In all that
has been said so far, it is plain that we have in fact two distinct notions of
number, which dynamically interact in experience.

For example
the number 3 can be given an independent (cardinal) existence or alternatively
an interdependent (ordinal) type definition.

Thus, from
the first perspective, 3 is viewed in quantitative terms, which can be expressed
in terms of its component units as 1 + 1 + 1 .So the independent (absolute)
units are all homogeneous in nature (thereby lacking any qualitative distinction).

However 3
can also be given an interdependent ordinal type definition, which is expressed
in terms of its component members as 1

^{st}+ 2^{nd}+ 3^{rd}. Thus the units here are of an interdependent (relative) nature (thereby lacking quantitative distinction).
I have
dealt with this latter ordinal (Type 2) notion of number in my previous entries. This established the vitally important fact that associated with it is a simple
function (Zeta 2) with a corresponding set of zeros. These in effect show how to
convert Type 2 qualitative type meaning in a (reduced) Type 1 quantitative
manner. So the various prime roots of 1 (excluding 1) can thereby be indirectly
used to uniquely express its various ordinal members.

So again
the 3 roots of 1 can be used to express the unique ordinal nature of 1

^{st}2^{nd}and 3^{rd}members (in the context of 3). However since all ordinal type relationships necessarily entail the fixing of position with respect to one member in an independent fashion, one solution i.e. 1 is thereby trivial in this respect!
Once again
therefore the importance of the (unrecognised) Zeta 2 zeros is that they enable the
unique conversion of the qualitative
(Type 2) ordinal nature of number in an indirect quantitative (Type 1) manner.

It is very
important to appreciate this fact, as the Zeta 1 (Riemann) zeros can be then shown to
play a direct complementary role with respect to number.

The
general Zeta 2 equation can be expressed as follows:

ζ

_{2}(s) = 1 + s^{1 }+ s^{2 }+ s^{3 }+….. + s^{t – 1 }(with t prime). However ultimately this can be extended to all natural numbers.
Now this can equally be written as

ζ

_{2}(s) = 1 + s^{– 1 }+ s^{– 2 }+ s^{– 3 }+….. + s^{– (t – 1) }
Therefore
the zeros for this function are given as

ζ

_{2}(s) = 1 + s^{– 1 }+ s^{– 2 }+ s^{– 3 }+….. + s^{– (t – 1) }= 0.^{}
The corresponding Zeta 1
equation, i.e. the Riemann zeta function can be expressed by the infinite
equation.

ζ

_{1}(s) = 1^{– s }+ 2^{– s }+ 3^{– s }+ 4^{– s }+…..
And the zeros for this function are
given by

ζ

_{1}(s) = 1^{– s }+ 2^{– s }+ 3^{– s }+ 4^{– s }+….. = 0.
Notice the complementarity as
between both expressions!

Whereas the first expression
represents a finite, the second represents an infinite series of terms.

Then whereas the natural numbers 1,
2, 3, …. represent dimensional powers in the Zeta 2, they represent base
quantities with respect to the Zeta 1 and in reverse fashion whereas the
natural numbers represent base quantities with respect to the Zeta 1, they
represent dimensional powers with respect to the Zeta 2.

Now in the case of the Riemann
(Zeta 1) function we start with the notion of numbers, expressed as the unique
product of primes (representing base quantities) as independent
entities.

However this begs the obvious
question of how to express the corresponding interdependence of these numbers
in their consistent interaction with respect to the overall number system!

In other words, in dynamic
interactive terms, quantitative independence (with respect to individual
numbers) implies the opposite notion of qualitative interdependence (with
respect to their overall relationship with each other).

Thus the Riemann (Zeta 1) zeros represent
an infinite set of paired numbers that uniquely expresses in Type 2 fashion the qualitative
interdependent nature of the number system.

So just as the Zeta 2 zeros, as we
have seen convert (Type 2) qualitative ordinal type natural number notions in a
(Type 1) quantitative manner, in reverse fashion, the Zeta 1 zeros convert (Type
1) quantitative cardinal type prime notions in a (Type 2) qualitative manner.

Indeed we could rightly say -
though Conventional Mathematics has no way of adequately interpreting this
notion - that the Zeta 1 zeros express the collective ordinal nature of the
primes (with respect to the natural number system).

Indeed in even simpler terms,
the Zeta 1 zeros express the holistic basis of the cardinal number system.