Wednesday, April 20, 2011

The Trivial Zeros (4)

As is well known, Euler did great pioneering work on the Zeta Function (where it is defined for all real values of s > 1). s in this context representes the dimensional number (i.e. power or exponent) to which all the terms in the series are raised.

So for example when s = 2, we get

1/(1^2) + 1/(2^2) + 1/(3^2) + 1/(4^2) +…..

= 1 + 1/4 + 1/9 + 1/16 +……

Now Euler was able to show that the sum of these terms converged to (pi^2)/6.

This result is very revealing for not alone does the zeta function converge to a finite quantitative value (when defined in 2-dimensional terms) but the result is closely related to pi.

Now in quantitative terms pi represents the relationship between the circular circumference and its line diameter.

In corresponding qualitative fashion pi represents the relationship as between circular and linear type understanding.
And as we have seen the very attempt to rationally convey the nature of 2-dimensional interpretation requires both circular and linear type understanding!

When we consider the Euler Function for values of s < 1, it seemingly breaks down as the series diverges to infinity.

So for example when s = – 2, we get

1^2 + 2^2 + 3^2 + 4^2 +…..

= 1 + 4 + 9 + 16 +….. which clearly diverges in quantitative terms (to infinity).

However by defining s (the dimensional power) for zeta in complex terms, through a process of analytic continuation, Riemann was able to extend the domain of definition of the Riemann Hypothesis to all values of s (with the exception of 1).

So from this new extended perspective when s = – 2, the value of the function = 0.

Now on the face of it this result is hugely problematic in quantitative terms.

Remember from a standard linear interpretation, which defines quantitative values, when s = – 2, the value of the Zeta Function diverges to infinity.
However from this new perspective provided by Riemann, the value of the same Zeta Function = 0.

So the question arises - or at least should arise - as to how we can reconcile the two results.

The truly remarkable answer that unfolds is that whereas the divergent result for the Function corresponds to standard quantitative interpretation, the corresponding convergent result = 0 corresponds to the alternative unrecognised qualitative interpretation.

Put another way, all meaningful quantitative results are based qualitatively on linear interpretation whereby the values are reduced in default 1-dimensional terms.

Interestingly therefore the only value for which the Riemann Function is undefined is where s = 1.

As the quantitative result diverges here, the corresponding qualitative interpretation (which is identical in this one case to the quantitative) must also diverge.

However in all other cases, an alternative qualitative interpretation for the Function can be given in accordance with the dimension (power) of s.

And as both interpretations are paradoxical in terms of each other, this entails that if the Function diverges (from a standard quantitative perspective), it necessarily converges from the alternative qualitative perspective.

We have already given meaning to the negative sign of the 2nd dimension as corresponding to the purely intuitive appreciation of the complementarily of real opposite poles (such as external and internal).

So when we understand the Riemann Zeta Function from a qualitative perspective, with interpretation according to the negative sign of the 2nd dimension, this corresponds to pure intuitive recognition that is empty of form. Thus numerically this is represented by 0.

Therefore the remarkable finding we have found is that the first trivial zero - and by extension all trivial zeros - relate to a qualitative rather than quantitative interpretation of numerical values.

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