So for example when s = 2, ζ(s) = πs/6.
It is also well known, such a relationship does not apply to positive odd integers of s, where a closed form numerical solution apparently does not apply.
So a definite asymmetry applies with respect to the behaviour of ζ(s) with respect to positive even and positive odd integers respectively.
One simple way of appreciating this difference is through reference to the nature of even numbered and odd numbered roots of 1.
Where even numbered roots are concerned, full complementarity exists with respect to the roots, where half can be exactly matched against the other half (through multiplication by – 1).
So for the relatively simple case of the 4 roots of unity, we can divide the roots into two halves, with 1 and i in one half, matched in complementary fashion by – 1 and – i in the other.
The implication of this is then easy to illustrate in geometrical terms, for when we then represent the four roots as equidistant points on the circle of unit radius (in the complex plane), opposite roots can be connected by straight line diameters (through the central point).
So we can appreciate here the important relationship as between the the circular circumference and these line diameters, with π representing the ratio as between both.
However where odd numbered roots are concerned such complementarity does not exist, with one i.e. + 1 always in a sense isolated to a degree from the other roots which exist as complex conjugate pairs.
So for example the 3 roots of 1 would be represented as + 1, – .5 + .866i and – 5 – .866i respectively.
Therefore in geometrical terms, no means exist for directly connecting opposite points - where again these roots are represented as equidistant points on the unit circle - through straight line diameters. So the direct connection with π for odd-numbered roots therefore does not arise.
However there is a much deeper reason why these various roots of 1 are so relevant in this case, which goes to the heart of the very nature of number.
As we have seen, in conventional mathematical terms, number is treated in an absolute linear - literally 1-dimensional - fashion. So for example, when two numbers i.e. 2 and 3 are multiplied together, the transformation involved in treated in a merely reduced quantitative fashion.
So in conventional terms 2 * 3 = 6. Now, expressed more clearly, from this perspective, 21 * 31 = 61 (with the 3 numbers now all treated as lying on the real number line).
However, as I have continually stated on this forum, properly speaking, number has two aspects (Type 1 and Type 2) which continually interact with each other in a truly interactive relative manner. So all numbers thereby possess both a quantitative aspect of relative independence and a qualitative aspect of relative interdependence respectively. And whereas the quantitative aspect directly corresponds with rational interpretation (in an analytic fashion) the qualitative aspect directly corresponds with intuitive appreciation (in a holistic manner), which then indirectly can be given a refined rational interpretation in circular i.e. paradoxical terms.
So the constant appeal I make to complementary type relationships, which can only be properly appreciated in a dynamic relative context, points directly to holistic - rather than strict analytic - interpretation!
So whereas in Type 1 terms, a number such as 2 is defined as 21, with 2 here as base number having (from one valid reference point) a quantitative meaning, in Type 2 terms, 2 is now defined in inverse fashion as 12, with 2 now as dimensional number, having - relatively - a qualitative meaning.
One simple way of appreciating this distinction as between the two aspects of number is the acceptance that both the cardinal and ordinal appreciation of number must be given - relatively - distinct interpretations. When this is not done - as unfortunately is the unquestioned practice in conventional mathematical terms - ordinal meaning, which points to a unique relationship as between the individual members of a number group, is invariably reduced in a merely quantitative (i.e. cardinal) manner.
With this in mind, let me briefly explain the profound qualitative significance of the Type 2 aspect.
So 2 now (i.e. 12) has a holistic meaning, representing direct intuitive appreciation of the interdependence of two units (as complementary in positive and negative terms with each other).
I have frequently explained that this holistic appreciation is implicit in our very ability to understand that left and right turns at a crossroads have a merely relative meaning (depending on the direction from which they are approached).
However the deeper significance is that all understanding is necessarily conditioned by external (objective) and internal (subjective) aspects, which are complementary in dynamic fashion with each other. Therefore the "objective" external notion of a number such as "2" has no strict meaning in the absence of the corresponding "subjective" internal mental perception of the number "2". So the two aspects are - relatively - positive and negative with respect to each other, with both continually switching as reference frames (external and internal) likewise continually switch in experience.
And it is this continual switching which implicitly enables our very ability to distinguish as between the cardinal and ordinal notion of number!
However the huge unaddressed problem is that a highly reduced - merely absolute - interpretation is then explicitly offered as the "correct" formal mathematical interpretation.
This then has profound implications for proper appreciation of the Zeta 1 (Riemann) function, which when grasped, properly explains the relationship of the function (for positive even integers of s) to corresponding powers of π.
So what is now required is a new much more refined way of expressing the Zeta 1 function that incorporates both Type 1 and Type 2 aspects!
To illustrate, we will initially concentrate on the well-known case of ζ(s), where s = 2.
In conventional terms, this is interpreted in an (absolute) quantitative Type 1 fashion,
So ζ(2) = 1/12 + 1/22 + 1/32 + ...
Thus we obtain,
ζ(2) = 1/1 + 1/4 + 1/9 + ...
However, we will now introduce the Type 2 aspect, and as in this case all numbers are raised to the power of 2, this means that each term (already defined in a Type 1 manner) must be associated with its Type 2 counterpart.
So in this more refined formulation,
ζ(2) = (1/1)1 * 12 + (1/4)1 * 12 + (1/9)1 * 12 + ....
Thus the function is now defined with respect to the Type 1 aspect (interpreted in a linear quantitative manner) and a Type 2 aspect (interpreted - indirectly - in circular qualitative fashion).
Again the qualitative interpretation of 2 refers to the recognition of + 1 and – 1 as relatively complementary (and thereby interdependent with each other).
Then when we attempt to express this in a linear 1-dimensional fashion, by obtaining the two roots of 1, we again get + 1 and – 1 (but now in a relatively independent fashion).
And this is intimately tied up with the very way in which we implicitly form appreciation of ordinal numbers!
When we intuitively understand + 1 and – 1 as interdependent, this likewise enables us to appreciate 1st and 2nd as potentially interchangeable. So again left and right turns at a crossroads (which could be represented in numerical terms as 1st and 2nd) are potentially interchangeable, depending on the direction from which they are approached.
However in any actual context, 1st and 2nd are given a definite location. So again, if one in actual terms approaches a crossroads (heading N) left and right (and thereby 1st and 2nd) in this context have a definite meaning.
So when we look at the Zeta 1 function in this revised dynamic interactive manner, it carries both a linear quantitative and circular - indirectly - qualitative meaning (as the paradoxical rational expression of intuitive understanding).
So again as we have seen the numerical result of this function (for s = 2) can be expressed as a relationship involving the square of π.
The holistic qualitative expression of this same relationship entails appreciation, in this 2-dimensional context, of the direct relationship as between linear (quantitative) and circular (qualitative) meaning.
And properly understood, both the analytic and holistic interpretations are intimately related to each other.
Without the holistic appreciation (through the Type 2 aspect of the number system), there would be no way of coherently explaining why the quantitative result should entail a relationship involving the square of π.
And without corresponding quantitative appreciation, it would be equally difficult to coherently explain how the qualitative necessarily entails a relationship as between circular and linear aspects.