly separated) as analytic and the latter intuitive extreme (where quantitative and qualitative are explicitly understood as interdependent with each other) as holistic interpretation, respectively.

And again such holistic interpretation is then indirectly conveyed in a circular (paradoxical) rational manner.  

So what I am at pains to illustrate here, in repetitive detail, is the truly vital key point, that appropriate interpretation of the Riemann zeta function requires a dynamic interactive approach, incorporating both analytic and holistic aspects of interpretation.


Now in introducing the eta function, we have been able to demonstrate in the last blog entry that

1 – 1 + 1 – 1  + ...  = 1/2.

And we have shown how this result can be interpreted from matching analytic and holistic perspectives.

The question arises as to how we can then "convert" this result given for the alternating sequence of terms of the eta function to a corresponding result for the non-alternating sequence of terms of the zeta function. 

Now the customary manner of explaining this is as follows

ζ(s)  = 1/1^s  + 1/2^s + 1/3^s + 1/4^s + ^...  , and

η(s) = 1/1^s – 1/2^s  + 1/3^s – 1/4^s  + ^...

Thus η(s) = ζ(s) – 2(1/2^s + 1/4^s + 1/6^s + 1/8^s  + ^...) = ζ(s){1 – (2 * 1/2^s)}

Therefore ζ(s) = η(s)/(1 – 1/2^{s – 1}).

So in this case where s = 0,

ζ(s) = η(s)/(1 – 2) =  – 1/2.

This non-intuitive result (from the customary analytic perspective) arises due to the unique manner in which finite and infinite notions are juggled together with each other.

However once again the Zeta 2 function can be used to throw light on the precise nature of what is involved here!

Now where s_{2  = 2,}

1 + 2 + 4 + 8 +  = 1/(1 – 2) = – 1.

Now this seems initially to represent a completely non-intuitive result.

However to see what properly is involved, imagine a large cake which is divided into two equal portions. So let  one of these portions = 1.

This portion represents 1/2 of the total cake (and 1 portion).  

Now with respect to the remaining slice let is divide again into 2 halves. Now take

one of these portions (= 1/4 of total cake) and = 1/2 of the first portion and divide it into 2 equal slices.

Then with respect to the remaining slice again divide into 2 halves (= 1/8 of total cake) and = 1/4 of the first portion and now divide it into 4 equal slices.

Then with respect to the remainder of the cake again divide it in two halves (= 1/16 of total cake) and = 1/8 of first portion and now divide it into 8 equal slices. And continue on indefinitely in this manner!

Now if we look at the (part) fractions we obtain the infinite series 1/2 + 1/4 + 1/8 + 1/16 + …,   which in the limit = 1.

However if we look at the whole slices generated at each stage, we obtain 1 + 2 + 4 + 8 + … = – 1.

The significance of this negative result can be approached from another fascinating angle.

Let us start with the whole number (i.e. integer) “2”!

Now expressed more fully in Type 1 terms, 2 = 2¹. In other words 2 as a number on the real number line is - literally - 1-dimensional, so that it is expressed with respect to the (default) positive dimensional value of 1.

However, when we express 2 with respect to the corresponding negative dimensional value of 1, a fascinating transformation is involved. So 2 ^{– 1} = 1/2 i.e. (1/2)¹.

In other words by switching from + 1 to  – 1 with respect to the dimensional value of a number we thereby switch from whole to part recognition.

Again, in holistic terms, + implies the act of positing a phenomenal object (in a conscious manner). By contrast, – implies the corresponding act of negating the phenomenal object (in an unconscious manner).

Thus the role of the unconscious is vital in enabling one to switch, in every possible context, from the recognition of whole to corresponding recognition of part (and then in reverse manner from recognition of part to the the corresponding recognition of whole).

However though the unconscious is clearly vital in enabling such dynamic switching to take place (i.e. from whole to part and from part to whole) in the dynamics of understanding, this is completely lost in conventional interpretation where a reduced - merely conscious - explanation is given.

So 2 is interpreted as a number on the real line i.e. 2¹. Likewise 1/2 is interpreted as a number on the real line i.e. (1/2)¹. So in effect a reduced merely quantitative interpretation is given with respect to both numbers, where whole notions, which are inherently qualitative in nature, are reduced to parts (so that the whole in any context is thereby expressed as the sum of its quantitative parts).

Now with respect to our cake example,

1/2 + 1/4 + 1/8 + … = 1, concurs with conscious linear notions (where the whole is reduced in quantitative terms to the parts).

However  1 + 2 + 4 + 8 + … = – 1 concurs with unconscious linear notions, where part notions are now interpreted as wholes.

So the significance of  – 1 in this result is that it concurs directly with unconscious intuitive type recognition (where the conventional relationship as between whole and part is reversed).

Thus again the result of 1 in the first series i.e. + 1 concurs with conventional notion of the whole (in this quantitative context) as the sum of its constituent parts;

However the result of – 1 in the second series, concurs with the unconventional reverse notion of the part (in a distinctive qualitative context) as the sum of its constituent wholes.

And it requires true holistic appreciation (based directly on the unconscious aspect of understanding) to make intuitive sense of this latter result.

Earlier in this entry we have shown that

ζ(s) = η(s)/(1 – 1/2^{s – 1}).  

And as we now see, (1 – 1/2^{s – 1}) is directly related to the Zeta 2 function,

i.e.  ζ(s₂) =  1 + s₂¹ + s₂²  + s₂³ + ... = 1/(1 – s₂), where s₂ = 1/2^{s – 1}. 

Extending this realisation, all the results of the Riemann zeta function for ζ(s) (where s > 1) make intuitive sense from the analytic perspective (based directly on conscious recognition).

However corresponding results of the function for ζ(s) (where s < 0 and also the limiting case where s = 0) only make intuitive sense from the holistic perspective (based directly on unconscious recognition).

And as there are important links through the functional equation for values of ζ(s) and ζ(1 – s), this implies that it explicitly requires both analytic and holistic recognition to properly interpret the Riemann zeta function.