We concluded yesterday that the numerical values of the Riemann Zeta Function for every finite value of s (with the exception of 1) on the real axis, can be given - relatively - both a quantitative (specific) and qualitative (holistic) interpretation. This means that when we change reference frames, numerical values now have a quantitative (holistic) and qualitative (specific) interpretation.
So depending on context, numerical values can be given both specific and holistic interpretations. Likewise, again depending on context, corresponding psychological interpretation of such values can be given both specific and holistic interpretations. And though this may not indeed be clear yet without further ample illustration, it is absolutely key to appreciating the true nature of the Riemann Zeta function (and indeed its associated Riemann Hypothesis).
Also we mentioned that the Riemann Functional Equation establishes an important complementary relationship as between values for ζ(s) on the RHS > .5 and corresponding values for ζ(1 - s) on the LHS < .5.
What this means in effect is that when we define a specific quantitative value for ζ(s) with respect to a linear frame of reference on the RHS (where s > .5) the corresponding value with which it is related for ζ(1 - s) on the LHS (where s < .5) is necessarily defined with respect to an alternative circular frame of reference.
So for example the value for ζ(2) on the RHS = (pi^2)/6 is defined in terms of standard linear interpretation.
The corresponding value through the Riemann Functional Equation on the LHS is
ζ(- 1) = - 1/12. Though this indeed has a specific numerical value, the result has no meaning in terms of the standard linear interpretation of a series. So what has happened is that the frame of reference has shifted here to a distinctive circular mode of interpretation!
Therefore though the quantitative results with respect to both series can be given specific values they correspond to differing modes of interpretation. So in the language that I customarily now use, (pi^2)/6 corresponds to a Type 1, with - 1/12 in relative terms, corresponding to a distinctive Type 2 interpretation.
So once again we can see the insuperable difficulties that Conventional Mathematics faces in appreciating the true nature of the Riemann Hypothesis. Because it is defined in merely Type 1 terms, it has no way of properly interpreting all the numerical values on the LHS of the Function (for s < .5) and thereby has no means of interpreting what the whole Function is really about!
When I spent some time reading about the Riemann Hypothesis I was amazed at how little attention was actually given to philosophical interpretation of the strange values that are thrown up for the Function through analytic continuation.
This lack of proper understanding is carefully hidden behind a forbidding wall of technical jargon regarding the nature of continuation. So we hear a lot about differing domains of definition, holomorphic and meromorphic functions, uniquely defined values etc. all of which of course at the Type 1 level is perfectly valid.
However, the reality remains that mathematicians are still unable to provide a satisfactory explanation with respect to half the values defined by the Riemann Zeta Function! And the reason is that is requires the (unrecognised) Type 2 aspect of Mathematics for such interpretation!
And finally we saw in yesterday's blog that both quantitative and qualitative interpretations - which are separated for other values - ultimately coincide where s = .5!
This means that for all the points that lie on the imaginary line drawn through .5, quantitative and qualitative likewise coincide.
And as all the non-trivial zeros (of the Zeta Function) necessarily relate to points on this imaginary line if the Riemann Hypothesis is true, this then implies a coincidence of both quantitative and qualitative aspects with respect to the primes.
In other words the Riemann Hypothesis is the condition necessary to ensure that we can perfectly reconcile the individual nature of each prime (that is finite) with the overall collective nature of the primes (in infinite terms). And this ultimately points to an ineffable state!
Now in looking more carefully at the Riemann Zeta Function (and of course the non-trivial zeros) we will first demonstrate in more detail the manner in which quantitative and qualitative interpretations operate.
In doing this we will initially consider values of the Function on the RHS of the real axis for s > 1.
We will then look at its complementary partner values with respect to the Function where s < 0.
And finally we look at the all-important critical strip as between 0 and 1 (where ultimately in the complex plane all the non-trivial zeros lie).
Our first task seems relatively easy and uncontroversial as Euler had already mapped out the Function on the real axis (for s > 1) long before Riemann.
So the first thing we can determine is that the value of the Function (for all values > 1) results in convergent series with finite values.
However even here some subtlety with respect to interpretation of numerical results for the Function is required.
Now, it is important to keep reminding ourselves that finite and infinite constitute distinctive concepts, which properly relate to quantitative and qualitative understanding respectively.
So the finite strictly relates to actual (specific) numbers, whereas the infinite properly relates to the (holistic) potential that is inherent in all numbers. However, as I have frequently stated, the very essence of the linear approach is that it reduces the infinite in actual terms, effectively treating it as a linear extension of the finite!
And this is amply demonstrated by values for the Zeta Function (where s > 1).
So for example to illustrate with the series (where s = 2) we generate a sequence of actual (specific) terms 1 + 1/4 + 1/9 + 1/16 + ...
It quickly becomes apparent that the size of these terms diminishes very rapidly with their sum converging towards a limiting value. So, after some point (which we can arbitrarily choose) the addition of further terms makes little difference to the overall sum, and we then conclude that the infinite set of terms thereby converges to the same limit.
I have already made the observation that intuition - insofar as it is informally used in Type 1 Mathematics - does so in a supporting role (so that formally it can be ignored).
And that is precisely what happens here.
Now if we want to express in a subtler manner the nature of convergence with respect to infinite type series, we rationally interpret that the series approaches some limiting value, and then intuit that the infinite series of terms will attain the same value.
From a conventional Type 1 perspective, the final step in moving from the finite to the infinite conclusion is missed, with the whole process misleadingly interpreted in rational terms!
You might consider that such subtlety in interpretation is unnecessary and constitutes merely pin pricking.
However you would be wrong! for it is the very identification of what is involved here, that will enable us to make the decisive leap in interpreting corresponding values for the Zeta Function for s < 0.
For example when s = - 1, we generate the sum of series of the natural numbers i.e. 1 + 2 + 3 + 4 + .....
Now from the standard linear approach, it is quite obvious that this series will diverge in value so that we cannot give its sum a finite value.
Yet, in the context of the Riemann Zeta Function, it acquires the rational finite result of - 1/12.
And as the Riemann Functional Equation establishes an important relationship as between such values on the LHS (with no meaningful interpretation in linear terms) and corresponding values on the RHS (with a finite linear interpretation) then we cannot even properly understand these values on the RHS) without properly understanding corresponding values on the LHS.
And once again, because of the absence in formal terms of a holistic qualitative (Type 2) aspect to Conventional Mathematics, it thereby cannot provide such interpretation.
So, the deeper significance of what I am attempting through this exploration of the Riemann Zeta Function is the key fact that Conventional Mathematics is sadly lacking as a comprehensive means of overall interpretation.
And its limitations are especially exposed in tackling the Riemann Hypothesis.
Thus, properly understood, the two aspects of Mathematics (quantitative and qualitative) dovetail together in perfect harmony with respect to appreciation of the Riemann Hypothesis. And this synchronised use of Type 1 and Type 2 Mathematics, which I am attempting to demonstrate here, represents but the most preliminary introduction to Type 3 Mathematics (which is what Mathematics should truly be about).
So to sum up this contribution. The finite and infinite notions correspond to two distinctive notions, which are quantitative and qualitative with respect to each other.
Conventional Mathematics - by definition - applies a strictly linear (1-dimensional) approach, where effectively the infinite notion is reduced in a finite manner (or alternatively where the qualitative aspect is reduced to the quantitative).
And then, we get numerical results for the Zeta series, which seemingly correspond with common sense (that is likewise based on linear interpretation).
So Conventional Mathematics is based on 1-dimensional interpretation (where qualitative is reduced to quantitative). However I have already made the important observation that an unlimited set of alternative dimensional interpretations of mathematical reality exist corresponding to all other numbers ≠ 1.
And here quantitative and qualitative - while being necessarily related - maintain a distinctive identity.
This likewise entails that in all such cases, finite and infinite notions of number likewise maintain a relatively distinct identity.
And where this is the case, the actual finite nature of series does not necessarly concur directly with their infinite values.
And when we come to investigate values of the Zeta Function (for s < 1) this is especially relevant.
So again (for s = - 1) the actual nature of the terms in the series 1, 2, 3, 4, etc. would strongly suggest that its sum diverges to infinity (from a linear perspective).
However this is not the case with the infinite sum = - 1/12!
Thus, we will need to investigate this numerical value in the alternative manner suggested (where linear notions of interpretation thereby have to be abandoned).