In my last blog entry, I defined the simple Functional Equation for the Zeta 2 Function i.e.
ζ2(1/s) = 1 – ζ2(s)
This implies that ζ2(1/s) = ζ2(s) = 1/2 (when both are equal).
Of course a similar type of condition applies to the Zeta 1 Function.
In this case ζ1(s) = ζ1(1 – s) = 0, when real part of s = 1/2.
Now by appreciating the precise significance of this equality in the context of the Zeta 2 Function, we can learn a great deal more regarding the true significance of the Riemann Hypothesis (where all non-trivial zeros are postulated to lie on the imaginary line through 1/2).
Indeed the well-known Eta 1 Function (as the alternating counterpart to the Zeta 1 Function where s = 0 ) can be defined where s = 0 as,
1 – 1 + 1 – 1 + ....... .
Now this in turn corresponds to the Zeta 2 Function
i.e. ζ2(s) = 1 + s1 + s2 + s3 +... = 1/(1 – s) where s = – 1
So ζ2(– 1) = 1 – 1 + 1 – 1 + ....... = 1/2.
The key significance of this value for s (i.e. – 1) is that it represents the simplest of all the Zeta 2 non-trivial zeros.
These non-trivial zeros again correspond to the n – 1 roots of 1, where n is prime (i.e. all roots other than the default root = 1).
The n – 1 roots of 1 correspond to the n – 1 solutions of the finite equation,
1 + s1 + s2 + s3 +.... + sn – 1 = 0.
Then as explained in the previous blog entry, when we keep repeating (with regular cycles of n terms) these n – 1 roots will likewise act as solutions to the infinite Zeta 2 equation
i.e. ζ2(s) = = 1 + s1 + s2 + s3 +... = 0.
However when this is given a linear expression i.e. where the series can be increased by 1 one term at a time (rather than a regular cycle of terms) it will acquire a finite value.
We can illustrate this most easily with reference to the first of the non- trivial zeros i.e. – 1.
This arises as the 2nd root of 1 and represents the solution to the finite equation
1 + s = 0.
Then when we repeat with regular cycles of 2, – 1 is equally the solution to the infinite Zeta 2 expression,
1 + s1 + s2 + s3 +... = 0;
However from the formula, the value of 1 + s1 + s2 + s3 +... = 1/(1 – s).
Thus when s = – 1, 1/(1 - s) = 1/2!
Now this can be easily explained as an average of circular and linear type interpretations of the series.
Once again for s = – 1 ,
ζ2(– 1) = 1 – 1 + 1 – 1 + .......
If we take these terms in regular cycles of 2 (with complementary pairing of positive and negative values for 1),
ζ2(– 1) = 0.
However if we allow linear progression (with the series increasing by 1 term at a time) the value will keep alternating equally as between 0 and 1.
Thus the value of 1/2 in the formula relates to this linear interpretation of the alternating infinite series whose average value = 1/2.
So once again for the circular grouping of complementary even number of terms
ζ2(– 1) = 0.
However then in linear terms for an odd number of terms ζ2(– 1) = 1.
Thus the significance of 1/2 in this context is that it represents the mid-point in quantitative (analytic) terms as between circular ( = 0) and corresponding linear ( = 1) interpretation.
More crucially, from a corresponding holistic perspective it represents a perfect balance as between quantitative (cardinal) and qualitative (ordinal) type interpretation with respect to the number system.
And once again this is the significance of 1/2 in the context of the Riemann Hypothesis (with respect to the Zeta 1 Function).
It simply represents the condition for the ultimate identity of both the cardinal and ordinal aspects of the number system!
However this can only be properly understood in the context of an inherently dynamic mathematical approach (that gives equal emphasis to its Type 1 (cardinal) and Type 2 (ordinal) aspects.
Conventional Mathematics is quite unsuited for appreciation of this central issue as - by its very nature - it reduces qualitative to mere quantitative interpretation!
So we have illustrated the significance of 1/2 with respect to the Zeta 2 Function, in the simplest situation where the finite equation,
1 + s1 + s2 + s3 +.... + sn – 1 = 0, has just one non-trivial solution (as the 2nd root of 1). However all other situations can then be seen as extensions of this simplest case in a crucial respect i.e. that the sum of the n – 1 roots of 1 (excepting the default root of 1) = – 1.
Thus the circular interdependence of these n – 1 roots is demonstrated through their group sum = – 1.
So in this manner, 1/2 remains the value that ensures the equal emphasis - in both quantitative and qualitative terms - as between number (both with respect to linear interdependence and circular interdependence).
It must be remembered that this simplest case of the Zeta 2 series where s = – 1,
i.e. 1 – 1 + 1 – 1 + ....... provides an easy means for obtaining a numerical value for s = 0 (with respect to the Zeta 1 Function).
Indeed in an earlier blog entry, I showed in this manner how to obtain the corresponding Zeta 1 values for ζ1(0), ζ1(– 1), ζ1(– 2) and ζ1(– 3).
Just one more important point at this stage!
As we know with respect to the Zeta 1,
ζ1(0) = 1 + 1 + 1 + 1 +..... = – 1/2.
Then with respect to the Zeta 2 when s = 1,
ζ2(1) = 1 + 1 + 1 + 1 +.....
Though this from a merely quantitative perspective might seem to represent exactly the same expression as ζ1(0), it remains undefined with respect to the Zeta 2 Function!
Now the key explanation for this seeming anomaly is that the dimensional value to which 1 is raised = 0, with respect to the Zeta 1, whereas the base value in the case of the Zeta 2 = 1.
So we now can see that the crucial condition for both the Zeta 1 and Zeta 2 Functions remaining undefined is that the dimensional and base values in both cases respectively = 1.
And as the linear type interpretation that characterises Conventional Mathematics is precisely defined in qualitative terms by the dimensional number of 1, and the circular type interpretation that characterises Holistic Mathematics by the base number of 1, this means that neither the Zeta 1 nor Zeta 2 Functions can be properly appreciated in this manner. In other words proper interpretation entails the dynamic relationship of both types of meanings!
The crucial point is that - properly understood - all mathematical understanding is inherently dynamic (with two-way interaction as between both its quantitative and qualitative aspects).
When seen in this light the existing problem with what we misleadingly identify as Mathematics, could not be more fundamental (with the true nature of the Riemann Hypothesis remaining undefined from this limited perspective).