We
ended yesterday's entry with the "converted" holistic expression for ζ(– 3), i.e.

1

^{13}/5(e^{2π }– 1) + 2^{13}/5(e^{4π }– 1) + 3^{13}/5(e^{6π }– 1) + ... = 1/120.
Perhaps
this is better written as,

2.1

^{13}/10(e^{2π }– 1) + 2.2^{13}/10(e^{4π }– 1) + 2.3^{13}/10(e^{6π }– 1) + ... = 1/120.
And
then the corresponding "converted" holistic expression for ζ(– 5),

= 2.1

^{13}/21(1 – e^{2π}) + 2.2^{13}/21(1 – e^{4π}) + 2.3^{13}/21(1 – e^{6π}) + .... = – 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) = 2.691.1

^{13}/2730(e^{2π }– 1) + 2.691.2^{13}/2730(e^{4π }– 1) + 2.691.3^{13}/2730(e^{6π }– 1) + ...
= 691/32760.

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.

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