Thus again in the best known case where s = 2
ζ(2) = 1/12 + 1/22 + 1/32 + 1/42 + … = 4/3 * 9/8 * 25/24 * 49/48 * … = π2/6
And as we have seen the general form of this relationship i.e. kπs will continue to hold when we eliminate on the RHS (product over primes expression) an individual term or series of terms (based on a particular prime or series of primes) and then in corresponding fashion eliminate with respect to the LHS (sum over integers expression) the collection of terms where this prime or series of primes operate(s) as shared common factor(s).
Now this is already known since the time of Euler from an analytic (quantitative) perspective.
However what is fascinating - and indeed very important - is the fact that a convincing holistic (qualitative) mathematical rationale can likewise be given for the relationship.
And from the comprehensive dynamic interactive perspective of understanding, both analytic and holistic interpretations are necessarily involved.
When we look at π from the conventional quantitative perspective its value is explained as the (perfect) relationship of the circular circumference to its line diameter.
However what is not all properly realised is that - as with all mathematical symbols - a corresponding holistic (qualitative) interpretation can be given.
So from this perspective π represents the (perfect) relationship of circular to linear type understanding.
And this is precisely what I have been striving to explain in the last few entries where the very interpretation of the related product over primes and sum over the integers expressions (which necessarily must hold with respect to the Riemann zeta function and all subsequent L-function) in fact entails in dynamic interactive terms the two way relationship of circular to linear type understanding.
So the standard analytic (quantitative) interpretation of the primes and natural numbers is to literally treat all members as lying on the real line.
However the largely unrecognised holistic (qualitative) interpretation of the primes and natural numbers entails treating them as inter-related points on the unit circle (in the complex plane).
Therefore for example with respect the first prime i.e. 2 is viewed in analytic (quantitative) terms as a point on the real number line (measuring from the origin two units).
So again this is the cardinal notion of number as composed of independent homogeneous units that corresponds to standard rational interpretation.
However 2 is viewed in corresponding holistic (qualitative) terms as composed in ordinal terms of unique 1st and 2nd members that are freely interchangeable with each other.
Whereas the direct qualitative recognition of this ordinal relationship requires intuitive insight (as a direct representation of psycho-spiritual energy), indirectly it is then represented in quantitative fashion as two equidistant points on the unit circle (in the complex plane) i.e. + 1 and – 1 respectively.
When we for example one recognises that that the two turns at a crossroads can potentially be left or right (depending on the direction from which the crossroads is approached) then one is I fact implicitly giving recognition to the holistic (qualitative) notion of 2.
So if from one direction of approach the left turn is designated as + 1 then in this context the right is thereby – 1 (not a left turn). If however from the opposite direction of approach the right turn is now designated as +1, then in this context the left is – 1 (not a right turn).
Thus potentially before a direction of approach is given, left and right turns have a purely circular (paradoxical) meaning, which directly equates with holistic (qualitative) understanding.
However once a fixed direction of approach is given then left and right acquire in actual terms as separate independent (as two unambiguously distinct turns + 1 and + 1).
However potentially before a direction of approach is given left and right remain interdependent as, relatively, + 1 and – 1 respectively.
So when reason and intuition are properly integrated in understanding both analytic (quantitative) and holistic (qualitative) aspects of all mathematical relationships must be explicitly incorporated.
And again we can see the enormous reductionism which defines conventional mathematical interpretation i.e. where the qualitative aspect though of a radically distinctive intuitive nature is reduced in a merely rational manner.
Alternatively we can say that both linear and circular aspects must be perfectly combined (where circular in this context represents the indirect quantitative means of representing understanding that is directly of an intuitive i.e. qualitative nature)
So to properly understand the Riemann Zeta function and the fact that ζ(s) = kπs , where s is an even positive integer, it is not sufficient in the manner of Euler to provide a merely analytic (quantitative) interpretation of this relationship; equally in fact it is important to provide the corresponding holistic (qualitative) interpretation of the relationship, which I have been outlining in the last few entries.
For in fact, properly understood in dynamic interactive terms both interpretations are in truth inextricably inter-related.
Finally with respect to this contribution I wish to provide a simple holistic explanation as to the perplexing fact, of which Euler was well aware (in quantitative terms) that where s is an positive odd integer that no relationship of the form ζ(s) = kπs exists.
The key to the holistic appreciation of this apparent conundrum is the very notion of complementarity that always implies for any positive designation an exactly matching negative designation.
We can see this very simply with respect to the root structure of number.
As we have seen the n roots of 1 can be represented as n equidistant points on the unit circle (in the complex plane).
Now when n is even any root (with respect to one quadrant of the circle) can be directly connected through a diameter line drawn through to the centre of the circle to another root (in the opposite quadrant).
We can see this for example very simply with respect to the 4 roots of 1.
The two real roots + 1 and – 1 can be connected through the (horizontal) diameter line, whereas the two imaginary roots + i and – i can likewise be connected through the (vertical) diameter line.
And where this is the case - as with all even numbered roots - a direct complementary relationship thereby exists as between opposite roots.
However where we have an odd number of roots this is not the case.
Again we can see this simply in the case of the 3 roots of 1, where there is no way of connecting the different roots through a straight line diameter.
So in the case of odd numbered roots, a directly complementary relationship does not exist as between opposite roots. However when n is very large a relationship that progressively better approximates to perfect complementarity exists.