We have looked at the apparent fact that all the Riemann zeros lie on an imaginary line (drawn through .5 on the real axis).
Therefore in dynamic interactive terms, the "real" nature of the
natural number line i.e. where all real numbers are viewed in linear rational
terms (as lying on the 1-dimensional line) is complemented by the
"imaginary" nature of a corresponding number line on which all the
Riemann zeros are postulated to lie.
An as we have seen - again in dynamic interactive terms - when the interpretation
of the "real" number line takes place in the conventional analytic
manner (based on the assumed independence of number), then the interpretation
of the "imaginary" number line (containing the Riemann zeros) should
then rightly take place in a holistic manner (where the corresponding interdependence
of number - which appears paradoxical in analytic terms - is now equally
Looked at from a psychological experiential perspective, this implies that
both conscious (analytic) and (unconscious) holistic appreciation of number be brought into a dynamic equilibrium with each other (the ultimate
nature of which is truly ineffable).
Expressed more simply, this implies the balanced recognition of both reason
and intuition with respect to all number relationships.
Because of the reduced nature of accepted mathematical interpretation, number
is treated solely with respect to its quantitative (independent) nature that is
viewed in an absolute manner.
However there is always - inescapably - an unrecognised qualitative aspect
to recognition, where one accepts that numbers can be consistently
related with each other (i.e. as interdependent with each other).
In conventional mathematical terms this qualitative aspect is blindly
assumed to be consistent with the quantitative aspect (in a static absolute
manner) which strictly speaking is a completely untenable position.
So once we recognise the equal importance of both independent and interdependent
aspects, we must then treat number in a dynamic interactive
fashion with complementary aspects that are quantitative (independent) and qualitative
(interdependent) with respect to each other.
So in this dynamic context, the key issue for Mathematics is that consistency
can be maintained as between both quantitative and qualitative aspects.
And this consistency requires than a complementary holistic linear
formulation of number exists that complements the accepted analytic linear interpretation.
And this again is the statement of the Riemann
Hypothesis with the additional requirement that the imaginary line - containing
the Riemann zeros - passes through .5 on the real axis!
Once again this clearly cannot be proven through conventional mathematical
methods (as its axioms already blindly assume consistency).
So its truth depends on acceptance of the twin complementary
nature of the number system, which ultimately reflects an initial act of faith
in the subsequent consistency of the whole mathematical enterprise.
We also know that for every "positive" expression of a Riemann
zero i.e. a + it, a corresponding "negative" expression equally
exists i.e. a - it and again the assumption of the Riemann Hypothesis is that a
Now one might query as to what the second "negative" version of
each zero refers!
To briefly recap, I have suggested that the the frequency of the Riemann
zeros coincides very closely to the manner in which the natural factors of
numbers accumulate as we move up the number scale. And just as these
factors progressively increase (as we move higher up the linear scale to n), likewise it
is similar with the trivial zeros moving on a corresponding "circular" scale
up to t where n = t/2π.
However, we are referring solely
here to the "positive" zeros!
One must remember however
that in dynamic interactive terms, switching of reference frames continually
Therefore when we associate
the "positive" zeros with the holistic interpretation of the zeros
(that complement the analytic interpretation of the natural number system) then
the "negative zeros coincide with the corresponding analytic interpretation
of the zeros (that complement the corresponding holistic interpretation of
the natural number system).
And of course what is
"positive" and "negative" in this dynamic interpretation is
merely relative depending on context.
Therefore the fact that all
Riemann zeros are postulated to have both a positive and negative equal identity on
the imaginary line simply reflects the fact that all these zeros can be given -
in dynamic interactive terms - both a holistic and analytic interpretation