When I initially decided to tackle the Riemann Hypothesis, one key issue intrigued me with respect to negative values of the Riemann Zeta Function.
For example when s = – 2,
ζ(– 2) = 0.
However this seemed to me very puzzling as,
ζ(– 2) = 1/1 –
2 + 1/2 – 2 + 1/3 – 2 + .........
= 1 + 4 + 9 + ......
Thus in terms of conventional analytic interpretation, this series diverges to infinity.
Yet, paradoxically in terms of the Riemann zeta function its value is given as 0!
Though typically, negative values of s are derived through the accepted process of analytic continuation, I could find no convincing mathematical explanation given anywhere as to why such a quantitative series can lead to two utterly distinct values.
Eventually I managed to come up with an explanation, which however required re-interpretation of the Riemann Zeta Function in a radical new manner.
What I concluded was that in effect all mathematical relationships can be given two complementary interpretations that are analytic and holistic with respect to each other.
Then with respect to the Riemann zeta function, where the value of s is positive and > 1, a quantitative value arises that concurs directly with standard analytic interpretation.
However where the value of s is negative and < 1, a quantitative value arises that, in complementary fashion, concurs directly with holistic interpretation.
Thus the significance of the Riemann zeta function where for any value of ζ(s), a corresponding value can be given for ζ(1 – s), arises from the fact that it directly relates quantitative values, with an analytic interpretation on the RHS of the real axis, with corresponding values with a complementary holistic interpretation on the LHS.
It is here that the many years work I had devoted to holistic mathematical interpretation paid great dividends.
So I had already holistically defined the dimension 2 in a circular manner as representing the complementarity of opposite poles ( + 1 and – 1).
However this holistic meaning of 2 (as dimension) in turn can be given positive and negative values (i.e. as + 2 and – 2 respectively).
When + 2 is used, this signifies the (indirect) rational attempt to translate the paradoxical notion of complementary opposites (i.e. that simultaneously are both positive and negative with respect to each other)
When – 2 is used this signifies however the direct intuitive realisation of this relationship (which intrinsically is of a qualitative nature). So such intuitive realisation is thereby literally nothing in quantitative terms (which implies the holistic meaning of 0).
I was helped towards this realisation through a deep immersion in the works of St. John of the Cross who deals intensively with the process of "passive" purgation" in the spiritual life.
The goal of such purgation is to cleanse the mind of indirect conscious attachment to symbols that intrinsically are designed to serve a nondual spiritual purpose.
And the goal of such cleansing - as put so starkly by St. John - is to achieve "nada" (i.e. a state of nothingness or spiritual emptiness).
Thus in the context of mathematical understanding, such contemplative cleansing would be designed to reveal the pure holistic meaning of symbols (that are directly understood in an intuitive manner).
Thus, when correctly understood, all mathematical symbols have both rational (analytic) and intuitive (holistic) meanings with - in dynamic experiential terms - a dynamic interaction continually taking place as between both aspects. However in conventional mathematical terms the intuitive (holistic) aspect is reduced to rational (analytic) interpretation in an absolute fashion!
So when one holistically interprets the value of the Riemann zeta function that corresponds to – 2, this implies appreciation that relates to the direct intuitive realisation of the complementarity of two opposite poles (which is 0 in quantitative terms).
Thus through Riemann's functional equation, we can directly relate analytic quantitative values on the RHS of the real axis (where s > 1) with corresponding holistic quantitative values on the LHS (corresponding to 1 – s).
Crucially in the critical region (where 0 < s < 1), values arising share both analytic (quantitative) and holistic (qualitative) characteristics.
The significance therefore of the condition (to which the Riemann Hypothesis relates) that s = .5, is that through the Riemann function, this now signifies the vital condition where both analytic (quantitative) and holistic (qualitative) values coincide.
The additional requirement that the value of the zeta function = 0 (for all values of s) then directly relates to the non-trivial zeros of the function.
Thus meaningful interpretation of the Riemann zeta function (and associated Riemann Hypothesis) requires a dynamic interactive appreciation entailing the two-way complementary relationship of numerical values (that can be given both analytic and holistic meanings).
From this dynamic perspective the non-trivial zeros represents those points (represented on the imaginary axis trough .5) where both the analytic (quantitative) and holistic (qualitative) interpretation of mathematical values directly coincide.
Alternatively this could be expressed as those points where both randomness and order dynamically coincide with respect to operation of the number system.