Thus if x =
1, then x – 1 = 0 and
therefore (x – 1)

^{n}= 0. (1)
Equally if
x = 1, then x

^{n}= 1 so that x^{n }– 1 = 0. (2)
Both relationships conform to the use of conventional
accepted mathematical procedures.

However, (x
– 1)

^{n}= 0 and x^{n }– 1 = 0 are no longer equivalent expressions from this perspective!
So, in fact the attempted reconciliation of (1) and (2)
requires a much more refined mathematical interpretation that entails the
combined interplay of both quantitative (analytic) and qualitative (holistic)
aspects that are dynamically complementary with each other.

Then in yesterday’s blog entry, I extended (1) to
consideration of a particular class of the more general case, where (x – k)

^{ n}= 0, with k an integer > 1 (and n = 2).
Finally, I also considered the related case of (x + k)

^{n}with k an integer ≥ 1 (and again n = 2).
So now in this entry, we will switch to consideration of
(2).

So now we start with x = k. Therefore x

^{n }= k^{n }so that here x^{n }– k^{n}= 0,
And initially we will confine ourselves to one particular
class of this general expression (where n = 2).

Thus x

^{2 }– k^{2}= 0.
And when k = 2, this implies that x

^{2 }– 4 = 0.
Then the unique number sequence, associated with this
polynomial equation is,

1, 0, 4, 0, 16, 0, 64, …

Then ignoring the terms with 0 we obtain the (infinite) sum
of reciprocals of these numbers to obtain 1 + 1/4 + 1/16 + 1/64 + … = 4/3.
(i.e. r = 1/4)

Then when k = 3, this implies that x

^{2 }– 9 = 0.
Then the unique number sequence, associated with this
polynomial equation are,

1, 0, 9, 0, 81, 0, 729, …

Then again ignoring terms in 0, the (infinite) sum of
reciprocals of these numbers are

1 + 1/9 + 1/81 + 1/729 + … = 9/8 (i.e. r = 1/9)

And using just one more case to illustrate, when k = 5 (the next prime), this
implies that x

^{2 }– 25 = 0.
The unique number sequence then associated with this
equation is

1, 0, 25, 0, 625, 0, …

And again ignoring terms in 0, the corresponding (infinite)
sum of reciprocals of these terms is

1 + 1/25 + 1/625 + … = 25/24 (i.e. r = 1/25).

Now if we examine the product over primes expression for the
Zeta 1 function, where s = 2, then

ζ

_{1}(2) = 4/3 * 9/8 * 25/24 * … = π^{2}/6
In other words, each individual term of the Zeta 1 (Riemann) function
i.e. ζ

_{1}(2), corresponds to a Zeta 2 function, that is directly related to the simple polynomial expression x^{n }– k^{n}= 0 (where n = 2) .
And this correspondence can be fully generalised for all
values of n > 1.

So for example,

ζ

_{1}(3) = 8/7 * 27/26 * 125/124 * … = 1.20205693…
And when n = 3, x

^{3 }– k^{3}= 0.
Thus where k = 2, x

^{3 }– 8 = 0.
And the unique numbers associated with this polynomial
equation are

1, 0, 0, 8, 0, 0, 64, …

Therefore, again ignoring the terms in 0, the (infinite) sum
of reciprocals of this number sequence,

= 1 + 1/8 + 1/64 + … = 8/7 (i.e. r = 1/8)

And as we can see this is the first term in the product
sequence for ζ

_{1}(3)!
All the other terms of ζ

_{1}(3) will correspond to the (infinite) reciprocal sum of the unique numbers (excepting 0’s) related to the polynomial expression x^{n}– k^{n}= 0 (n = 3; k prime).
And this can be readily extended to all positive integer values of ζ

_{1}(s), where s > 3, with corresponding Zeta 2 expressions for each individual terms related to x^{n}– k^{n}= 0 (where n > 3).
However there is an important additional point to be made
here, which is crucial for proper understanding the true nature of
the product over primes expressions, associated with the Zeta 1 function (where
s is an integer > 1).

In establishing the equivalence of each individual term of
the Zeta 1 with a corresponding Zeta 2 expression, we edited out the role of
the 0’s (where meaningful reciprocals cannot be given).

However, the key point about the 0’s is that they point
directly to the holistic - rather than analytic - interpretation of multiplication.

So for example, in conventional mathematical terms, when 2
(or more) numbers are multiplied together, interpretation is given solely in a
reduced analytic i.e. quantitative manner.

So 2 * 3 = 6, with each number interpreted in a 1-dimensional
manner (as a point on the real number line).

However, properly understood, a qualitative transformation is
also necessarily involved. Thus in geometrical terms, we can easily appreciate
how 2 * 3 leads to a rectangular figure (that is correspondingly measured in square
i.e. 2-dimensional units).

Thus, though the result = 6 (from a quantitative perspective),
a qualitative transformation has also taken place (from 1-dimensional to 2-dimensional)
in the nature of the units involved.

However this in itself represents but an analytic interpretation
of such dimensional transformation.

A much subtler holistic transformation is also required,
which I will now again briefly illustrate.

Imagine we have two rows of coins (with 3 in each row).

In conventional mathematical terms the independent nature of each unit is
solely recognised!

So 3 = 1 + 1 + 1 (with the units independently interpreted in
a homogeneous manner that lacks any qualitative distinction).

Now we could independently add up the units (in each row) to
achieve a total of 6.

However the very basis of multiplication is to recognise a
common similarity as between the rows (or equally the columns).

Thus when we recognise for example that the two rows are
similar (with 3 units in each row), we can use the operator 2 with 3 to more
quickly attain our results.

So 2 * 3 = 6.

However the crucial point here - which is completely
overlooked in conventional mathematical interpretation - is that we have
thereby moved from the quantitative (analytic) notion of number independence to
the corresponding qualitative (holistic) notion of number interdependence.

Thus without both the analytic notion of number units as independent
of each other, and the holistic notion of numbers as interdependent with each
other (i.e. in sharing a similar identity) the very notion of multiplication
cannot be meaningfully interpreted.

So the very number sequences that we generate in the above
examples (all entailing 0’s), when properly interpreted, imply the holistic aspect
related to multiplication. And the higher the

dimension, the more holistic is the corresponding interpretation involved (indirectly signified by the increased repetition of 0’s in the unique number sequences generated).

dimension, the more holistic is the corresponding interpretation involved (indirectly signified by the increased repetition of 0’s in the unique number sequences generated).

Properly interpreted therefore the key significance of the
dual relationship as between the sum over natural numbers and product over
primes expressions of the Zeta 1 (Riemann) function - and indeed the entire
plethora of associated L functions - relates to the fact that addition and
multiplication are, relatively, analytic (quantitative) and holistic
(qualitative) with respect to each other.

And this can only be properly appreciated in a dynamic interactive manner, where both aspects - of equal importance - are understood as complementary with each other

If you can even vaguely appreciate the significance of what is stated above, then you will have no option but to conclude that the present accepted understanding of number (and all associated
number relationships) is simply not fit for purpose.

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