We are starting on this process of explaining how the numerical values that are given for the Riemann Zeta function i.e. the Zeta 1 function (for values of s < 0) are calculated. And the complementary Zeta 2 Function, which I introduced earlier will prove invaluable in this regard.
Before proceeding, we need to return a little more to the precise relationship as between Zeta 1 and Zeta 2.
What is interpreted as a dimensional number in Zeta 1 is interpreted in inverse complementary fashion as a base number in Zeta 2. And likewise what is interpreted as a base number (in Zeta 1) is now interpreted as a dimensional number (in Zeta 2).
So an expression such as a^s in Zeta 1 becomes s^a (from the corresponding Zeta 2 perspective). And of course the relationship here is always as between quantitative and qualitative (and qualitative and quantitative).
This thereby implies with respect to interpretation, a complementary relationship as between linear and circular (and circular and linear) respectively.
Also, what is positive as dimensional value with respect to Zeta 1, is negative with respect to Zeta 2 (and vice versa).
So a^s in Zeta 1 equates with s^(- a) in Zeta 2; also in the form that is more directly suited to interpretation of the LHS of Zeta 1, a^(- s ) with respect to Zeta 1, equates with s^a with respect to Zeta 2.
There is also another feature of difference that requires explanation.
When one switches from Zeta 1 to Zeta 2, comparing the respective dimensions in both cases, what is – s with respect to Zeta 1, is in fact a + 2 with respect to Zeta 2. So for example to give meaning to the numerical result for ζ (- 1) we need to examine the structure corresponding to 3 as dimension (and 1/3 as root) in Zeta 2.
So in the notation I have been using, in this respect ζ (- 1)1 corresponds with ζ(3) 2. (When not restricted to the limitations of conveying notation in a blog, I would rather define the two Functions by attaching subscripts (1 and 2) to the Zeta symbol!)
So with respect to the absolute nature of the dimensional number, in each case we increase by 2 when switching from Zeta 1 to Zeta 2.
Now the first contribution to this gap of 2 arises from the that in the case where both Functions are identical, the dimensional number for Zeta 1 is 0 and Zeta 2 is 1.! And this relationship is then enshrined in the Zeta Function, where ζ(s) = ζ(1 - s).
So, this would explain the need to increase by 1. However it has to be remembered that the original equation from which the Zeta 2 is derived is simply that for which the roots of unity are calculated.
i.e. 1 – s^n = 0.
Then to derive the Zeta 2 expression we have to divide this expression by the 1st root (1 – s) = 0.
This would explain the need to add 2 (rather than 1) to the absolute value of the dimensional power in Zeta 1 to get the comparable dimensional value (with its corresponding roots) that applies in Zeta 2!
Now with a view to explaining the precise nature of interpretation that is required to understand Zeta 1 values for s < 0, it is of special importance to understand the value associated with ζ(0).
Using the Functional Equation, ζ(0) on the LHS of the equation, as ζ(1 - s), is directly linked with ζ(s) on the RHS (where s = 1).
It might seem surprising that ζ(0) can be linked with the one value of the Function where it is not defined i.e. ζ(1), but in fact for this very reason it is the most important of all values in appreciating the true relationship as between quantitative and qualitative!
Remember that a complementary relationship properly connects the two Zeta values!
Thus, from a Type 3 mathematical perspective, the very reason why ζ(1) is not defined is because this uniquely, is the one place where a total separation of quantitative from qualitative type interpretation occurs. So 1 as dimension - when defined in its qualitative sense - as repeatedly stated, entails the reduction of qualitative to quantitative type interpretation.
So 1-dimensional interpretation is defined by the total separation – in formal terms – of quantitative from qualitative meaning.
Therefore from the complementary perspective of the Type 3 approach, ζ(0) is thereby defined in terms of the perfect complementarity of both quantitative and qualitative!
So in other words in going from ζ(1) to ζ(0), we have gone from interpretation with respect to the extreme linear to corresponding interpretation of the extreme circular position.
In fact properly understood this should provide deep insight once again into the true nature of the Riemann Hypothesis. For bounded by s = 1 and s = 0, is the famous critical region, within which all the non trivial zeros are known to lie.
Thus, as s = 1 and s = 0 provide the boundaries as between extremes with respect to both linear and circular interpretation respectively, this thereby entails that all values within these bounds simultaneously combine both linear (quantitative) and circular (qualitative) aspects. Outside of these bounds though all values - except for ζ 1) – possess both quantitative and qualitative aspects, they do so in a relatively separate fashion so that an aspect on one side of the Functional Equation can always be matched with a corresponding complementary aspect on the other.
However within the critical region, inevitably a degree of interdependence necessarily attaches to such values. And then the Riemann Hypothesis is based on all non-trivial zeros lying on the straight line that divides this critical region in half!
So once again, it is perhaps easy in this context to appreciate its true significance as the condition where both (linear) quantitative and (circular) qualitative aspects with respect to the primes (and natural numbers) are now identical!
It requires a very refined form of understanding to properly appreciate the true nature of ζ(0).
When we put let s = 0 in the Functional Equation, we obtain,
1^0 + 1^0 + 1^0 + 1^0 +…..
Now properly this means that we should interpret this sum of terms according to the qualitative dimension that corresponds to s = 0.
However from a reduced Type 1 mathematical perspective, 0 (as power of 1) has no distinctive quantitative meaning.
So 1^0 + 1^0 + 1^0 + 1^0 +…… is quickly reduced in 1-dimensional terms to
1^1 + 1^1 + 1^1 + 1^1 +…… i.e. 1 + 1 + 1 + 1 + … ,which clearly from this perspective diverges to infinity.
However, when properly considered, 0 as a qualitative dimension, properly involves the total interdependence of both linear and circular notions (which are treated as totally independent in linear terms).
Now, one way of visually this is as a geometrical circle with its line diameter drawn. The point at the centre of the line is equally the point at the centre of the circle. So the identity of line and circle is then, literally, this non-dimensional point!
As we have seen to consider the dimension 0 in its true qualitative context, we need to switch to interpretation of 2 as dimension in the Zeta 2 formulation. And this equates directly with the two roots of 1 (in quantitative terms) which geometrically are represented by the circle and its line diameter.
We dealt in detail earlier with the notion of 2 as dimension. This combines what is linear with respect to isolated reference frames (as with a turn on a road that is either left or right), with what is circular and interdependent (as when two turns at a crossroads are considered as necessarily left and right in relation to each other).
Now if we can represent this in qualitative number terms by letting 1 represent the (linear) isolated pole, while 1 – 1 represents the (circular) complementarity of opposite poles.
So combining both we then have ( 1 – 1) + 1 + (1 – 1) + 1 +…… as representing the true expression of ζ (0) which can be given an equally matching quantitative and qualitative interpretation (which are literally identical in this case).
And this then corresponds with the Zeta 2 interpretation where 2 (as dimensional number is matched in complementary fashion with its 2 corresponding roots).
Now in conventional Type 1 Mathematics, this latter expression for ζ(0) is pragmatically arrived at through defining a new Eta Series where the terms alternate in a merely quantitative manner.
So η (0) = 1 - 1 + 1 – 1, + …….
Now though alternating, from the conventional this initially might initially seem somewhat easy to interpret.
So if we take an even number of terms the sum of the series = 0; however if we take an odd number the sum = 1.
So as odd and even would have an equal probability of occurring we could therefore arrive at a single unambiguous answer for the series by getting the mean of the two results = 1/2.
However lying behind alternating terms are profound qualitative issues of an ordinal nature!
Strictly speaking therefore, we could arrive at a limitless number of results for this series depending on the ordering of the terms. So the reason in this case why we get the answer of 1/2 is because we are using the configuration that is consistent with the interpretation of 0 as a dimensional number.
It should also be borne in mind that qualitative issue of the manner of pairing terms, is confined solely to infinite series. Where the sum of terms is finite the ordinal issue of how the terms are ranked makes no difference to the quantitative result!
Again this should suggest that the infinite notion is qualitatively distinct from the finite (requiring thereby a distinctive means of interpretation). But somehow all these key issues are conveniently glossed over in Conventional terms.
So modern Mathematics is misleadingly considered a rigorous discipline. Well, certainly from my perspective it is anything but rigorous!
From the quantitative perspective 1/2 represents the mean of terms (1 - 1) taken as complementary) pairings and single positive terms (1). From the corresponding qualitative perspective 1/2 represents a perfect balance as between linear (1), where opposite poles are taken as separate and (1 - 1), where both poles are now understood as complementary. So this golden mean can qualitatively be expressed as 1/2.
So from the perspective of 0 (as a qualitative dimension) we have arrived at a numerical result which is intuitively meaningful (in the context of using the appropriate dimension for its interpretation).
There is yet another angle on this which further illuminates the nature of 0 as dimension!
Positive and negative signs make no difference where 0 is concerned. Now the deeper significance of this (from a Type 3 perspective) is that both rational (linear) and circular (holistic) type understanding are thereby identical in terms of 0 (as qualitatively interpreted).
This result of 1/2 is also deeply significant as it is in fact qualitatively identical with the famed condition for the Riemann Hypothesis.
So the true value for ζ(0) i.e. 1/2 as interpreted using the numerical configuration corresponds with the dimension that is appropriate for interpretation of 0.
We also know that the Riemann Hypothesis states the condition that all the non-trivial solutions for the Zeta Function lie on real line = 1/2.
What this implies in both situations is an ineffable state, pre-existing finite phenomena.
Once phenomenal activity unfolds both linear and circular (quantitative and qualitative) aspects must always be to a degree separated. So a condition that requires their mutual identity cannot have any strict phenomenal meaning!
It also implies that corresponding true awareness of the Riemann Hypothesis requires ultimately entering a pure contemplative state that is likewise ineffable!
However we are not quite finished yet!
We have now achieved the numerical result for ζ(0) in its correct qualitative context (where the result is in accordance with the interpretation of 0 as dimension). However Type 1 Mathematics by its nature requires expressing results in the standard 1-dimensional manner (as qualitatively interpreted).
Therefore a further conversion process is required to change this result (from the context where it is intuitively meaningful) to the standard 1-dimensional format. And here, it is then rendered intuitively meaningless in quantitative terms (from this new perspective).
Now most of the books that I have read on this procedure of obtaining the conversion from Eta to corresponding Zeta value describe it as a “trick”. However that simply represents an unsatisfactory explanation of what occurs.
So we start with the expression for ζ(s). Then with respect to each even term we subtract twice its value from the original expression to obtain the corresponding Eta expression, where each even term is negative.
Now dividing the even terms by 2^s with respect to the infinite interpretation of the series, we can obtain the original Zeta series and can thereby form an expression through which ζ(s) can be calculated.
So therefore {ζ(s)– 2[1/(2^s)]}[(ζ(s)]} = η(s)
Then ζ(s){[(1 – 1/2^)](s – 1)} = η(s)
So ζ(s) = η(s)/{[(1 – 1/2)]^(s – 1))}
So where s = 0, as in the present case,
ζ(0) = 1/2/{(1 – 2^(- 1)} = 1/2(– 1) = –1/2
So by applying this "trick" and dividing the Eta value by thereby obtain the corresponding Zeta value for s = 0 = – 1/2.
What it actually implies is a means of converting numerical values that can indeed be given an intuitively meaningful interpretation (in accordance with the appropriate logical interpretation that defines the qualitative dimension to which they relate) to the standard linear format for which no intuitively meaningful interpretation can be given.
And this is likewise true for all values of the Zeta Function (for s < 1).
These numerical values actually relate (in complementary fashion) to the qualitative interpretations that are in accordance with the particular value of s (< 0) that occurs in the expression. And the key to unlocking the meaning inherent in these dimensions lies in the Zeta 2 - rather the Zeta 1 - Function.
So when these values are converted back to standard linear expression (in accordance with Type 1 Mathematics) their true nature remains masked and they thereby appear intuitively meaningless from this perspective. And this is no small matter as it applies to the major portion of the Riemann Zeta Function (for s < 1).
Indeed one could validly argue that even the remaining values (for s > 1) cannot be fully interpreted without first unravelling the nature of remaining values (for s < 0).
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