Wednesday, August 1, 2018

Dynamic Appreciation of Nature of Riemann Hypothesis

Imagine we are presented with the first of the Riemann zeros i.e. 1/2 + 14.134725 …, (or equally 1/2  14.134725 …), without knowing the origin of the value and asked to find a solution to the following infinite equation,

k1– (1/2 + 14.134725 …) + k2– (1/2 + 14.134725 …) + k3– (1/2 + 14.134725 …) + k4– (1/2 + 14.134725 …) + …   = 0,

where k1, k2, k3, k4, … are complex numbers, then by lucky chance we might be able to find that when k1, k2, k3, k4, … represent the natural numbers 1, 2, 3, 4 , … respectively, that the value of the equation = 0.

And imagine in turn if we were presented in turn with all of the known Riemann zeros and asked to solve the same equation for each one individually, we might then find that in every case when k1, k2, k3, k4, …, represent the natural numbers 1, 2, 3, 4 , … respectively, that again the value of the equation = 0.

So this would indeed be remarkable! For on the one hand we would have a list of known exponents (i.e. dimensional numbers) that all lie on the same imaginary line (through 1/2).

Then on the other hand we would have an infinite series i.e. the natural numbers all on the real line (through 0) that provide in each case a solution to the equation.

However there would still remain a significant problem in that we would not be able to guarantee in any case that perhaps that another set of solutions for k1, k2, k3, k4, …  (with some or all not necessarily not lying on the real line) might likewise provide a solution to the equation.

Also we could not guarantee that for some future imaginary value for the exponent (not presently known) of a known zero, that is placed on the same imaginary line through 1/2, that k1, k2, k3, k4, …, would again always represent the natural numbers 1, 2, 3, 4 , … in providing a solution to the equation.

This would of course be a problem if a zero were in fact to lie off the imaginary line (through 1/2).

Therefore we could not guarantee that all the imaginary parts (of known zeros) would be associated with the natural numbers.
In other words another unique set of numbers for k1, k2, k3, k4, … would be associated with a zero where the imaginary part is off the line through 1/2 (shared by all other zeros).

Now of course customarily in looking at this issue we start with the natural numbers and then attempt to derive the solution as a common shared (negative) exponent.

So therefore when we have

1–s + 2–s + 3–s + 4–s + … = 0,

The first solution for s, i.e. the first Riemann zero is 1/2 + 14.134725 … (or alternatively
1/2  14.134725 …).

However as we know, there is - in the absence of proof of the Riemann Hypothesis - no guarantee that a future zero might lie off the imaginary line.

However when looked at correctly i.e. from a dynamic interactive perspective, there is no reason to necessarily start with the second formulation of the equation, with the natural numbers as given data (where we solve for the common dimensional value).

Equally we could start from the first formulation of the equation where each dimensional value is given (and we solve for an infinite set of associated numbers).

When expressed in a psychological manner what this entails is as follows.
Because of the strong rational conscious bias of present Mathematics we take the existence of the natural numbers as necessarily all existing on the real number line as a given unquestioned assumption.

Though we are not explicitly aware of this crucial fact, when we then solve for the set of exponents (as the Riemann zeros of the equation) we are implicitly looking for the unconscious basis that is consistent with our assumption of the real numbers (all lying on the real line).

We can therefore only correctly assume that all natural numbers lie on the real line if the corresponding set of Riemann zeros lie on the same imaginary line (through 1/2).

Expressed equally in a slightly different manner, for the analytic (quantitative) aspect of mathematical understanding to be consistent (that all real numbers exist on the same real number line) with the holistic (qualitative) aspect of mathematical understanding, all the Riemann zeros must correspondingly lie on the same imaginary line (though 1/2).
Or again in an equivalent manner, (conscious) rational interpretation of number relationships should be consistent with (unconscious) intuitive appreciation of such relationships. So consistency with respect to both aspects  - which are necessarily distinct from each other - are equally important.   

As we know all composite natural numbers represent the product of unique prime factors.

So 6 in common understanding  = 2 * 3 Now customarily we express the composite.

However if one considers fro example a table with width 2 metres and length 3 metres, the area of the table = 6 square metres.

Thus in multiplying the two numbers as well as the quantitative change in units, there is also a qualitative change in the nature of the units (i.e. from 1-dimesnional to 2 –dimensional).

However remarkably this crucial fact is ignored in the conventional interpretation of multiplication. So the quantitative transformation in units aspect is solely considered with the corresponding qualitative change (in the nature of units) simply reduced in a linear quantitative manner.

So strictly speaking we are not entitled to make the assumption that composite numbers lie on the same line (as the primes from which they are derived) without in turn justifying the validity of reducing the qualitative aspect (of dimensional change) to the quantitative aspect (of homogeneous linear measurements).

And perhaps surprisingly the condition for making this reduced assumption is that the Riemann zeros - which represent in an indirect quantitative manner the holistic (qualitative) nature of the natural number system - all lie on the same imaginary line (through  1/2). 

Indeed in holistic terms the imaginary line represents the indirect rational means of representing the qualitative aspect of the number system!

So again we can only make the assumption that all the natural numbers lie on the (1-dimensional) real line, if in turn all the Riemann zeros lie on the same imaginary line.

Equally from the complementary opposite perspective we can only assume that all the Riemann zeros lie on the same imaginary line (through 1/2) if the natural numbers (as products of the primes) all lie on the real line (through 0). 

So from the dynamic interact perspective the Riemann Hypothesis, which already assumes the consistency of the real number line, cannot be proven (or strictly disproven).

We can say however say that the Riemann Hypothesis (in the assumption that all the Riemann zeros lie on the imaginary line through 1/2) is consistent with the corresponding assumption of all the natural numbers lying on the same line.

Likewise we can say that the assumption that all the natural numbers lie in the same real line is consistent with the truth of the Riemann Hypothesis.

However we cannot strictly prove these propositions. Thus their acceptance ultimately requires faith in the consistency of the overall number system.    

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