We ended yesterday's entry with the "converted" holistic expression for ζ(– 3), i.e.
113/5(e2π – 1) + 213/5(e4π – 1) + 313/5(e6π – 1) + ... = 1/120.
Perhaps this is better written as,
2.113/10(e2π – 1) + 2.213/10(e4π – 1) + 2.313/10(e6π – 1) + ... = 1/120.
And then the corresponding "converted" holistic expression for ζ(– 5),
= 2.113/21(1 – e2π) + 2.213/21(1 – e4π) + 2.313/21(1 – e6π) + .... = – 1/252.
Intriguingly, the "converted" holistic expression for ζ(– 1) is exactly the same as that for ζ(– 13)
= – 1/12.
In this respect, it is just like a clock that at both at 1.00 hours and at 13.00 hours will show the same time (representing 1 AM and 1 PM respectively).
Though the early results for the smaller negative odd integer values of ζ(1 – s) involve the simple reciprocal of a whole number, later results look much more unwieldy with the numerator now attaining an increasingly large size. as can be seen here.
However in principle they can all be equally expressed in terms of the standard holistic "conversion" for ζ(– 13).
So ζ(– 11) = 2.691.113/2730(e2π – 1) + 2.691.213/2730(e4π – 1) + 2.691.313/2730(e6π – 1) + ...
I now wish to comment on the true qualitative holistic significance of these number transformations.
So again from the standard linear perspective the series for ζ(– 1) seems utterly straightforward, representing the familiar sum of the natural numbers i.e. 1 + 2 + 3 + 4 + ...
However, though we expect the result of this infinite series to diverge - in conventional terms - to
∞, in fact according to the Riemann zeta function, the result = – 1/12.
The implications here could not be more significant, for when one properly understands the true reason for this "strange" result it then becomes readily apparent that our present interpretation of number operations (such as addition) is simply not fit for purpose.
In the conventional approach both the quantitative and qualitative aspects of number understanding are formally abstracted from each other leaving the misleading impression that number operations can be understood in an absolute quantitative manner (without the need for any qualitative considerations).
So, again in conventional terms, the natural numbers are treated in an absolute independent manner (with respect merely to their quantitative identity).
However this position is strictly quite untenable for if numbers were truly independent of each other in an absolute fashion, then it would not be possible to understand them in relation to each other.
So this all important aspect of number interdependence - which inherently is of a qualitative nature - in conventional mathematical terms is simply reduced in a quantitative manner.
And this reduced quantitative way of thinking leads directly to a reduced notion of the whole as merely the sum of its constituent parts.
Thus, from this perspective, when for example we attempt to sum an infinite series, we do not give this series a proper collective identity (as befits its infinite status) but rather merely a part identity as the aggregate of its individual finite elements.
And we are so attuned to reducing whole to part notions in conventional mathematical terms that we no longer even question this approach.
However to coherently interpret the Riemann zeta function, we need to radically change the prevailing mathematical orthodoxy.
For rather than number misleadingly being treated with with respect to an absolute quantitative identity, number must now be understood in truly relative terms (with explicitly recognised quantitative and qualitative aspects).
In recent weeks, I have been showing how a hidden qualitative aspect must be introduced to coherently interpret values of the Riemann zeta function for values of s > 1.
So as well as the recognised linear element, an unrecognised circular aspect is likewise involved.
However because intuitively meaningful quantitative results emerge for the function (within this range of s) we misread the dynamic relationship as between both linear and circular aspects of the number system that is properly involved, thereby reducing interpretation in a linear quantitative manner.
However a mirror image picture then emerges with respect to values of the function for s < 0.
Again, both quantitative and qualitative aspects are involved. However, here the numerical result that is given primarily represents a true whole (i.e. qualitative) interpretation of the series (which is then indirectly converted in a quantitative manner).
Thus because of the complete lack of a recognised holistic aspect in conventional mathematical terms, the results that arise must necessarily remain non-intuitive from a limited linear perspective.
So in the most beautiful manner possible the Riemann zeta function - when correctly understood - shows how both quantitative (analytic) and qualitative (holistic) aspects fundamentally interact in a dynamic two-way relative fashion throughout the number system.