Where s = 0 (1)

As is well known, with respect to Riemann's functional equation, there is an important link as between values of ζ(s) where s > 1 and corresponding values of ζ(1 – s) where s < 0.

Now again it is vital to appreciate the true dynamic relative nature of the Riemann zeta function in order to appreciate the corresponding complementary nature of these values.

However, in this regard it can be instructive to look at the two values (i.e. 1 and 0) that lie on the thresholds as it were of applicable values of s.

As we have seen, when s = 1, ζ(1) diverges in value to infinity. And in yesterday's blog entry, we looked at the deeper significance of this result. Every numerical expression has both a unique quantitative aspect, which is defined ultimately with respect to values expressed with respect to the default dimensional value of 1 and a unique qualitative aspect, defined ultimately with respect to a default base value of 1 (raised to other dimensional values ≠ 1).  

The one exception therefore arises when values are initially expressed with respect to the default dimension 1. Here no distinction can be made as between the quantitative and qualitative aspect with the qualitative then reduced in absolute fashion to the quantitative.

So the deeper reason why the Riemann zeta function remains undefined where s = 1, is because this relates to the one exception where the Zeta 1 function cannot be uniquely defined - in relative terms - with respect to both its Type 1 (quantitative) and Type 2 (qualitative) aspects.

Thus the absolute nature of ζ(1) in Type 1 quantitative terms, thereby removes the possibility of a unique dynamic relationship as between both Type 1 (quantitative) and Type 2 (qualitative) aspects.

However, when s = 1,  ζ(1 – s) = ζ(0) is now defined in dynamic relative terms as s no longer represents the default dimensional value = 1 (which defines the absolute quantitative approach).

However because of the complementary nature of ζ(s) and ζ(1 – s) with respect to the Riemann Zeta function,  ζ(0) will now lie at the opposite extreme in representing a purely relative notion of number i.e. both linear (quantitative) and circular (qualitative) notions are inextricably linked.

From one perspective, this can easily be seen from the form of the various terms of ζ(0).

So  ζ(0)  = 1/1⁰ + 1/2⁰ + 1/3⁰ + ...

Then in Type 1 (quantitative) terms, this expression would be written as 1¹ + 1¹ + 1¹ + ..., (where each term is ultimately expressed with respect to the default dimensional value of s = 1).

In Type 2 (qualitative) terms it would be written as,

 ζ(0) = 1⁰ + 1⁰ + 1⁰ + ..., (where the default base value of 1 is raised to dimensional values other than 1).

So with respect to dimensional values for the Type 1 and Type 2 aspects, we have 1 and 0 (which are holistically related to linear and circular notions of interpretation respectively).


However we can perhaps get a better intuitive notion of this inevitable link as between linear and circular notions by looking initially at the alternating (Eta) function,

η(1) = 1 – 1 + 1 – 1 + ...

Now if one attempts to add up the terms of this series, a problem inevitably arises, for when an even number of terms is involved, η (1) = 0; however when an odd number of terms is involved, η (1) = 1.

As the probability of an even number or an odd number of terms is similar, one way of resolving this dilemma is by obtaining the average of the two results = (0 + 1)/2 = 1/2.

Indeed this is precisely the result suggested for the Zeta 2 function where s₂ = – 1.

So  ζ(s₂)  = ζ(– 1) = 1 – 1 + 1 – 1 + ...   = 1/1 – s₂)  = 1/2. 

However, once again there is a deeper underlying meaning to this result.  

The circular logical approach - as the indirect rational expression of intuitive type appreciation - is ultimately characterised, in holistic terms, by the complementarity of opposite poles. 

Indeed the roots of 1 (which numerically represent this approach) ultimately can always be combined so that we obtain 1 – 1 = 0. 1 in this holistic context represents the positing (+) of one pole in conscious terms;  – 1 then represents the dynamic negation of this same pole in an unconscious manner. This (unconscious) negation of what has been (consciously) posited then leads to directly to intuitive recognition which indirectly is conveyed in rational terms through a circular logical approach.  

So from a holistic perspective, the sum of terms of η(1) represents an expression that keeps alternating as between circular interpretation (based on the complementarity of opposite relative poles) and linear interpretation (based on just one absolute pole).  

And in analytic terms, both approaches seem equi-probable with no reason for favouring one over the other.

Therefore, we can attempt to holistically solve this problem by expressing appropriate interpretation as representing an equal balance (i.e. average) as between circular (qualitative) and linear (quantitative) type understanding.

So 1/2 in this context carries a distinct qualitative connotation as representing the perfect balance as between two unique manners of interpretation.

And we have now indeed explained this result (1/2) from both the quantitative (analytic) and qualitative (holistic) perspectives.