^{s}(where k is a rational fraction).

Thus again
in the best known case where s = 2

ζ(2) = 1/1

^{2}+ 1/2^{2 }+ 1/3^{2 }+ 1/4^{2 }+ … = 4/3 * 9/8 * 25/24 * 49/48 * … = π^{2}/6
And as we
have seen the general form of this relationship i.e. kπ

^{s}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 1

^{st}and 2^{nd}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) = kπ

^{s }, 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) = kπ

^{s }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.

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