As is well known, Riemann defined an important equation with respect to the Riemann Zeta Function whereby for any value of s with respect to ζ (s) the corresponding value of ζ (1 – s) can be obtained from the Functional Equation.
So for example when s = 2, ζ (2) = π2/6, then through the Functional Equation
ζ ( – 1) = – 1/12.
As we have seen, there are in fact two complementary aspects to the Zeta Function, with the Riemann Zeta Function relating to Zeta 1, which - in the notation that I adopt - is ζ1(s).
Therefore in this context, Riemann's Functional Equation establishes a relationship as between ζ1(s) and ζ1(1 – s).
Now for s > 1, the infinite sum of terms of ζ1(s) results in a finite value value that is meaningful in terms of linear type interpretation.
ζ1(s) = 1/1s + 1/2s + 1/3s + 1/4s + ....
So for example,
ζ1(2) = 1 + 1/4 + 1/9 + 1/16 + .... = π2/6 = 1.644934...
However clearly the corresponding value through the Functional Equation for
ζ ( – 1) = – 1/12, does not conform to linear type interpretation.
Thus from this linear perspective,
1/1– 1 + 1/2– 1 + 1/3– 1+ 1/4– 1 + .... = 1 + 2 + 3 + 4 + .... which diverges to infinity.
However when we interpret this expression from a circular - rather than linear - perspective it can then be demonstrated how this series obtains a meaningful finite value (i.e. – 1/12).
Now switching from the quantitative (cardinal) to the qualitative (ordinal) interpretation of number, implies a corresponding switch from linear to circular type interpretation i.e. from the Type 1 to the Type 2 aspect of the number system.
So when properly interpreted in dynamic interactive terms, the Riemann Functional Equation establishes the appropriate relationship (with respect to the Zeta 1 Function) as between the Type 1 (cardinal) and Type 2 (ordinal) aspects of the number system.
Then when ζ1(s) = ζ1(1 – s) = 0, the real part of s = .5!
Thus the requirement that all the non-trivial zeros of the Zeta 1 Function lie on a line through .5, points to the key fact that it is at this point (on the real axis) and only this point that both the Type 1 (cardinal) and Type 2 (ordinal) aspects of the number system are identical.
Thus the famous Riemann Hypothesis can be seen thereby as the necessary condition for the ultimate identification of both the quantitative (Type 1) and qualitative (Type 2) aspects of the number system.
However we can equally approach this mutual identification of the quantitative and qualitative aspects of the number system from the perspective of the Zeta 2 Function (which in some important respects is easier to understand than Zeta 1).
So once again the Zeta 2 Function is initially defined with respect to a finite series.
So ζ2(s) = 1 + s1 + s2 + .....+ sn – 1
However this can then be extended to an infinite sequence of terms in two ways.
(i) in the accepted linear manner where we keep adding on single additional terms in a quantitative manner.
(ii) in the corresponding circular manner where we keep extending the series through taking regular groups of n terms while using a modular (clock) arithmetic as the appropriate means of interpreting the value of the series.
Now when s > 1, the infinite series for ζ2(s) will converge to a finite value according to the standard linear method of interpretation.
For example when s = 1/2,
ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ..... = 2.
However in the case of Zeta 2, if the series converges for s, then it will diverge for 1/s in linear quantitative terms.
Therefore when now s = 2,
ζ2(2) = 1 + 2 + 4 + 8 + .... which diverges to infinity in linear terms.
However if we now view this relationship in a circular manner, the series will indeed converge.
So for example 2 in this context would be represented 2 by modular or clock arithmetic (where 2 would be represented as 0).
So in terms of one cycle,
ζ2(2) = 1 + 2 = 1 (in modular arithmetic terms).
Thus when we extend this to an infinite sequence of terms (through orderly groups of 2) the value of the series will remain unchanged as 1!
1/(1 – s) = 1 + s1 + s2 + s3 + ......
Now, when – 1 < s < 1, this takes on a finite value in the normal linear quantitative) interpretation of a series.
However when s > 1 or < – 1, the infinite series will diverge from this same linear perspective.
However when we now apply the alternative circular perspective (implying modular arithmetic) the infinite series acquires a finite result for these values.
For example when s = 2, the LHS of the equation = 1/(1 – 2) = – 1.
Interestingly the value obtained using clock arithmetic = 1.
However whereas the LHS value is defined in linear terms, the corresponding RHS value is defined in a complementary circular manner.
Now, in the very dynamics of understanding one moves from linear (quantitative) to circular (qualitative) understanding through the negation of what is linear.
Therefore to express the circular result (on the RHS) in an appropriate linear manner we must - literally - negate this value to obtain – 1.
Then when s = 3, in modular (clock terms) again the sum of the infinite series (this time for groupings of 3 terms) = 1.
However we have now two non-trivial values (in the opening finite sequence of terms i.e. 1 + s1 + s2 . So the average value = 1/2 which when expressed in appropriate linear terms = - 1/2.
And again the LHS = 1/(1 – 3) = – 1/2.
We can now define the simple - though remarkable - Functional Equation for the Zeta 2 expression as follows:
ζ2(1/s) = 1 – ζ2(s) which holds for all values of s except where s = 1.
Here we have an important similarity with the Zeta 1 Function which likewise is defined for all values on the complex plane except s = 1.
The reason for this is again striking. Quite obviously when s = 1, no distinction can be made as between ζ2(s) and ζ2(1/s). So we lack the means therefore of establishing a distinctive relationship as between the linear (quantitative) and circular (qualitative) explanations of values.
So once again - when appropriately interpreted in dynamic interactive terms, the Zeta 2 Functional Equation is revealed as an expression that establishes intimate connections as between both the quantitative and qualitative type interpretation of number.
There is another striking aspect worth noting.
Clearly in multiplicative terms s * (1/s) = 1.
Well, remarkably the Zeta 2 Functional Equation expresses the corresponding combination of Functions (based on s) in terms of addition.
Thus ζ2(s) + ζ2(1/s) = 1.