Interesting Relationships Continued

Initially we started with the utterly simple equation x = 1, to quickly show how it possesses both linear (particle) and circular (wave) expressions, that correspond in fact to two distinct notions of mathematical dimensions.

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

Equally if x = 1, then xⁿ = 1 so that xⁿ – 1 = 0. (2)

Both relationships conform to the use of conventional accepted mathematical procedures.

However, (x – 1)ⁿ = 0 and xⁿ – 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 xⁿ = kⁿ so that here xⁿ – kⁿ = 0,

And initially we will confine ourselves to one particular class of this general expression (where n = 2).

Thus x² – k² = 0.

And when k = 2, this implies that x² – 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 x² – 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 x² – 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

ζ₁(2) = 4/3 * 9/8 * 25/24 * …    = π²/6

In other words, each individual term of the Zeta 1 (Riemann) function i.e. ζ₁(2), corresponds to a Zeta 2 function, that is directly related to the simple polynomial expression xⁿ – kⁿ  = 0 (where n = 2) .

And this correspondence can be fully generalised for all values of n > 1.

So for example,

ζ₁(3) = 8/7 * 27/26 * 125/124 * …    = 1.20205693…

And when n = 3, x³ – k³  = 0.

Thus where k = 2, x³ – 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 ζ₁(3)!

All the other terms of ζ₁(3) will correspond to the (infinite) reciprocal sum of the unique numbers (excepting 0’s) related to the polynomial expression xⁿ – kⁿ = 0 (n = 3; k prime). 

And this can be readily extended to all positive integer values of ζ₁(s), where s > 3, with corresponding Zeta 2 expressions  for each individual terms related to xⁿ – kⁿ = 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).

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.