Odd Numbered Integers (6)

There are interesting features to the rational fractions generated by the Riemann Zeta Function (for negative odd integers of s) which have a fascinating qualitative significance.

When s = - 1, the result of the Function = - 1/12; then the value falls up to s = - 5 with the results for s = - 3 and s - 5, 1/120 and - 1/252 respectively. Then the absolute magnitude of the fraction starts to continually increase in an accelerating fashion.

Now it might appear that there is no discernible pattern to these values.

However a series of approximating formulaes can be generated that express the value of successive ratios of the Riemann Function (for the negative odd integers), and also the difference of successive ratios, and the difference of the differences of successive ratios. See Approximation Formulae for Negative Odd Integers of the Zeta Function. What is fascinating is that all these approximating formulae are based on s and pi and in the last case simply on pi.

In fact the difference of the difference of successive ratios can be approximated by the simple expression 2/(pi^2) based on absolute values for the Zeta fractions.

To see what this means we can illustrate with reference to the first four values of s = -1, -3, -5 and -7.

So {[Zeta (-7)/Zeta (-5)] - [Zeta (-5)/Zeta (-3)]} -

{[Zeta (-5)/Zeta (-3)] - [Zeta (-3)/Zeta (-1)]} is approximated by 2/(pi^2).

This approximation .20 is already correct to two significant figures.

Now in the region of Zeta values around 50, the approximation has already greatly improved to the extent that it is correct to 11 significant figures!


The significance of the pi connection here in qualitative terms can be easily explained.

For the positive even integers the results of the Riemann Zeta Function can be exactly expressed in terms of formulae involving pi. This in turn is due to the fact that these represent states of integration (entailing the full harmonisation of linear with circular type understanding).

For the negative odd integers the results can be approximated in terms of formulae entailing pi. And this approximation greatly improves as the absolute value of s increases!
What this entails from a psychological perspective is that, even though a degree of broken symmetry necessarily attaches to the negative odd values of s, that with higher numbered dimensions (in absolute terms) the differentiated element of understanding becomes so refined that it can be scarcely distinguished from (even) integral dimensions.

Thus the higher numbered dimensions of understanding are so refined and dynamic that differentiation (in explicit terms) becomes indistinguishable from integration.


The corresponding situation from a physical perspective is that interaction becomes so dynamic that material particles cannot be explicitly distinguished from (pure) energy.