Now
properly understood - in dynamic interactive terms – these two Functions are
complementary with each other.

Once again the Zeta 1 Function (i.e. the standard Riemann Function) is given in conventional terms as the infinite series,

1

^{–s }+ 2

^{–s }+ 3

^{–s }+ 4

^{–s }+……..

where each of the natural numbers is raised to the negative of the same dimensional value (i.e. s).

By contrast
the Zeta 2 Function is given as the finite series,

1 + s

^{1 }+ s

^{2 }+ s

^{3 }+….. + s

^{t – 1 }(with t prime),

where in
reverse, s as the quantitative value is raised to each of the
natural values (up to t – 1).

The value of s will of course be distinct for each Function!

The value of s will of course be distinct for each Function!

So through
each Function we have the expression of the natural number system (both in
finite and infinite terms) alternating in each case as between dimensional values
(that are qualitative) and - relatively - base number values (of a quantitative nature).

Now the
zeta zeros (Zeta 1 and Zeta 2 respectively) represent the solutions of each
Function, where the sum of terms in the series expression = 0.

Thus when we look at these solutions in a
dynamic interactive manner, the crucial key as to their nature is revealed.

Putting it
simply, the zeta zeros represent the solutions for which both the quantitative
and qualitative aspects of the number system are perfectly reconciled (from two
complementary perspectives).

In
fact it is easier to initially appreciate the precise nature of such holistic
solutions from the Zeta 2 perspective (which we will return to shortly).

So the Zeta
2 solutions show how the internal nature of each prime number is comprised
of a series of individual natural numbers in ordinal terms, with the
(indirect) quantitative nature of each of these natural numbers (in isolation)
perfectly reconciled in a holistic manner with the overall qualitative interdependence
of this prime number grouping.

Now the
uniqueness of each prime number in this context is that all its natural number ordinal
members (except 1) are uniquely defined in an (indirect) quantitative manner.
In other words these prime number roots are unique for each prime number group.

Thus for
example 5 is a prime number. Now by definition 5 as a group is composed of its
5 individual members which are 1

^{st}, 2^{nd}3^{rd}, 4^{th}and 5^{th}in natural number ordinal terms. Each of these ordinal members can then be indirectly defined in a quantitative (relatively independent) manner through the corresponding 5 roots of 1.
Then when we
combine these roots to demonstrate their corresponding relative interdependence
(in a qualitative manner) their sum = 0.

So in this
way we can demonstrate a unique holistic balance for each prime number as
between its individual members (in quantitative terms) and their overall
collective interdependence (in a qualitative manner).

We are
accustomed to interpreting prime numbers in an absolute analytic fashion as the
independent quantitative building blocks of the natural number system.

So from
this perspective, the qualitative relational aspect of the primes become fully
separated and then fully reduced to the quantitative aspect.

And once
again the analytic aspect of the number system is defined by such absolute
reductionism (whereby it is formally interpreted in a merely quantitative
manner)!

However the
holistic aspect is altogether distinctive in nature (and lying at the other
extreme) where quantitative elements, while maintaining their independent
identity in a relative manner, are yet seen - relatively - as fully interdependent
with each other in qualitative collective terms.

And it is vital
to grasp that zeta zeros directly represent the latter holistic interpretation
of number!

And so once
again this is exactly why the very nature of these zeros is - necessarily - completely
missed through the conventional mathematical approach.

We cannot therefore
hope to appreciate the holistic nature of number (entailing the balanced interaction
of its quantitative and qualitative aspects) in an analytic manner (that is
formally based on mere quantitative interpretation).

Now I have
spelt out in some detail here the holistic nature of the Zeta 2 zeros.

Though we
will return to this in future entries in this “holistic” series on my blog, the
general nature of the - greater recognised - Zeta 1 zeros can easily be
expressed.

Just as the
Zeta 2 zeros are defined for the internal ordinal natural members of each individual
prime, the Zeta 1 zeros are defined externally for the number system as a whole
with respect to the collective nature of the primes with respect to the
(composite) cardinal natural numbers.

So the non-trivial
zeros of the Zeta 1 function ensure a complementary type of holistic harmony as
between the qualitative interaction of primes and the quantitative natural
numbers resulting.

Thus the Zeta
1 and Zeta 2 serve as complementary mirrors to each other.

In the case
of the Zeta 2 each cardinal prime is defined in an ordinal manner through a
finite set of natural number members (1, 2, 3…p).

In the case
of Zeta 2, the cardinal natural numbers are defined in terms of the ordinal
activity of the prime number groupings that uniquely define their nature.

When one
can dynamically interpret these relationships simultaneously in both Zeta 1 and
Zeta 2 terms, the primes and natural numbers are revealed as perfect mirrors of
each other (in an ultimately ineffable manner). And this is because the Zeta
zeros likewise serve as perfect holistic mirrors of each other.

Another interesting
feature worth emphasising here is that the true holistic nature of the zeta
zeros (Zeta 1 and Zeta 2) implies that they in fact represent energy states.

Once again
the conventional analytic interpretation of number is based on the apparent
existence of numbers absolutely existing as rigid immutable forms.

So in
analytic quantitative terms “2” for example is interpreted as a fixed
number symbol.

However
just as matter can be transformed into energy (and energy into matter), it is
exactly similar in mathematical terms.

So the
understanding of numbers existing as absolute forms represents the extreme analytic
interpretation (where the qualitative aspect is reduced to the quantitative).

However
when, at the other extreme, we understand numbers in pure holistic terms, they are
understood as representing energy states.

Of course both
of these aspects necessarily interact to a degree in experience, representing the inherent dynamic nature of number.

So here
number forms in rational (analytic) terms interact with number energy states in
a (holistic) intuitive manner leading to a continual dynamic transformation in the
experience of number.

Of course
in formal terms such experience is greatly misrepresented in a mere rational
(analytic) manner.

Recent
strong correlations as between the Zeta 1 zeros and energy states in quantum
physics suggest that in some sense the zeros do indeed represent energy states!

However
none of this can have intuitive resonance within the restricted conventional manner in which
number is presently interpreted.

However it
makes perfect sense when one interprets the zeros in an appropriate holistic
manner.

However what is still being completely missed is that from a dynamic perspective, physical and psychological aspects are complementary.

However what is still being completely missed is that from a dynamic perspective, physical and psychological aspects are complementary.

So not
alone do the zeta zeros represent physical energy states, they equally
represent psychological - or more correctly - psycho-spiritual energy states.

Therefore, correctly
understand the zeta zeros (in appropriate terms) represent psycho-spiritual
energy states that have complementary correlates in nature (in a psycho-physical
manner).

And of
course in dynamic interactive terms, physical nature is necessarily understood
in a psycho-physical manner!

The
spiritual traditions often make the distinction as between dual and nondual
reality.

Dual
reality - literally - relating to “twoness” refers to analytic type understanding,
where the poles of understanding (such as internal/external and quantitative/qualitative
are clearly separated.

Nondual
reality - literally not-two - refers to holistic type understanding, where the
opposite poles of understanding (that necessarily condition all experience) are
understood as complementary and ultimately identical in a spiritual manner.

So in these
terms, the zeta zeros relate directly to a pure nondual intuitive type
appreciation at the deepest possible level of the ultimate interdependence of
both the qualitative and quantitative (and indeed internal and external) aspects
of the number system.

Now as with
all phenomenal experience, holistic appreciation is necessarily of a relative
approximate nature (which entails a degree of supporting analytic
activity).

So the zeta
zeros can indeed be encoded in a phenomenal numerical manner.

However it
is important to remember that their inherent nature is holistic and thereby
they relate to the most dynamically interactive experience that is
humanly possible, while remaining in the world of phenomena. In this sense they are
like the most fleeting of virtual particles.

We are a
long, long way in our present evolution from being able to appreciate their inherent nature truly in the
appropriate interactive manner (which will entail the most advanced states of
spiritual contemplative development).

However we
have come far enough to begin to understand their true nature, which is
as far away as is possible from the rigid analytic notion of number that dominates conventional mathematical interpretation.

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