We now come to the qualitative interpretation of the Riemann Zeta Function for corresponding negative odd integer values i.e. where s = - 1, - 3, - 5,...
Now we can initially relate several interesting facts with respect to the numerical nature of these results.
1) Though indirectly these numerical values have indeed a discernible quantitative form, in direct terms they are qualitative in nature. Once again the Riemann Zeta Function for negative odd integer values of s, diverges to infinity.
So these alternative results that are obtained (which are finite) relate directly to the alternative qualitative - as opposed to quantitative - interpretation of number.
2) The numerical values (i.e. for the negative odd integers of s) are always rational in nature e.g. - 1/12, 1/120, - 1/252,...etc.
3) These numerical values consistently alternate as between their positive and negative expressions.
4) Though initially the magnitude of these rational values decreases (in absolute terms), the value then increases and indeed steadily accelerates with respect to the magnitude involved. These values in fact can be approximated very closely with simple formulae that are very revealing as regards their nature.
We will first deal here with the qualitative nature of these numerical results attempting to explain what is entailed.
Though the same principle is indeed involved for negative odd integer values, it may be easier to appreciate what is involved for s = 0.
Some time ago I set about trying to resolve this for myself.
Though the Zeta Function for s = 0 in linear terms leads to 1 + 1 + 1 + 1 + ...
(which in standard terms diverges to infinity) we can make progress by considering the alternative Eta Function where the terms alternate so that we obtain
1 - 1 + 1 - 1,...
Now this series has apparently two values. When we take an even number of terms the series sums to zero. However if we take an odd number of terms it sums to 1.
So one way of resolving the issue to get a single unambiguous answer is to obtain the mean of the two results = 1/2.
Then by a simple equation whereby the Zeta Function can be expressed in terms of the Eta, we can obtain an unambiguous result for the Zeta where for s = 0, the value is -1/2.
Now the qualitative interpretation of the Eta series (from which the final Zeta result is obtained) is very revealing.
When we combine two terms i.e. 1 - 1, we are in fact combining complementary opposites. However with odd terms we always have a single linear term left over.
So the Eta Function corresponds in qualitative terms with the consistent movement from (circular) complementarity to (linear) separation.
So in qualitative terms, the significance of the numerical value 1/2 that is obtained from this series relates to the fact that we are obtaining an even balance as between the two logical systems (linear and circular).
Now once again, the Zeta Series for s = 0 properly diverges (when interpreted in standard linear rational terms).
Quite literally therefore this convergent result of 1/2 thereby refers to interpretation that is - literally - only half rational (with an equal emphasis on circular intuitive understanding - that indirectly corresponds to the complementarity of opposites).
Therefore we can refer to the numerical value of the Eta Function as representing a certain degree of rationality in interpretation. It is likewise similar with respect to the derived Zeta Function value (which properly represents a derived degree of rationality with respect to interpretation).
The important point to grasp here is that in the conventional linear approach - where only quantitative interpretation is formally allowed - a solely rational logic is used (that is linear in nature).
However once we admit the relevance of both circular - as well as linear - logic, then interpretation of numerical values has a qualitative (as well as quantitative) significance.
So in this case the appropriate way of interpreting the Zeta result is as representing the degree of rationality required for its qualitative interpretation.
And as always, psychological and physical reality are complementary. Therefore such numerical results must thereby have a relevance for the physical world. One obvious connection is the rational nature of quantum mechanical behaviour that involves integral or half integral values.
Thus with respect to the subatomic physical world we have the dynamic interaction of both material particles (that are relatively independent) and energy states (representing their interdependence). This likewise represents the interaction of linear type (independent) and circular type (interdependent) behaviour. So the numerical values here in a physical context would then relate to degrees of phenomenal identity (which would be lower where dynamic interactivity is especially strong).