I had already accepted that Mathematics (in its current form) is not fit for purpose some 40 years ago while at University.
At that time - and for many years later - I would have couched my arguments in a somewhat philosophical manner, regarding the nature of finite and infinite and how Conventional Mathematics necessarily entailed a reduction - in any context - of infinite with finite - or alternatively from the other perspective - finite with infinite notions.
The first form is evident in the manner in which series are viewed, where in effect the infinite notion is treated as a linear extension of what is finite. This for example is the case with convergent and divergent series. So in the first when a finite series converges on a limiting value, the finite notion is then extended in linear terms to the infinite. Alternatively when it diverges away from a limiting value, again this notion, derived from finite experience, is extended in linear terms to the infinite.
However though seems to accord with common sense intuition, a fundamental inconsistency is thereby involved.
Likewise we can have the opposite case, where what is true for the infinite (in the reduced sense that it is used in Mathematics) is then considered true for the finite case. So, whenever a theorem has been proven for the general case (i.e. as applying to "all") is then assumed to apply in a finite context to specific examples. So for example the Pythagorean Theorem (that in a right angled triangle the square on the hypotenuse is equal to the sum of squares on the other two sides) has been proven as true in "all" cases (as infinite). It is then assumed to apply in any individual case (which is finite). However again a subtle - but fundamental - inconsistency is involved.
Now practicing mathematicians might be inclined to dismiss all this as mere philosophical supposition (thereby remaining fully committed to the status quo).
However recently, I have come to realise that the same basic argument can be expressed in a manner that mathematicians will find much harder to refute, relating to the distinction as between the cardinal and ordinal use of number. And it is the very clarification of these issues that played a large role in my own development in leading me on to discover the true relationship of prime to natural (and natural to prime) numbers.
This basically involves interpreting number consistently both with respect to cardinal and ordinal usage. And when this is done, the wonderful mystery of how the prime numbers simply are a reflection of the natural (and the natural simply in turn a reflection of the primes) is then revealed in all its glorious splendour!
We can indeed express the basic argument very simply!
With respect to Conventional Mathematics, number cannot be consistently interpreted in both a cardinal and ordinal fashion! Indeed this is the root of the uncertainty principle I have mentioned on several occasions recently, and lies at the centre of all mathematical procedures.
Now the fundamental philosophical reason behind this is that cardinal and ordinal refer - in relative terms - to the quantitative and qualitative aspects of number respectively (which in turn reflects the distinction as between finite and infinite notions).
The corollary of this is that a mathematical approach that does not in fact recognise a clear qualitative distinction (with respect to finite and infinite notions) likewise cannot deal satisfactorily with the distinction as between the cardinal and ordinal use of number. So these two notions of number - though fundamentally distinct - are continually confused with each other in conventional mathematical interpretation.
When we use the cardinal notion of number in the conventional mathematical fashion, it implies an absolute whole existence.
So in this sense the number 5 does not contain unique parts. So if we were to crack open the the number 5 as it were - say - with reference to a collection of 5 objects - we could represent each object without qualitative distinction as 1 (in a strictly quantitative manner).
Thus 5 = 1 + 1 + 1 + 1 + 1. So here we have the very hallmark of the linear approach whereby whole numbers are treated as the collection of unitary parts (without any qualitative distinction).
Indeed it is this very approach that is the basis of the string theory approach in physics where the fundamental "objects" of nature are viewed in terms of 1-dimensional strings (without qualitative distinction).
It is this cardinal mindset therefore that enables conventional mathematicians to view prime numbers as the "basic building blocks" of the number system. So the primes are treated somewhat as individual atoms without further substructure. Then the natural numbers arise through a unique product of such constituent prime factors.
However the fundamental problem with viewing numbers in a merely cardinal sense is that we thereby treat them in an absolute independent manner.
The very capacity to combine numbers with each other (as in the derivation of natural from prime number factors) is that it presupposes a relational capacity (which is of a qualitative nature).
When we allow for this qualitative relational aspect of numbers, then we necessarily must interpret them in an ordinal - rather than strict - cardinal sense.
Thus such a relational aspect thereby implies the ordinal use of number! And on investigation this proves to be quite problematic (with no fixed identity).
For example when we use 2 in an ordinal sense to imply second, its meaning depends on the context (which can vary endlessly). So with 2 items the second item may seem unambiguous (as the second of 2). However if 2 is used as a ranking with respect to 3 items it now changes (as the second of 3). And of course, as we can rank 2 in an unlimited manner, the relational meaning of 2 is likewise unlimited.
So we can see here that the ordinal notion of a number is of a qualitative relational nature with its precise interpretation depending on context.
Now once we accept this relational ordinal notion as in this case with respect to 2), it poses a fundamental problem for the absolute notion of number as cardinal.
Quite simply if we define cardinal numbers in an absolute independent sense, then it is strictly impossible to subsequently relate them to other numbers!
So coming back to the conventional view of primes as building blocks. If we define the primes initially in cardinal terms, then the unique relational capacity as the product of prime numbers, cannot come from their cardinal identity! In other words we cannot therefore explain the derivation of natural numbers (from prime factors) in a strictly quantitative manner!
And then there is the further problem that the very ability to meaningfully refer to the primes already implies an ordinal number ranking that is natural!
So when we refer to 7 for example as the 4th prime, the very notion of a (composite) natural number is already inherent in the prime, even though we had sought earlier to explain prime numbers as the "building blocks" of the natural numbers.
Thus when we subject it to close scrutiny, the conventional treatment of number is full of confusion where no clear distinction is made as between its respective cardinal and ordinal use.
And again, the key problem is the reductionist attempt to define number merely with respect to its quantitative aspects.
We can see this dilemma in an even better light when we look at number in a circular context with respect to the Type 2 system (where the qualitative nature of number is explicitly recognised).
On a personal level, I struggled with this issue for many years before reaching a satisfactory resolution. For example in referring to 2 as dimension, I was conscious of the fact that I kept using it in two different ways (without fully reconciling thr relationship between them).
So for example the 2nd dimension of 1, in this Type 2 context (as 1^2) implied in complementary quantitative fashion the 2nd root of 1 (as 1^1/2) = - 1.
However I also frequently used 2 in a cardinal sense which then implied the 2 roots of 1 (i.e. 1^1 and 1^2) respectively.
However I was aware of an inherent problem here for in the context - say - of the 3 roots of 1, we would also have a 2nd root (as the 2nd of 3). However this would have a different meaning than the 2nd root (in the context of 2 roots). And therefore likewise in complementary manner the notion of 2 as 2nd would have a distinctive meaning in the context of 3 dimensions!
And then in clarifying this distinction, I realised that it inevitably meant that the cardinal notion of number would likewise have a merely relative meaning in this context.
So for example when I refer to the 2 roots of unity, these roots ( + 1 and - 1) have a cardinal meaning (with the second of these i.e. 2 as ordinal in the sense of second of 2) corresponding with - 1 (in cardinal terms).
However if I now refer to the 3 roots of unity, these roots + 1, {-1 + 3^(1/2)i}/2 and {-1 - 3^(1/2)i}/2 have a cardinal meaning. But now the second of these (i.e. 2 as 2nd of 3) is associated with a different cardinal number expression (as the 2nd corresponding root).
So strictly speaking once we properly unravel the nature of cardinal and ordinal interpretation, both can change identity! In other words, correctly understood, the very nature of number, in this interactive context (of cardinal and ordinal meaning) is of a merely relative nature.
So from this context the very approach in Conventional Mathematics (in explicitly recognising solely the quantitative aspect) appears as simple untenable. In other words number has both quantitative and qualitative aspects (which dynamically interact in understanding).
So we originally - in a linear context - cracked open the prime number 5 to find it made up of homogeneous units i.e. devoid of qualitative significance.
However now from the complementary circular perspective, when we crack open 5 (as representing both the qualitative dimensional meaning of 5 and its corresponding 5 roots of 1 in quantitative terms) we find that each of these constituent ordinal units (as the 1st, 2nd, 3rd, 4th and 5th roots respectively) is uniquely distinct.
Furthermore we can see here clearly the manner in which the natural numbers (as ordinal) are contained in the primes.
So of course the upshot of all this is that both the primes and the natural numbers have both cardinal and ordinal interpretations which ultimately are purely interchangeable. And it is in this recognition that the mystery of the primes (which is inseparable from the mystery of the natural numbers) is ultimately resolved.
And on reflection the implications of this are even graver for Conventional Mathematics than I had once imagined.
I have recently come to the realisation that even with respect to the Type 1 aspect of Mathematics there are two distinct approaches.
The Conventional approach is explicitly based on absolute quantitative type interpretation which strictly is quite untenable in giving a satisfactory interpretation of the nature of number.
The alternative Type 1 approach - which is what I would advocate - while concentrating mainly on the quantitative properties of numbers, crucially interprets them in a merely relative manner (thus implicitly recognising their equal qualitative nature).
And then when later in the Type 3 approach the incorporation of Type 1 and Type 2 is required, the Type 1 aspect can then be readily adapted to this task.
Thursday, March 15, 2012
Tuesday, March 13, 2012
Interpreting Numerical Result for ζ(0)
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).
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).
Monday, March 12, 2012
Interpreting Number for s < 0
I explored the qualitative nature of number as dimension in previous blogs especially with respect to 2 and 4.
However the more general point I was making is that associated with every number as dimension is a unique holistic qualitative manner of overall mathematical interpretation.
Now the basic nature of all these dimensions is that both circular and linear understanding are combined in a manner (that is unique for each dimension). Well strictly, I should say unique for prime number dimensions. So we saw in relation to the interpretation of 4 that two of these dimensions (i.e. relating to the real polarities) had already been encountered in dealing with 2 as dimension.
When one considers then that Conventional Mathematics is - in formal terms - confined to interpretation with respect to the qualitative nature of 1 as dimension, one begins to appreciate how limited the scope of such mathematical understanding truly is.
In fact - what we conventionally term - Mathematics represents in fact the extreme special case, where quantitative is divorced from qualitative meaning. For all other dimensions, both quantitative and qualitative are combined with each prime dimensional number representing a unique configuration with respect to the relationship between the two aspects.
However as qualitative and quantitative are necessarily complementary, this means that that we can equally define an unlimited set of alternative number systems where again each prime dimensional number defines a unique system.
So again Conventional Mathematics represents the extreme case where number is defined merely with respect to its quantitative aspect.
However we have seen how number can equally be given an ordinal (qualitative) meaning.
So the initial starting point in coming to appreciation of these alternative number systems (which comprise the LHS of the Riemann Zeta Function for values of s < 0) is that number is now used in a manner where various configurations with respect to both cardinal and ordinal interpretation are used.
So the common sense notion of number (consistent with Type 1 Mathematics) actually implies a pure cardinal system of interpretation. So when we are led to quickly see that the sum of a series such as 1 + 2 + 3 + 4 +..... diverges, it is because we are interpreting number is a pure cardinal (quantitative) manner.
However we could equally give a coherent meaning to the same series as
1 + 2 + 3 + 4 + ..... = 0 by interpreting terms in a pure ordinal sense.
For example is these numbers are now used to represent for example the successive roots of 1, the sum of thee roots (in cardinal terms) = 0 with 1, 2, 3, simply used to qualitatively rank their respective order.
We could also again with respect to roots define the number expression
1 * 2 * 3 * 4 * ... * n = 1 representing the fact that the product of the n roots of 1 = 1.
Indeed if we now view the sum of the natural numbers from a pure ordinal (rather than cardinal perspective) we can validly say that,
1 + 2 + 3 + 4 +.... = 1.
In other words, we are here defining number in a pure Type 2 sense where the expression means that though the numbers are being added, the qualitative dimension from which they are interpreted (i.e. 1) remains the same.
So more fully we are viewing,
1^1 + 2^1 + 3^1 + 4^1 +.... with respect to the unchanging dimensional number (rather than the base quantities)
So in giving these few examples, we have already given the sum of the natural numbers 1 + 2 + 3 + 4 +....., 3 distinctive interpretations.
However in the latter two cases we are dealing with pure ordinal type meaning which universally applies (in these respective contexts). So no variation from a quantitative perspective thereby takes place.
However when we look at the result of the Riemann Zeta Function where s = - 1, we once again get yet another sum of the natural numbers,
i.e. 1 + 2 + 3 + 4 +..... = - 1/12
So in this case a non-trivial quantitative result emerges.
What this suggests therefore is that we are no longer dealing with interpretation of number, in either a pure cardinal or ordinal sense, but rather in a manner where elements of both types of appreciation are combined.
And once again associated with each negative number as dimension is a unique quantitative finite expression that emerges from the Function.
Now interpreted from a strict cardinal perspective, all these results would diverge to infinity.
However the important thing to remember is that this cardinal interpretation is associated solely with 1 as a dimensional number.
So the very reason why we are getting non-intuitive numerical results from the Function on the LHS, is that we are now interpreting number in terms of the exact configurations (of cardinal and ordinal respectively) that are associated with each dimensional number.
This means in effect that each result for the Function (for s < 0) reflects its own unique manner of interpretation.
Thus for example the result of the Function for s = - 3 i.e. 1/120 in this case represents the unique configuration of cardinal and ordinal aspects that are associated with - 3 as a dimension.
Now when we used 3 as a dimensional number on the RHS for s > 1, we were able, in all cases to derive intuitively meaningful finite results by treating number in a merely cardinal manner!
So from a qualitative perspective all these results can be expressed - literally - in a reduced linear manner (as merely quantitative).
However this is clearly not the case on the LHS (when negative values for s are used).
So what we require here is a switch from the Zeta 1 Function to the Zeta 2, which is the true home for the mix of both quantitative and qualitative type appreciation from which the numerical results are derived.
So we will look carefully in future blog entries at how this process unfolds.
However the more general point I was making is that associated with every number as dimension is a unique holistic qualitative manner of overall mathematical interpretation.
Now the basic nature of all these dimensions is that both circular and linear understanding are combined in a manner (that is unique for each dimension). Well strictly, I should say unique for prime number dimensions. So we saw in relation to the interpretation of 4 that two of these dimensions (i.e. relating to the real polarities) had already been encountered in dealing with 2 as dimension.
When one considers then that Conventional Mathematics is - in formal terms - confined to interpretation with respect to the qualitative nature of 1 as dimension, one begins to appreciate how limited the scope of such mathematical understanding truly is.
In fact - what we conventionally term - Mathematics represents in fact the extreme special case, where quantitative is divorced from qualitative meaning. For all other dimensions, both quantitative and qualitative are combined with each prime dimensional number representing a unique configuration with respect to the relationship between the two aspects.
However as qualitative and quantitative are necessarily complementary, this means that that we can equally define an unlimited set of alternative number systems where again each prime dimensional number defines a unique system.
So again Conventional Mathematics represents the extreme case where number is defined merely with respect to its quantitative aspect.
However we have seen how number can equally be given an ordinal (qualitative) meaning.
So the initial starting point in coming to appreciation of these alternative number systems (which comprise the LHS of the Riemann Zeta Function for values of s < 0) is that number is now used in a manner where various configurations with respect to both cardinal and ordinal interpretation are used.
So the common sense notion of number (consistent with Type 1 Mathematics) actually implies a pure cardinal system of interpretation. So when we are led to quickly see that the sum of a series such as 1 + 2 + 3 + 4 +..... diverges, it is because we are interpreting number is a pure cardinal (quantitative) manner.
However we could equally give a coherent meaning to the same series as
1 + 2 + 3 + 4 + ..... = 0 by interpreting terms in a pure ordinal sense.
For example is these numbers are now used to represent for example the successive roots of 1, the sum of thee roots (in cardinal terms) = 0 with 1, 2, 3, simply used to qualitatively rank their respective order.
We could also again with respect to roots define the number expression
1 * 2 * 3 * 4 * ... * n = 1 representing the fact that the product of the n roots of 1 = 1.
Indeed if we now view the sum of the natural numbers from a pure ordinal (rather than cardinal perspective) we can validly say that,
1 + 2 + 3 + 4 +.... = 1.
In other words, we are here defining number in a pure Type 2 sense where the expression means that though the numbers are being added, the qualitative dimension from which they are interpreted (i.e. 1) remains the same.
So more fully we are viewing,
1^1 + 2^1 + 3^1 + 4^1 +.... with respect to the unchanging dimensional number (rather than the base quantities)
So in giving these few examples, we have already given the sum of the natural numbers 1 + 2 + 3 + 4 +....., 3 distinctive interpretations.
However in the latter two cases we are dealing with pure ordinal type meaning which universally applies (in these respective contexts). So no variation from a quantitative perspective thereby takes place.
However when we look at the result of the Riemann Zeta Function where s = - 1, we once again get yet another sum of the natural numbers,
i.e. 1 + 2 + 3 + 4 +..... = - 1/12
So in this case a non-trivial quantitative result emerges.
What this suggests therefore is that we are no longer dealing with interpretation of number, in either a pure cardinal or ordinal sense, but rather in a manner where elements of both types of appreciation are combined.
And once again associated with each negative number as dimension is a unique quantitative finite expression that emerges from the Function.
Now interpreted from a strict cardinal perspective, all these results would diverge to infinity.
However the important thing to remember is that this cardinal interpretation is associated solely with 1 as a dimensional number.
So the very reason why we are getting non-intuitive numerical results from the Function on the LHS, is that we are now interpreting number in terms of the exact configurations (of cardinal and ordinal respectively) that are associated with each dimensional number.
This means in effect that each result for the Function (for s < 0) reflects its own unique manner of interpretation.
Thus for example the result of the Function for s = - 3 i.e. 1/120 in this case represents the unique configuration of cardinal and ordinal aspects that are associated with - 3 as a dimension.
Now when we used 3 as a dimensional number on the RHS for s > 1, we were able, in all cases to derive intuitively meaningful finite results by treating number in a merely cardinal manner!
So from a qualitative perspective all these results can be expressed - literally - in a reduced linear manner (as merely quantitative).
However this is clearly not the case on the LHS (when negative values for s are used).
So what we require here is a switch from the Zeta 1 Function to the Zeta 2, which is the true home for the mix of both quantitative and qualitative type appreciation from which the numerical results are derived.
So we will look carefully in future blog entries at how this process unfolds.
Sunday, March 11, 2012
The Startling (Unrecognised) Significance of the Riemann Functional Equation
The Riemann Functional Equation establishes an important relationship as between positive and negative values of s for the Zeta 1 Function.
It can be expressed in several ways.
One that I especially like is that given in John Derbyshire's "Prime Obsession" on P. 147:
ζ(1-s) = 2^(1-s)π^(-s)sin{[(1-s)/2]π}(s-1)!ζ(s).
Thus, if we calculate a result for the Function with respect to a positive value of s on the RHS of the Function, we can equally calculate a corresponding value with respect to the negative value of s, i.e. 1 - s.
So for example if we have already calculated ζ(2), then through the formula we can equally calculate ζ(- 1).
However a very significant problem arises with respect to the Function for negative values of s, in that the results seem somewhat meaningless from a conventional linear perspective (which properly constitutes Type 1 Mathematics).
For example the Function for s = - 1, generates the sum of the natural numbers
1 + 2 + 3 + 4 +........
Now in conventional (Type 1) terms this series clearly diverges to infinity.
However the result as derived from the Functional Equation = - 1/12.
We will give a coherent interpretation of what this numerical result really signifies in a future entry. But firstly we have to place it within its proper general context.
Let us return to the Zeta 1 and Zeta 2 formulations of the Function, corresponding to the (linear) Type 1 and the (circular) Type 2 aspects of Mathematics respectively.
So Zeta 1 can be expressed as the infinite series
1/1^s + 1/2^s + 1/3^s + 1/4^s +.......
Now if we invert each term of this Function we then get
1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) +......
Zeta 2 by comparison is expressed by the inverse series (with respect to both base and dimensional numbers as
1 + s^1 + s^2 + s^3 + s^4 + .....
Now when we let s = 0 (as dimensional power) in the Zeta 1 and s = 1 (as base quantity) in the Zeta 2, both Functions appear as identical.
So for s = 0, Zeta 1 gives
1^0 + 2^0 + 3^0 + 4^0 + .....
= 1 + 1 + 1 + 1 + ....
Likewise for s = 1, in Zeta 2, we have
1 + 1^1 + 1^2 + 1^3 + 1^4 +......
= 1 + 1 + 1 + 1 + .....
So in this special case where both Zeta 1 and Zeta 2 appear identical, s = 0 (as dimension) in the former case and s = 1 (as base quantity) in the latter.
Therefore Zeta 2 is connected with Zeta 1 through the simple transformation s = 1 - s (where it is understood that s now switches as between dimensional power and base quantity).
Now to distinguish both Functions, let
Zeta 1 = ζ(s)1 and Zeta 2 = ζ(s)2.
Then in this special special case we have ζ(s)2 = (1 - s)ζ(s)1. So the Functional Equation established by Riemann (with its adjustment terms) can be used to extend this result for all other values of s (but now properly incorporating both Zeta 1 and Zeta 2).
Thus if we interpret - as is the convention - the Functional Equation using solely Zeta 1 where s = 0, then all the non zeta terms cancel out and we are left,
ζ(s) = (1 - s)ζ(s), i.e. the trivial identity
ζ(s) = ζ(s).
Now the important point I am making is that appropriate interpretation of the Riemann Functional Equation requires appreciation that both sides of the equation actually point to two Functions (Zeta 1 and Zeta 2 respectively) that bear a complementary relationship with each other.
So when we interpret quantitative symbols and numerical results in a linear fashion on the RHS in accordance with Zeta 1, this implies that corresponding appropriate quantitative interpretation on the LHS requires a complementary circular interpretation in accordance with Type 2.
Likewise when we give symbols a qualitative circular type interpretation, on the RHS in accordance with Zeta 1 (which I have already commenced in the Riemann Function blogs) this implies that when we come to the LHS, this now implies a complementary quantitative interpretation.
So the startling truth regarding the Riemann Zeta Function, is that its proper appreciation requires incorporation of two Functions, Zeta 1 and Zeta 2, which then operate in a perfect complementary manner with respect to interpretation on both sides of the Functional Equation.
Though the Functional Equation associated with Riemann (and perhaps to a degree Euler) is a marvellous discovery in its own right, unfortunately it completely masks the true nature of the Zeta Function (and of course its associated Riemann Hypothesis) by attempting to convey it merely in quantitative terms, from a Type 1 mathematical perspective. Though it is certainly valid from this context, it leaves the essential nature of both the Function and Hypothesis completely unexplained.
For when we understand correctly, we realise that the Functional Equation (incorporating both Zeta 1 and Zeta Functions) establishes a perfect complementary relationship as between both quantitative and qualitative type meaning (which is inherent in the very nature of the primes).
So, quantitative interpretation with respect to the RHS (in accordance with Zeta 1) is matched by corresponding qualitative interpretation on the LHS (in accordance with Zeta 2).
Likewise qualitative interpretation on the RHS (in accordance with Zeta 1) is matched by quantitative interpretation on the LHS (in accordance with Zeta 2).
And we could equally apply Zeta 2 to interpretation of the RHS with then complementary interpretation through Zeta 1 on the LHS.
So when we read the existing Riemann Function appropriately, we realise that through switching from from RHS (using Zeta 1) to LHS, that both the base quantities and dimensional powers (as defined on the RHS), thereby likewise switch with each other, so that in fact we are now interpreting in accordance with the complementary Zeta 2 formulation!
And if this seems a lot to take in, well its true significance is that what we presently define as Mathematics is simply not fit for purpose.
Using my own terminology, Conventional Mathematics is still rooted in an absolute version of the Type 1 (quantitative) approach.
But Mathematics equally needs a Type 2 aspect in a direct qualitative interpretation of the nature of mathematical symbols. And only then can a truly comprehensive Type 3 approach emerge, where both Type 1 and Type 2 are now understood in dynamic interactive terms as fully complementary with each other.
And if ever a problem needed this Type 3 approach for proper comprehension, the Riemann Zeta Function (and associated Hypothesis) certainly qualifies - pardon the pun - as a prime candidate!
It can be expressed in several ways.
One that I especially like is that given in John Derbyshire's "Prime Obsession" on P. 147:
ζ(1-s) = 2^(1-s)π^(-s)sin{[(1-s)/2]π}(s-1)!ζ(s).
Thus, if we calculate a result for the Function with respect to a positive value of s on the RHS of the Function, we can equally calculate a corresponding value with respect to the negative value of s, i.e. 1 - s.
So for example if we have already calculated ζ(2), then through the formula we can equally calculate ζ(- 1).
However a very significant problem arises with respect to the Function for negative values of s, in that the results seem somewhat meaningless from a conventional linear perspective (which properly constitutes Type 1 Mathematics).
For example the Function for s = - 1, generates the sum of the natural numbers
1 + 2 + 3 + 4 +........
Now in conventional (Type 1) terms this series clearly diverges to infinity.
However the result as derived from the Functional Equation = - 1/12.
We will give a coherent interpretation of what this numerical result really signifies in a future entry. But firstly we have to place it within its proper general context.
Let us return to the Zeta 1 and Zeta 2 formulations of the Function, corresponding to the (linear) Type 1 and the (circular) Type 2 aspects of Mathematics respectively.
So Zeta 1 can be expressed as the infinite series
1/1^s + 1/2^s + 1/3^s + 1/4^s +.......
Now if we invert each term of this Function we then get
1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) +......
Zeta 2 by comparison is expressed by the inverse series (with respect to both base and dimensional numbers as
1 + s^1 + s^2 + s^3 + s^4 + .....
Now when we let s = 0 (as dimensional power) in the Zeta 1 and s = 1 (as base quantity) in the Zeta 2, both Functions appear as identical.
So for s = 0, Zeta 1 gives
1^0 + 2^0 + 3^0 + 4^0 + .....
= 1 + 1 + 1 + 1 + ....
Likewise for s = 1, in Zeta 2, we have
1 + 1^1 + 1^2 + 1^3 + 1^4 +......
= 1 + 1 + 1 + 1 + .....
So in this special case where both Zeta 1 and Zeta 2 appear identical, s = 0 (as dimension) in the former case and s = 1 (as base quantity) in the latter.
Therefore Zeta 2 is connected with Zeta 1 through the simple transformation s = 1 - s (where it is understood that s now switches as between dimensional power and base quantity).
Now to distinguish both Functions, let
Zeta 1 = ζ(s)1 and Zeta 2 = ζ(s)2.
Then in this special special case we have ζ(s)2 = (1 - s)ζ(s)1. So the Functional Equation established by Riemann (with its adjustment terms) can be used to extend this result for all other values of s (but now properly incorporating both Zeta 1 and Zeta 2).
Thus if we interpret - as is the convention - the Functional Equation using solely Zeta 1 where s = 0, then all the non zeta terms cancel out and we are left,
ζ(s) = (1 - s)ζ(s), i.e. the trivial identity
ζ(s) = ζ(s).
Now the important point I am making is that appropriate interpretation of the Riemann Functional Equation requires appreciation that both sides of the equation actually point to two Functions (Zeta 1 and Zeta 2 respectively) that bear a complementary relationship with each other.
So when we interpret quantitative symbols and numerical results in a linear fashion on the RHS in accordance with Zeta 1, this implies that corresponding appropriate quantitative interpretation on the LHS requires a complementary circular interpretation in accordance with Type 2.
Likewise when we give symbols a qualitative circular type interpretation, on the RHS in accordance with Zeta 1 (which I have already commenced in the Riemann Function blogs) this implies that when we come to the LHS, this now implies a complementary quantitative interpretation.
So the startling truth regarding the Riemann Zeta Function, is that its proper appreciation requires incorporation of two Functions, Zeta 1 and Zeta 2, which then operate in a perfect complementary manner with respect to interpretation on both sides of the Functional Equation.
Though the Functional Equation associated with Riemann (and perhaps to a degree Euler) is a marvellous discovery in its own right, unfortunately it completely masks the true nature of the Zeta Function (and of course its associated Riemann Hypothesis) by attempting to convey it merely in quantitative terms, from a Type 1 mathematical perspective. Though it is certainly valid from this context, it leaves the essential nature of both the Function and Hypothesis completely unexplained.
For when we understand correctly, we realise that the Functional Equation (incorporating both Zeta 1 and Zeta Functions) establishes a perfect complementary relationship as between both quantitative and qualitative type meaning (which is inherent in the very nature of the primes).
So, quantitative interpretation with respect to the RHS (in accordance with Zeta 1) is matched by corresponding qualitative interpretation on the LHS (in accordance with Zeta 2).
Likewise qualitative interpretation on the RHS (in accordance with Zeta 1) is matched by quantitative interpretation on the LHS (in accordance with Zeta 2).
And we could equally apply Zeta 2 to interpretation of the RHS with then complementary interpretation through Zeta 1 on the LHS.
So when we read the existing Riemann Function appropriately, we realise that through switching from from RHS (using Zeta 1) to LHS, that both the base quantities and dimensional powers (as defined on the RHS), thereby likewise switch with each other, so that in fact we are now interpreting in accordance with the complementary Zeta 2 formulation!
And if this seems a lot to take in, well its true significance is that what we presently define as Mathematics is simply not fit for purpose.
Using my own terminology, Conventional Mathematics is still rooted in an absolute version of the Type 1 (quantitative) approach.
But Mathematics equally needs a Type 2 aspect in a direct qualitative interpretation of the nature of mathematical symbols. And only then can a truly comprehensive Type 3 approach emerge, where both Type 1 and Type 2 are now understood in dynamic interactive terms as fully complementary with each other.
And if ever a problem needed this Type 3 approach for proper comprehension, the Riemann Zeta Function (and associated Hypothesis) certainly qualifies - pardon the pun - as a prime candidate!
Saturday, March 10, 2012
Unveiling the Zeta 2 Function
Yesterday I mentioned how deviations could be eliminated with respect to the Zeta 2 Function. (We will outline the precise nature of this Function in a moment).
Now once again, this Function provides a method of measuring the distribution of natural numbers among the primes (as the reverse of the Zeta 1 where the measurement is with respect to the distribution of the primes among the natural numbers).
So associated in this second Function with each prime number is a set of roots which ranges over the natural numbers in ordinal terms from 1 up to and including the prime number in question.
These complex valued roots can for convenience be broken into cos and sin parts with measurements taken in an absolute fashion.
This enables us therefore for any prime number p, to calculate the mean value of the sum of both cos and sin parts.
This value (for both sin and cos) approaches 2/π (i/log i) quite quickly. Also both cos and sin values line up on either side of this value in a manner where the ratio of cos to sin - again in absolute terms - quickly approaches .5.
We then saw how deviations could be eliminated in a manner that involves the square of the prime numbers.
In fact what this demonstrates - in a manner that is the counterpart to the non-trivial zeros for Zeta 1 - is that every prime number (as an individual member) potentially makes a contribution to the elimination of this deviation. So once again we cannot divorce quantitative from qualitative considerations.
Again in the Zeta 1 case the non-trivial zeros embody in a quantitative manner, the holistic collective nature of the primes. They thereby eliminate deviations with respect to their general occurrence (among the natural numbers) and the precise individual identity of the primes.
Here we have the reverse situation where the individual primes eliminate the deviations with respect to the collective behaviour of the natural numbers (as ordinal prime roots of 1) which directly is of a qualitative nature.
However I only considered the cos aspect in yesterday's contribution. Though this real part of the roots does indeed make the more decisive contribution, the sin aspect is also involved. So just like in atomic physics particles also have a wave aspect, likewise, the cos aspect relating to individual primes also has a collective wave aspect that influence the deviation.
So in principle the complete explanation of the deviations involves consideration of both aspects!
We now will look more closely at the form of this Zeta 2 Function.
We start with the simple equation expression and in the manner of Zeta 1 we will use s (though this time representing the base rather than the dimensional number!)
So s^n = 1.
Therefore 1 - s^n = 0
Now (1 - s^n)/(1 - s) = 1 + s + s^2 + s^3 + s^4 +.....s^(n + 1) = 0.
Therefore - expressed in the standard reduced manner n → ∞, then,
(1- s^n)/(1 - s) = 1 + s + s^2 + s^3 + s^4 +..... = 0.
or alternatively,
(1 - s)(1 + s + s^2 + s^3 + s^4 +.....) = 0.
1 - s = 0 represents the 1st. where s = 1, and is always a solution.
In this sense it is trivial. Therefore if we divide by 1 - s,
We now have
1 + s + s^2 + s^3 + s^4 + ..... = 0. This expression contains all other non-trivial root solutions (i.e. except 1)
So this is Zeta 2. It is of course a well-known Function, but its hidden circular type properties hidden within Zeta 1 have been greatly overlooked!
Now the Zeta 1 Function by contrast is
1/1^s + 1/2^s + 1/3^s + 1/4^s +.......
If we invert each term of this Function we then get
1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) +......
Now there is an obvious complementary significance with respect to both Zeta 1 and Zeta 2 (which makes them identical in this sense)!
When s is used as base number in Zeta 2, it is used by contrast as a dimensional power in Zeta 1
Where the natural numbers are used as dimensional powers in Zeta 2, they are by contrast used as base numbers in Zeta 1!
Finally in their present form what is positive for the dimensional power in Zeta 2, is negative in Zeta 1 (and vice versa).
This is of great significance when we come to interpret numerical values on the LHS for s < 0 in the Zeta 1 (where they can be given no coherent quantitative interpretation)! In fact the reason is that values now actually confirm to the hidden Zeta 2 Function (which is indeed quantitatively defined for s < 0)!
Now you might remember that the very basis of my alternative (Type 2) number system requires such switching of base and dimensional numbers with respect to Type 1!
And this is precisely what happens here with these two formulations of the Zeta Function.
So, we can see that this alternative system, I have been dealing with in the past few days with a direct view to qualitative interpretation is simply the Type 2 counterpart of the Type 1 Riemann Function.
Thus in a sense it has the power - as we have seen - to turn everything on its head.
From a Type 1 perspective, we are accustomed to understanding one-way in sole consideration of the distribution of the individual (cardinal) primes among the natural numbers!
However, from the Type 2 perspective, we are led to see their hidden complementary aspect, where we now have the distribution of the individual (ordinal) natural numbers among the primes (as representing the prime roots of 1).
And this clearly brings out the key fact that there is a qualitative as well as quantitative aspect to the primes (and indeed also the natural numbers).
Now, when we then bring both Type 1 and Type 2 approaches together, we begin to appreciate that the prime and natural numbers are in fact complementary mirrors to each other and indeed are ultimately identical!
Again we can perhaps bring this out in a striking manner using our crossroads analogy.
Using - realtively - isolated frames of reference we can deal with Type 1 and Type 2 in seemingly unambiguous terms.
Thus from the Type 1 perspective, natural numbers are derived from the primes! From the Type 2 perspective, the primes are derived from the natural numbers (again as comprising the collection of roots of each prime number).
So from either perspective in isolation, unambiguous type connections seemingly can be made just as a person can unambiguously identify left or right turns using one isolated direction of movement.
However when we simultaneously bring together both aspects (as interdependent) all such unambiguous distinctions break down. So now depending on direction, what is left can be right and what is right can be left.
It is likewise exacty similar as regards the relationship as between the primes and natural numbers. When, as in a Type 3 approach, both aspects of interpretation (Type 1 and Type 2) are combined, the prime and natural numbers can no longer be clearly distinguished. So what we distinguish as prime and natural numbers are in fact just two aspects of the same mutual identity. What is prime from Type 1, is natural from Type 2; and what is natural from Type 2, is prime from Type 1.
And as Type 3 combines both aspects, the prime and natural numbers are understood as ultimately identical (in both quantitative and qualitative terms).
I mentioned before how in the Type 1 approach, each prime number (as cardinal) could be given a natural number ranking (as ordinal).
Then again in the Type 2 (now reversed) each natural number (as cardinal within this qualitative system) likewise can be given a prime number ranking (as ordinal).
So really what is prime or natural is purely dependent on arbitrary reference frames. So the two aspects, which in truth are ultimately identical as perfect mirrors to each other, only appear distinct as prime or natural numbers respectively, when we impose arbitrary reference frames for their interpretation!
Now once again, this Function provides a method of measuring the distribution of natural numbers among the primes (as the reverse of the Zeta 1 where the measurement is with respect to the distribution of the primes among the natural numbers).
So associated in this second Function with each prime number is a set of roots which ranges over the natural numbers in ordinal terms from 1 up to and including the prime number in question.
These complex valued roots can for convenience be broken into cos and sin parts with measurements taken in an absolute fashion.
This enables us therefore for any prime number p, to calculate the mean value of the sum of both cos and sin parts.
This value (for both sin and cos) approaches 2/π (i/log i) quite quickly. Also both cos and sin values line up on either side of this value in a manner where the ratio of cos to sin - again in absolute terms - quickly approaches .5.
We then saw how deviations could be eliminated in a manner that involves the square of the prime numbers.
In fact what this demonstrates - in a manner that is the counterpart to the non-trivial zeros for Zeta 1 - is that every prime number (as an individual member) potentially makes a contribution to the elimination of this deviation. So once again we cannot divorce quantitative from qualitative considerations.
Again in the Zeta 1 case the non-trivial zeros embody in a quantitative manner, the holistic collective nature of the primes. They thereby eliminate deviations with respect to their general occurrence (among the natural numbers) and the precise individual identity of the primes.
Here we have the reverse situation where the individual primes eliminate the deviations with respect to the collective behaviour of the natural numbers (as ordinal prime roots of 1) which directly is of a qualitative nature.
However I only considered the cos aspect in yesterday's contribution. Though this real part of the roots does indeed make the more decisive contribution, the sin aspect is also involved. So just like in atomic physics particles also have a wave aspect, likewise, the cos aspect relating to individual primes also has a collective wave aspect that influence the deviation.
So in principle the complete explanation of the deviations involves consideration of both aspects!
We now will look more closely at the form of this Zeta 2 Function.
We start with the simple equation expression and in the manner of Zeta 1 we will use s (though this time representing the base rather than the dimensional number!)
So s^n = 1.
Therefore 1 - s^n = 0
Now (1 - s^n)/(1 - s) = 1 + s + s^2 + s^3 + s^4 +.....s^(n + 1) = 0.
Therefore - expressed in the standard reduced manner n → ∞, then,
(1- s^n)/(1 - s) = 1 + s + s^2 + s^3 + s^4 +..... = 0.
or alternatively,
(1 - s)(1 + s + s^2 + s^3 + s^4 +.....) = 0.
1 - s = 0 represents the 1st. where s = 1, and is always a solution.
In this sense it is trivial. Therefore if we divide by 1 - s,
We now have
1 + s + s^2 + s^3 + s^4 + ..... = 0. This expression contains all other non-trivial root solutions (i.e. except 1)
So this is Zeta 2. It is of course a well-known Function, but its hidden circular type properties hidden within Zeta 1 have been greatly overlooked!
Now the Zeta 1 Function by contrast is
1/1^s + 1/2^s + 1/3^s + 1/4^s +.......
If we invert each term of this Function we then get
1^(- s) + 2^(- s) + 3^(- s) + 4^(- s) +......
Now there is an obvious complementary significance with respect to both Zeta 1 and Zeta 2 (which makes them identical in this sense)!
When s is used as base number in Zeta 2, it is used by contrast as a dimensional power in Zeta 1
Where the natural numbers are used as dimensional powers in Zeta 2, they are by contrast used as base numbers in Zeta 1!
Finally in their present form what is positive for the dimensional power in Zeta 2, is negative in Zeta 1 (and vice versa).
This is of great significance when we come to interpret numerical values on the LHS for s < 0 in the Zeta 1 (where they can be given no coherent quantitative interpretation)! In fact the reason is that values now actually confirm to the hidden Zeta 2 Function (which is indeed quantitatively defined for s < 0)!
Now you might remember that the very basis of my alternative (Type 2) number system requires such switching of base and dimensional numbers with respect to Type 1!
And this is precisely what happens here with these two formulations of the Zeta Function.
So, we can see that this alternative system, I have been dealing with in the past few days with a direct view to qualitative interpretation is simply the Type 2 counterpart of the Type 1 Riemann Function.
Thus in a sense it has the power - as we have seen - to turn everything on its head.
From a Type 1 perspective, we are accustomed to understanding one-way in sole consideration of the distribution of the individual (cardinal) primes among the natural numbers!
However, from the Type 2 perspective, we are led to see their hidden complementary aspect, where we now have the distribution of the individual (ordinal) natural numbers among the primes (as representing the prime roots of 1).
And this clearly brings out the key fact that there is a qualitative as well as quantitative aspect to the primes (and indeed also the natural numbers).
Now, when we then bring both Type 1 and Type 2 approaches together, we begin to appreciate that the prime and natural numbers are in fact complementary mirrors to each other and indeed are ultimately identical!
Again we can perhaps bring this out in a striking manner using our crossroads analogy.
Using - realtively - isolated frames of reference we can deal with Type 1 and Type 2 in seemingly unambiguous terms.
Thus from the Type 1 perspective, natural numbers are derived from the primes! From the Type 2 perspective, the primes are derived from the natural numbers (again as comprising the collection of roots of each prime number).
So from either perspective in isolation, unambiguous type connections seemingly can be made just as a person can unambiguously identify left or right turns using one isolated direction of movement.
However when we simultaneously bring together both aspects (as interdependent) all such unambiguous distinctions break down. So now depending on direction, what is left can be right and what is right can be left.
It is likewise exacty similar as regards the relationship as between the primes and natural numbers. When, as in a Type 3 approach, both aspects of interpretation (Type 1 and Type 2) are combined, the prime and natural numbers can no longer be clearly distinguished. So what we distinguish as prime and natural numbers are in fact just two aspects of the same mutual identity. What is prime from Type 1, is natural from Type 2; and what is natural from Type 2, is prime from Type 1.
And as Type 3 combines both aspects, the prime and natural numbers are understood as ultimately identical (in both quantitative and qualitative terms).
I mentioned before how in the Type 1 approach, each prime number (as cardinal) could be given a natural number ranking (as ordinal).
Then again in the Type 2 (now reversed) each natural number (as cardinal within this qualitative system) likewise can be given a prime number ranking (as ordinal).
So really what is prime or natural is purely dependent on arbitrary reference frames. So the two aspects, which in truth are ultimately identical as perfect mirrors to each other, only appear distinct as prime or natural numbers respectively, when we impose arbitrary reference frames for their interpretation!
Friday, March 9, 2012
The Hidden Zeta Function
In yesterday's blog entry, I mentioned how a complementary version of the Riemann Function exists. And we will see in a later entry how this alternative Function is - in a sense - already contained in the Riemann waiting to be unpacked.
And in reverse the Riemann Function is contained within this new Function likewise waiting to be unpacked. Indeed this is vitally necessary from both perspectives to demonstrate the true nature of the interaction as between quantitative and qualitative aspects.
Yesterday, I mentioned an equivalent Riemann Hypothesis with respect to this complementary Zeta Function. In fact to avoid confusing it with the recognised Function we will refer to it as Zeta 2 (with the existing Function Zeta 1). And the when finally both are brought together in full interactive glory we will refer to this combined Function as Zeta 3.
In fact this "new" Function is not new at all. Rather its newness related to its (unrecognised) intimate interdependence with the Zeta 1 Function (and the Zeta 1 with it in turn). So with the help of the Zeta 2 we will be able to resolve that old problem of giving a coherent explanation for values of the Zeta 1 Function (where s < 0).
Now once again the Zeta 1 Function is concerned with the precise prediction of the (cardinal) primes among the natural numbers.
Zeta 2 in reverse is concerned with precise prediction of the ordinal natural numbers among the primes.
In other words here we are looking at the distribution of the roots of 1 - which are listed in an natural number ordinal fashion - for each of the prime numbers.
So we have switched from a linear to a circular type perspective here.
As in the Zeta 1 case, the accuracy of the distribution quickly improves as the prime number (with respect to the extraction of all its natural numbered roots) increases.
Its value zones in on 2/π (= i/log i) just as the distribution of (cardinal) primes among the natural numbers approaches in complementary manner can be given as n/log n.
We also saw that the deviation from this number of the (absolute) cos value as a ratio of the corresponding (absolute) sin value quickly approaches .5, which in this context of the Zeta 2 Function is the complementary condition to the Riemann Hypothesis in Zeta 1!
I also spent some time working out how these deviations themselves are distributed and discovered an interesting pattern.
So concentrating here on the average cos deviation for some prime p1, if we wish to approximate the deviation for a larger prime p2 we multiply the existing deviation by the square of p1/p2.
So the new deviation d2 ~ d1 * (p1/p2)^2.
This simple procedure is surprisingly accurate though it does not fully predict the new deviation.
I will briefly illustrate!
We will start with the very simple case of the prime number 3 which has 3 corresponding roots i.e. the first second and third that are listed as 1, 2 and 3 in ordinal terms.
Now with respect to cos part, the 3 roots (expressed in absolute terms) are 1, 1/2 and 1/2. The sum of these roots is 2 so that the average is .66666666
The absolute deviation from 2/π (i/log i) = .0030046893...
Now I manually calculated all these deviation ratios for all prime root values (up to 127).
So if we wished to estimate the smaller deviation of the average cos value from 2/π where p2 = 127, then,
d2 ~ d1 * (p1/p2)^2
and d1 * (p1/p2)^2 = (.030046893...) * (3/127)^2
= .000016766...
Now given that we have based our calculation on an very early prime value 3, this compares extremely well with the true deviation = .00001623...
Of course if we estimated the deviation (in the case of 127) from a later deviation we would get a much better approximation.
Now we will later discover that a whole new world opens up from this absolute treatment of root values both with respect to the addition and multiplication of roots which can play a great role in understanding the true nature of the Riemann Zeta Function (i.e. Zeta 1).
However I mentioned that the Riemann Hypothesis lends itself primarily to qualitative interpretation so that one for example - from understanding of its nature - can establish in fact that in has no proof (in Type 1 terms) using quantitative Mathematics. However a full explanation does indeed require the complementary blending of both quantitative and qualitative aspects.
As I have repeatedly stated the Riemann Hypothesis points to the central aspect of the primes which is the ultimate identity of quantitative and qualitative aspects in an ineffable manner.
In fact when looked on appropriately, as simultaneous from both perspectives, the natural numbers and the primes are thereby understood as perfect mirrors of each other.
However this cannot be achieved through understanding based on just one logical system (as in Conventional Mathematics).
So we properly require two means of interpretation that are linear and circular (and circular and linear) with respect to each other. As we have seen linear logic is based on isolated (uni-polar) frames of reference that is ill-suited for appreciation of interdependence; circular logic is always based on the pairing of complementary opposites (in a bi-polar fashion).
So the appreciation of the simultaneous identity of both the natural and prime requires complex interpretation in a qualitative sense, where one can keep shifting as between real (1-dimensional) and imaginary (as the indirect rational expression of 2-dimensional) understanding respectively.
Now the sin part of a complex root is imaginary and the cos real.
So the condition of .5 simply relates to this fundamental relationship as between 1-dimensional and 2-dimensional appreciation. In other words, the mystery of the Riemann Hypothesis does not reside in appreciation in accordance with linear or circular understanding (as separate) but in the relationship between both, where ultimately they become identical as perfect mediators of an ineffable reality.
The point at the centre of a circle is equally the centre of its line diameter at the midpoint (1/2) of this diameter.
With Zeta 2 we are directly encountering the qualitative counterpart of this central point.
And in reverse the Riemann Function is contained within this new Function likewise waiting to be unpacked. Indeed this is vitally necessary from both perspectives to demonstrate the true nature of the interaction as between quantitative and qualitative aspects.
Yesterday, I mentioned an equivalent Riemann Hypothesis with respect to this complementary Zeta Function. In fact to avoid confusing it with the recognised Function we will refer to it as Zeta 2 (with the existing Function Zeta 1). And the when finally both are brought together in full interactive glory we will refer to this combined Function as Zeta 3.
In fact this "new" Function is not new at all. Rather its newness related to its (unrecognised) intimate interdependence with the Zeta 1 Function (and the Zeta 1 with it in turn). So with the help of the Zeta 2 we will be able to resolve that old problem of giving a coherent explanation for values of the Zeta 1 Function (where s < 0).
Now once again the Zeta 1 Function is concerned with the precise prediction of the (cardinal) primes among the natural numbers.
Zeta 2 in reverse is concerned with precise prediction of the ordinal natural numbers among the primes.
In other words here we are looking at the distribution of the roots of 1 - which are listed in an natural number ordinal fashion - for each of the prime numbers.
So we have switched from a linear to a circular type perspective here.
As in the Zeta 1 case, the accuracy of the distribution quickly improves as the prime number (with respect to the extraction of all its natural numbered roots) increases.
Its value zones in on 2/π (= i/log i) just as the distribution of (cardinal) primes among the natural numbers approaches in complementary manner can be given as n/log n.
We also saw that the deviation from this number of the (absolute) cos value as a ratio of the corresponding (absolute) sin value quickly approaches .5, which in this context of the Zeta 2 Function is the complementary condition to the Riemann Hypothesis in Zeta 1!
I also spent some time working out how these deviations themselves are distributed and discovered an interesting pattern.
So concentrating here on the average cos deviation for some prime p1, if we wish to approximate the deviation for a larger prime p2 we multiply the existing deviation by the square of p1/p2.
So the new deviation d2 ~ d1 * (p1/p2)^2.
This simple procedure is surprisingly accurate though it does not fully predict the new deviation.
I will briefly illustrate!
We will start with the very simple case of the prime number 3 which has 3 corresponding roots i.e. the first second and third that are listed as 1, 2 and 3 in ordinal terms.
Now with respect to cos part, the 3 roots (expressed in absolute terms) are 1, 1/2 and 1/2. The sum of these roots is 2 so that the average is .66666666
The absolute deviation from 2/π (i/log i) = .0030046893...
Now I manually calculated all these deviation ratios for all prime root values (up to 127).
So if we wished to estimate the smaller deviation of the average cos value from 2/π where p2 = 127, then,
d2 ~ d1 * (p1/p2)^2
and d1 * (p1/p2)^2 = (.030046893...) * (3/127)^2
= .000016766...
Now given that we have based our calculation on an very early prime value 3, this compares extremely well with the true deviation = .00001623...
Of course if we estimated the deviation (in the case of 127) from a later deviation we would get a much better approximation.
Now we will later discover that a whole new world opens up from this absolute treatment of root values both with respect to the addition and multiplication of roots which can play a great role in understanding the true nature of the Riemann Zeta Function (i.e. Zeta 1).
However I mentioned that the Riemann Hypothesis lends itself primarily to qualitative interpretation so that one for example - from understanding of its nature - can establish in fact that in has no proof (in Type 1 terms) using quantitative Mathematics. However a full explanation does indeed require the complementary blending of both quantitative and qualitative aspects.
As I have repeatedly stated the Riemann Hypothesis points to the central aspect of the primes which is the ultimate identity of quantitative and qualitative aspects in an ineffable manner.
In fact when looked on appropriately, as simultaneous from both perspectives, the natural numbers and the primes are thereby understood as perfect mirrors of each other.
However this cannot be achieved through understanding based on just one logical system (as in Conventional Mathematics).
So we properly require two means of interpretation that are linear and circular (and circular and linear) with respect to each other. As we have seen linear logic is based on isolated (uni-polar) frames of reference that is ill-suited for appreciation of interdependence; circular logic is always based on the pairing of complementary opposites (in a bi-polar fashion).
So the appreciation of the simultaneous identity of both the natural and prime requires complex interpretation in a qualitative sense, where one can keep shifting as between real (1-dimensional) and imaginary (as the indirect rational expression of 2-dimensional) understanding respectively.
Now the sin part of a complex root is imaginary and the cos real.
So the condition of .5 simply relates to this fundamental relationship as between 1-dimensional and 2-dimensional appreciation. In other words, the mystery of the Riemann Hypothesis does not reside in appreciation in accordance with linear or circular understanding (as separate) but in the relationship between both, where ultimately they become identical as perfect mediators of an ineffable reality.
The point at the centre of a circle is equally the centre of its line diameter at the midpoint (1/2) of this diameter.
With Zeta 2 we are directly encountering the qualitative counterpart of this central point.
Thursday, March 8, 2012
Alternative Riemann Hypothesis!
Yesterday I was dealing in my blog with the shadow approach to the relationship of the primes and natural numbers (using the Type 2 aspect of the number system). More correctly, it should be referred to (in qualitative terms) as the imaginary counterpart of the customary real Type 1 approach to the Riemann Zeta Function.
However we will gradually see that it is also subtly embedded - as it were - in the customary Zeta Function and that indeed it is vitally necessary to give meaning to the values of the Zeta Function for s < 0.
This alternative system though it has indeed a quantitative aspect - which I will presently deal with - is primarily geared to deal with the qualitative nature of the primes.
And as I have long maintained that the Riemann Hypothesis lends itself more readily to qualitative - rather than quantitative - appreciation it does indeed warrant further exploration. And here again - quite remarkably - we will quickly unravel both an alternative Prime Number Theorem and Riemann Hypothesis requiring nothing more, from a quantitative standpoint, than elementary Mathematics!
From the Type 1 perspective we start with the individual primes (as cardinal) and then attempt to express the natural numbers as entailing the product of a unique combination of these primes. However, when looked at more closely, the holistic capacity of primes with respect to the natural numbers is of a qualitative rather than quantitative nature. And appreciation of this requires looking at this relationship from the "imaginary" Type 2 perspective.
We then showed yesterday - in what initially is experienced as a quite remarkable discovery - that the relationship of the primes to the natural numbers is completely inverted from this alternative standpoint.
Here - viewing number now from an ordinal perspective - we start with the natural numbers (as representing the various roots of a given prime number). We then show how each prime is viewed as the unique product of all the natural numbers roots (from 1 up to and including the prime in question).
So to obtain the 17 roots of 1, we must raise the 17th root of 1 to each of the natural numbers from 1 to 17. So each natural number root here is qualitatively unique. However the prime number 17 (as number of roots) is of a cardinal nature.
So just, as properly speaking, the collective nature of the primes when using the Type 1 perspective, is of a qualitative, here in reverse, the individual nature of each prime (as comprising its 17 roots) is properly of a quantitative nature. This suggests that in the more comprehensive Type 3 approach, the two-way interdependence of both systems is required. So here the Type 2 would also be needed to explain fully the Type 1 and likewise the Type 1 to fully explain the Type 2, with eventually both seen as entirely interdependent.
So at this final stage, the primes and natural numbers are seen as dynamically identical, with the apparent difference between them due to the fact that their distinctive nature arises from attempting to look at them in relative separation, whereas their mutual identity requires using both lenses simultaneously (Type 3).
I will now mention briefly a result I came up up with some years ago. (I have dealt with it before in other blog entries but not specifically in this context!) I felt at the time that it was important, but had to wait some time before realising that it actually belongs to this Type 2 formulation of the Riemann Zeta Function.
I knew that the circular study of primes (as the prime roots of 1) should offer a complementary approach to the standard linear approach.
Now, the problem with these complex roots is that when one takes them as a collection i.e. all the roots of a prime number, they add up to 0 and when multiplied give a product of 1.
So the first insight was to decide to look at their numerical properties with respect to absolute values of the roots.
I then decided to sum up both sin and cos values separately to see if a different pattern applied to both.
So the idea was - say for the simple case of the 3 roots of 1 - to first get the sum, in absolute terms, of the three cos parts of the roots, followed by the sum of two sin parts (Note for prime valued roots there will always be one less sin value)!
Having obtained a total, I then divided by the prime number in question to get the mean of the 3 roots (with each part as separate).
I then continued on methodically in this fashion for each prime number, summing up all the roots (both cos and sin absolute values) correct to 8 and 9 places, before obtaining the average, going all the way to the 127 roots of 1.
It struck me quite quickly that the average for both cos and sin values was zoning in on a fixed value which I worked out was 2/pi.
I then made the interesting discovery that 2/pi can alternatively be expressed as i/log i.
Deriving this is quite simple:
e^(2iπ) = 1^1
Therefore, raising both sides to i,
e^(- 2π) = 1^i
Next raising each side to power of 1/4,
e^(- π/2) = i^i
Taking logs of each side,
- π/2 = i log i
π/2 = - i log i
Therefore
2/π = 1/(- i log i)
= i/log i
Now this bears direct comparison with the simple expression for the distribution of primes among the natural number i.e. n/log n.
In fact, it is the imaginary counterpart of the alternative real expression.
The idea of increasing to infinity is the Type 1 way of dealing with the relationship as between finite and infinite (in quantitative terms).
The notion of i, as a finite means of embodying the infinite in qualitative terms, is the corresponding Type 2 manner of incorporating infinite with finite notions.
So we have in fact a parallel prime number theorem here arising from the circular study of prime number roots.
So what we are saying is that when we obtain the average of the prime roots of 1 (measuring cos and sin values in absolute terms), the average of these prime roots sums ever closer to i/log i (i.e. 2/π) in both cases. 2/π = .636619772.. .
When I then studied the deviations of the average cos and sin values from 2/π, I made an interesting discovery, for it became quickly apparent that the cos average were always in excess of 2/π while the sin average was always less, with the ratio between cos and sin quickly approaching the value .5. Does this sound familiar?
Needless to say this ratio steadily improves as we get the average cos and sin values for higher numbered primes.
Though I thought it might be coincidental initially, I now have come to the opinion that in fact this does indeed represent an alternative way (from a Type 2 perspective) of stating Riemann's Hypothesis. As always there is a reversal in Type 2 terms. From the Type 1 perspective, calculations of the general distribution of primes according to formula n/log n, yield an approximation that steadily improves with larger n, while the .5 condition for the Riemann Hypothesis is given as a fixed value.
Here in reverse the value for general distribution of roots of prime number roots is given as a fixed value 2/π, while the deviations from 2/π, approximate ever closer to .5.
What is again remarkable however about this is that it represents a truly elementary approach (in contrast to the high duty Mathematics required in the Type 1 case).
Furthermore, its qualitative significance is much easier to appreciate (as befits a circular Type 2 approach).
So we will return again to this topic!
However we will gradually see that it is also subtly embedded - as it were - in the customary Zeta Function and that indeed it is vitally necessary to give meaning to the values of the Zeta Function for s < 0.
This alternative system though it has indeed a quantitative aspect - which I will presently deal with - is primarily geared to deal with the qualitative nature of the primes.
And as I have long maintained that the Riemann Hypothesis lends itself more readily to qualitative - rather than quantitative - appreciation it does indeed warrant further exploration. And here again - quite remarkably - we will quickly unravel both an alternative Prime Number Theorem and Riemann Hypothesis requiring nothing more, from a quantitative standpoint, than elementary Mathematics!
From the Type 1 perspective we start with the individual primes (as cardinal) and then attempt to express the natural numbers as entailing the product of a unique combination of these primes. However, when looked at more closely, the holistic capacity of primes with respect to the natural numbers is of a qualitative rather than quantitative nature. And appreciation of this requires looking at this relationship from the "imaginary" Type 2 perspective.
We then showed yesterday - in what initially is experienced as a quite remarkable discovery - that the relationship of the primes to the natural numbers is completely inverted from this alternative standpoint.
Here - viewing number now from an ordinal perspective - we start with the natural numbers (as representing the various roots of a given prime number). We then show how each prime is viewed as the unique product of all the natural numbers roots (from 1 up to and including the prime in question).
So to obtain the 17 roots of 1, we must raise the 17th root of 1 to each of the natural numbers from 1 to 17. So each natural number root here is qualitatively unique. However the prime number 17 (as number of roots) is of a cardinal nature.
So just, as properly speaking, the collective nature of the primes when using the Type 1 perspective, is of a qualitative, here in reverse, the individual nature of each prime (as comprising its 17 roots) is properly of a quantitative nature. This suggests that in the more comprehensive Type 3 approach, the two-way interdependence of both systems is required. So here the Type 2 would also be needed to explain fully the Type 1 and likewise the Type 1 to fully explain the Type 2, with eventually both seen as entirely interdependent.
So at this final stage, the primes and natural numbers are seen as dynamically identical, with the apparent difference between them due to the fact that their distinctive nature arises from attempting to look at them in relative separation, whereas their mutual identity requires using both lenses simultaneously (Type 3).
I will now mention briefly a result I came up up with some years ago. (I have dealt with it before in other blog entries but not specifically in this context!) I felt at the time that it was important, but had to wait some time before realising that it actually belongs to this Type 2 formulation of the Riemann Zeta Function.
I knew that the circular study of primes (as the prime roots of 1) should offer a complementary approach to the standard linear approach.
Now, the problem with these complex roots is that when one takes them as a collection i.e. all the roots of a prime number, they add up to 0 and when multiplied give a product of 1.
So the first insight was to decide to look at their numerical properties with respect to absolute values of the roots.
I then decided to sum up both sin and cos values separately to see if a different pattern applied to both.
So the idea was - say for the simple case of the 3 roots of 1 - to first get the sum, in absolute terms, of the three cos parts of the roots, followed by the sum of two sin parts (Note for prime valued roots there will always be one less sin value)!
Having obtained a total, I then divided by the prime number in question to get the mean of the 3 roots (with each part as separate).
I then continued on methodically in this fashion for each prime number, summing up all the roots (both cos and sin absolute values) correct to 8 and 9 places, before obtaining the average, going all the way to the 127 roots of 1.
It struck me quite quickly that the average for both cos and sin values was zoning in on a fixed value which I worked out was 2/pi.
I then made the interesting discovery that 2/pi can alternatively be expressed as i/log i.
Deriving this is quite simple:
e^(2iπ) = 1^1
Therefore, raising both sides to i,
e^(- 2π) = 1^i
Next raising each side to power of 1/4,
e^(- π/2) = i^i
Taking logs of each side,
- π/2 = i log i
π/2 = - i log i
Therefore
2/π = 1/(- i log i)
= i/log i
Now this bears direct comparison with the simple expression for the distribution of primes among the natural number i.e. n/log n.
In fact, it is the imaginary counterpart of the alternative real expression.
The idea of increasing to infinity is the Type 1 way of dealing with the relationship as between finite and infinite (in quantitative terms).
The notion of i, as a finite means of embodying the infinite in qualitative terms, is the corresponding Type 2 manner of incorporating infinite with finite notions.
So we have in fact a parallel prime number theorem here arising from the circular study of prime number roots.
So what we are saying is that when we obtain the average of the prime roots of 1 (measuring cos and sin values in absolute terms), the average of these prime roots sums ever closer to i/log i (i.e. 2/π) in both cases. 2/π = .636619772.. .
When I then studied the deviations of the average cos and sin values from 2/π, I made an interesting discovery, for it became quickly apparent that the cos average were always in excess of 2/π while the sin average was always less, with the ratio between cos and sin quickly approaching the value .5. Does this sound familiar?
Needless to say this ratio steadily improves as we get the average cos and sin values for higher numbered primes.
Though I thought it might be coincidental initially, I now have come to the opinion that in fact this does indeed represent an alternative way (from a Type 2 perspective) of stating Riemann's Hypothesis. As always there is a reversal in Type 2 terms. From the Type 1 perspective, calculations of the general distribution of primes according to formula n/log n, yield an approximation that steadily improves with larger n, while the .5 condition for the Riemann Hypothesis is given as a fixed value.
Here in reverse the value for general distribution of roots of prime number roots is given as a fixed value 2/π, while the deviations from 2/π, approximate ever closer to .5.
What is again remarkable however about this is that it represents a truly elementary approach (in contrast to the high duty Mathematics required in the Type 1 case).
Furthermore, its qualitative significance is much easier to appreciate (as befits a circular Type 2 approach).
So we will return again to this topic!
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