I have continually referred on these blogs to the enormous central weakness underlying all conventional mathematical interpretation i.e. its complete absence of any genuine notion of interdependence.

It seems quite remarkabe that it is left to someone like myself - who would be considered a complete outsider by the mathematics profession - to address this all important issue that is steadfastly ignored by its own practitioners.

By its very nature Mathematics is built on linear rational notions of fixed relationships in a static independent manner. It therefore can only deal with interdependent notions – which essentially relate to the qualitative holistic aspect of mathematical activity – in a reduced and distorted quantitative fashion.

Now this problem cannot be addressed through the development of ever more complex specialised procedures within the present accepted framework of Mathematics. In fact these will only serve to blind us further to the fundamental problem which persistently is avoided i.e. that for all its great successes, such Mathematics is built on highly limited assumptions.

So what in truth represents but an extreme – though admittedly very important – special case has been misleadingly elevated to represent all valid Mathematics.

However – when properly appreciated – the true nature of Mathematics is dynamic in nature and incomparably greater that what has yet been imagined. In fact instead of the existence of just one valid system (in absolute terms), potentially an unlimited number of alternative dynamic mathematical systems exist (each possessing an important relative validity).

As we have seen Conventional Mathematics represents the special case where interpretation is - literally - 1-dimensional (linear) in nature. And the very nature of this system is that qualitative meaning such as holistic interdependence - in every context - is thereby reduced to quantitative interpretation.

The very way of properly dealing with interdependent relationships entails a circular – rather than linear – approach that is inherently based on the dynamic notion of complementarity.

I have made the important observation before that it is only when the dimension = 1 that an absolute – rather than relative – interpretation of mathematical symbols occurs. And here the qualitative aspect is thereby reduced to the quantitative in a static fixed manner.

For all other dimensions ≠ 1 a truly relative interpretation of mathematical symbols ensues where quantitative (analytic) and qualitative (holistic) aspects interact with each other in a dynamic complementary manner.

As is well known the only value for which the Riemann Zeta Function is undefined is where the dimensional value s = 1. For all other dimensional values of s, he function is indeed defined. What this implies again is that the Riemann Zeta Function – when correctly interpreted - entails the relationship as between the analytic and holistic aspects of number. Therefore the one point where it is undefined is where a pure analytic interpretation takes place (as in Conventional Mathematics).

This is why I repeatedly stress that the true nature of the Riemann Zeta Function (and its accompanying Riemann Hypothesis) cannot be properly understood within the framework of Conventional Mathematics (which uniquely does not formally allow for a holistic interpretation).

The simplest dynamic interpretation entails 2 dimensions i.e. a 1st and 2nd respectively).

The structure of these dimensions is inversely related to the quantitative nature of the two roots of i.e. + 1 and – 1 respectively.

The corresponding qualitative interpretation entails the complementary (interdependent) relationship of these two dimensions.

Now the 1st dimension (i.e. + 1) relates to the standard linear rational approach (which literally posits phenomena in experience in a rational conscious manner (which is of an analytic quantitative nature). The 2nd dimension – by contrast negates conscious interpretation in recognition of an alternative holistic qualitative type appreciation that is inherently of an unconscious (intuitive) nature.

What the latter truly entails is that conscious understanding is always – by definition – conditioned by polar opposites interpretations (that are equally valid in nature).

Therefore to affirm interpretation (in respect of one arbitrary pole) is always limited as interpretation according to the opposite pole is equally valid. Thus in order to switch reference frames as between poles – which is the very means through which experience becomes dynamically interactive – we must first negate identification with arbitrary interpretation (based on one pole as reference point).

I have illustrated this countless times before in relation to directions at a crossroads. If we approach the crossroads from one arbitrary polar direction, (e.g. travelling “up” a road) left and right have an unambiguous linear interpretation. Likewise if approached from the opposite direction (travelling “down” the road again left and right have an unambiguous linear interpretation.

However in terms of each other both of these interpretations are clearly paradoxical. So whereas linear understanding always implies interpretation in terms of one arbitrarily fixed reference frame, corresponding appreciation of interdependence entails the simultaneous recognition of both reference frames.

Now again this explains exactly why Conventional Mathematics is lacking any true notion of interdependence! By definition such recognition – at a minimum – entails two separate poles (as possible reference frames).

The opposite pole to quantitative is qualitative! Therefore in this regard before we can truly recognise notions of interdependence then we must recognise two separate poles (as reference frames) that are quantitative and qualitative with respect to each other!

Then within each frame (considered in relative separation) unambiguous type linear results will apply in rational terms. However when one then simultaneously combines both reference frames in an intuitive complementary manner, deep paradox results.

The important point therefore is that the unambiguous understanding of (single) independent reference frames conforms to linear interpretation in rational terms. By contrast the paradoxical appreciation of (multiple) interdependent reference frames conforms to a uniquely distinct type of appreciation which in direct is always of an (unconscious) intuitive nature. However indirectly it can be conveyed in a circular rational manner (based on the complementarity of opposite poles such as quantitative and qualitative).

Now my own approach is deliberately designed to conform to this model.

Thus I start by defining two aspects of the number system Type 1 and Type 2.

Initially quantitative results can be obtained under both types (in relative separation). However when we combine both in a truly complementary manner, their relative interdependence (which truly reveals their qualitative holistic nature) then becomes apparent.

So we can see in this two-dimensional approach that we continually alternate between the 1st dimension of interpretation that is linear and the 2nd dimension that is circular.

This enables us to properly preserve therefore (linear) analytic notions of quantitative independence with corresponding (circular) holistic notions of qualitative interdependence.

Now in an earlier blog I demonstrated what in fact is a striking example of this dynamic interactive approach with respect to both the Zeta 1 and Zeta 2 Functions.

Once again both of these Functions can be studied in relative separation from each other yielding important results (of a quantitative linear kind).

However when we attempt to combine both (in an interdependent manner) the truly complementary behaviour of both becomes readily apparent. One could validly say therefore that qualitative holistic understanding resides in the very ability to recognise this rich network of interconnecting complementary relationships with respect to the Functions.

I will just elaborate again on an especially striking example from previous blogs (Complementary Zeta Functions 5 & 6) .

I established that the terms in the harmonic series (i.e. for the Zeta 1 Function, where s = 1) is composed of the values corresponding to the Zeta 2 Functions (less 1) that range over the reciprocals of all natural number values of s ≠ 1.

So notice the complementarity here! Each term (as part of the infinite Zeta 1 expression) represents the whole of a corresponding infinite Zeta 2 expression (less 1).

Likewise whereas the Zeta 1 is defined for s = 1, the Zeta 2 is defined in terms of all other natural numbers ≠ 1. And finally we have complementarity where a whole number power (for Zeta 1) is paired with its reciprocal values for Zeta 2.

The requirement to subtract 1 from each Zeta 2 expression is explained by the difference between addition and multiplication with the additive identity (leaving an expression unchanged) = 0 and the multiplication identity (likewise leaving an expression unchanged = 1).

The deeper significance of this can be explained as follows.

In the cardinal (Type 1) approach each natural number is defined by a unique combination of prime number factors (in quantitative terms).

However when we understand more fully we realise that each of these prime numbers represents a hidden sub-atomic world as it were (in Type 2 terms) where each prime number is already defined in a unique ordinal manner by all its natural number members.

In this sense the corresponding quantitative uniqueness of every natural number except 1 (in terms of its prime factors) is only possible because each of these prime numbers is itself uniquely defined in a qualitative ordinal fashion by its prime number members!

However an equally fascinating reverse form of complementarity exists.

This time we start with the Zeta 2 expression where s = 1, which contains the infinite sequence of terms 1 + 1 + 1 + 1 ..

We now look at the corresponding Zeta 1 Functions which now in reverse complementary fashion range over all the natural number values for s (except 1).

Then when we sum up this infinite sequence of Zeta 1 expressions (again subtracting 1 in each case) the resulting sum = 1.

Therefore in this case the whole infinite sum of all the Zeta 1 Functions (with 1 subtracted in each case) = each individual term in the Zeta 2 expression.

Again the deeper significance of this is that just as the subatomic qualitative natural number ordinal structure is internally contained in each prime (in cardinal terms) thereby ensuring the subsequent unique quantitative structure of the natural numbers, this atomic quantitative prime number structure is already contained in each ordinal natural number member (thus ensuring the subsequent uniqueness of its internal qualitative structure). So each unique (qualitative) part is contained in the collective (quantitative) whole; also the collective (quantitative) whole is contained in each unique (qualitative) part!

In other words through this set of complementary relationships we grow in appreciation of the ultimate truly paradoxical nature of the relationship between the primes and natural numbers.

Thus the great question regarding the number system is ultimately resolved whereby the primes and natural numbers are seen as perfect reflections of each other in a pure ineffable manner.

So we are led through ever more refined rational reflection in attempting to grasp the two-way relationship of the primes and natural numbers with each other, to finally let go of all remaining remnants of thought through finding the answer revealed in total mystery.

## Saturday, March 30, 2013

## Friday, March 29, 2013

### Brief Interlude

It may perhaps help to place some of my recent blogs on the intimate complementarity of the Zeta 1 and Zeta 2 Functions in perspective.

My overriding purpose in these blogs is simple.

Over the past 50 years or so I have reached the firm conclusion that – strictly speaking - Mathematics, as we know it, is simply not fit for purpose!

So in this context I am using the Riemann Hypothesis to illustrate the nature of this dilemma.

The true significance of the Riemann Hypothesis could hardly be more significant as it relates directly to the fundamental nature of our number system.

Now the inherent nature of this system - as indeed all mathematical activity - is truly dynamic with twin analytic (quantitative) and holistic (qualitative) aspects that continually interact in a complementary fashion with respect to each other.

Now as Conventionally Mathematics is formally interpreted in a merely reduced manner (that gives sole recognition to its quantitative aspect) not alone is the Riemann Hypothesis incapable of proof from within this perspective, more importantly it cannot even be properly understood in this manner!

Right at the heart of the conventional understanding exists a basic form of reductionism where the ordinal (i.e. qualitative) interpretation of number is assumed to be implied directly by its corresponding cardinal (i.e. quantitative) interpretation!

So we tend to look at the cardinal numbers as independent units in quantitative terms. However momentary reflection on the matter would suggest that corresponding ordinal interpretation entails a relationship between numbers (which is necessarily of a qualitative nature).

In actual experience the notion of number (such as 2 in this example) continually alternates as between its cardinal (quantitative) and ordinal (qualitative) meanings.

However far from realising the significance of this relative dynamic interaction, a highly reduced - and thereby utterly distorted - interpretation of the nature of number has come to dominate Western culture (and indeed other cultures) whereby number is misleadingly interpreted in an absolute static manner (with respect to its quantitative attributes).

Thus we mistakenly believe that the ordinal notion of the natural numbers for example such as 1st, 2nd, 3rd, 4th and so on are directly implied by the corresponding cardinal notions of 1, 2, 3, 4!

Indeed very often cardinal numbers are used directly to refer to ordinal rankings. So Tiger Woods is now once again no. 1 (i.e. 1st) in the PGA professional golf rankings, illustrating how the cardinal notion of 1 as a quantitative number is readily interchanged with the corresponding ordinal notion of 1st, as its relational qualitative counterpart!

However this fallacy i.e. of the ordinal being directly implied by its cardinal aspect, is immediately exposed when we attempt to explain the relationship of the primes to the natural numbers (from the conventional mathematical perspective).

Here the primes are viewed as the cardinal building blocks of the natural numbers so that each natural number represents a unique combination of prime factors.

However if we reflect on it for a moment a prime number strictly has no meaning in the absence of its ordinal natural number counterpart.

So for example the very recognition of 2 and 3 and 5 as prime numbers implies a natural number ordinal ranking among the primes of 1st 2nd and 3rd respectively. And if we assume that the ordinal is implied by its cardinal aspect, then this means that we must already assume the pre-existent identity of the cardinal numbers before we can even begin to explain their derivation from the primes!

This observation in fact implies a related circular notion of number (the significance of which is effectively unrecognised in conventional terms) that represents the appropriate (unreduced) appreciation of ordinal meaning.

Again even a little reflection on the matter will reveal how ambiguous is the ordinal notion of number! Clearly the meaning of 1st, 2nd, 3rd etc is of a relative nature depending on the size of the number group to which it refers.

So for example the 2nd of a group of 2 has a relatively distinct meaning from the second of a group of 3, 4, 5 etc. with potentially an unlimited range applying.

Therefore all natural numbers (in ordinal terms) have an unlimited range of potential meanings (depending on the size of the finite group to which they belong).

A little further reflection would clearly indicate that since cardinal and ordinal meaning are thereby in dynamic terms interconnected, that indirectly the cardinal number system is also of a merely relative nature!

In other words, in dynamic terms, all numbers have two complementary aspects that are - relatively - independent and also - relatively - interdependent with each other.

The static notion of numbers as abstract entities is therefore just an illusion (based on a reduced - merely quantitative – interpretation).

In my own dynamic treatment of number, I am therefore at pains to demonstrate that the number system is composed of two interacting aspects - which I refer to as Type 1 and Type 2 respectively.

The Type 1 is directly associated with the quantitative aspect (though indirectly through interdependence equally possessing a qualitative aspect).

The Type 2 is then directly associated with the qualitative aspect (though again through interdependence) indirectly possessing a quantitative aspect.

Then I refer to the full combined interaction of both aspects (Type 1 and Type 2) as Type 3.

Basically the Type 1 aspect treats numbers as homogeneous collective units in quantitative terms (thus allowing for no qualitative distinction as between individual units). So 3 as a cardinal number from this perspective = 1 + 1 + 1 (in quantitative terms).

From the Type 1 cardinal aspect the prime numbers represent the basic building blocks of the natural number system. All natural numbers (except 1) from this perspective represent a unique combination of prime factors.

The Type 2 aspect by contrast is based the notion of a number group as composed of unique individual members (in qualitative terms). Thus from this perspective 3 relates to its distinctive 1st, 2nd and 3rd individual members as ordinally defined.

From the Type 2 aspect the natural numbers represent the basic building blocks of each prime number (in ordinal terms). So every prime number (p) from an ordinal perspective is composed of a unique set of natural numbers 1st, 2nd, 3rd,...pth!

Then using these distinctions, I go on to define both Zeta 1 and Zeta 2 Functions with respect to uncovering the true nature of the Riemann Hypothesis. This relates to the ultimate identity of both the quantitative and qualitative aspects of number.

From this enlarged perspective we now have two complementary sets of (non-trivial) zeta zeros.

Indeed perhaps the best explanation of these incredibly significant sets of numbers can be given as follows.

The Type 1 zeta zeros indirectly represent the holistic aspect of the cardinal number system, whereas the Type 2 zeta zeros directly represent its corresponding holistic ordinal aspect.

Ultimately - which can be approximated in Type 3 terms where both systems simultaneously interact - both Type 1 and Type 2 zeros can be seen as fully complementary and indeed identical in an ineffable manner with the natural number system (in both cardinal and ordinal terms).

The very manner by which the quantitative and qualitative aspects of the number system interact is though the two-way interaction of the primes with the natural numbers (and the natural numbers with the primes), both of which are mediated through the two sets of zeta zeros respectively.

This leads to the remarkable conclusion that the hidden holistic activity of both sets of zeta zeros necessarily underlies all human experience in an innate fashion in the seemingly obvious identification of cardinal with ordinal meaning (i.e. 1st with 1, 2nd with 2, 3rd with 3 etc.)

Indeed even more remarkably this innate activity necessarily underlies all natural processes as the very means by which they acquire a phenomenal identity!

So what seems most accessible at a conscious level (appearing totally obvious) therefore remains least accessible in corresponding unconscious terms.

Mathematics is now in urgent need of addressing its great shadow i.e. the unrecognised holistic aspect of interpretation. As such understanding intimately affects all of the sciences - now greatly in need of a more integrated understanding - our very civilisation may now well depend on such holistic realisation occurring rapidly in the very near future.

My overriding purpose in these blogs is simple.

Over the past 50 years or so I have reached the firm conclusion that – strictly speaking - Mathematics, as we know it, is simply not fit for purpose!

So in this context I am using the Riemann Hypothesis to illustrate the nature of this dilemma.

The true significance of the Riemann Hypothesis could hardly be more significant as it relates directly to the fundamental nature of our number system.

Now the inherent nature of this system - as indeed all mathematical activity - is truly dynamic with twin analytic (quantitative) and holistic (qualitative) aspects that continually interact in a complementary fashion with respect to each other.

Now as Conventionally Mathematics is formally interpreted in a merely reduced manner (that gives sole recognition to its quantitative aspect) not alone is the Riemann Hypothesis incapable of proof from within this perspective, more importantly it cannot even be properly understood in this manner!

Right at the heart of the conventional understanding exists a basic form of reductionism where the ordinal (i.e. qualitative) interpretation of number is assumed to be implied directly by its corresponding cardinal (i.e. quantitative) interpretation!

So we tend to look at the cardinal numbers as independent units in quantitative terms. However momentary reflection on the matter would suggest that corresponding ordinal interpretation entails a relationship between numbers (which is necessarily of a qualitative nature).

In actual experience the notion of number (such as 2 in this example) continually alternates as between its cardinal (quantitative) and ordinal (qualitative) meanings.

However far from realising the significance of this relative dynamic interaction, a highly reduced - and thereby utterly distorted - interpretation of the nature of number has come to dominate Western culture (and indeed other cultures) whereby number is misleadingly interpreted in an absolute static manner (with respect to its quantitative attributes).

Thus we mistakenly believe that the ordinal notion of the natural numbers for example such as 1st, 2nd, 3rd, 4th and so on are directly implied by the corresponding cardinal notions of 1, 2, 3, 4!

Indeed very often cardinal numbers are used directly to refer to ordinal rankings. So Tiger Woods is now once again no. 1 (i.e. 1st) in the PGA professional golf rankings, illustrating how the cardinal notion of 1 as a quantitative number is readily interchanged with the corresponding ordinal notion of 1st, as its relational qualitative counterpart!

However this fallacy i.e. of the ordinal being directly implied by its cardinal aspect, is immediately exposed when we attempt to explain the relationship of the primes to the natural numbers (from the conventional mathematical perspective).

Here the primes are viewed as the cardinal building blocks of the natural numbers so that each natural number represents a unique combination of prime factors.

However if we reflect on it for a moment a prime number strictly has no meaning in the absence of its ordinal natural number counterpart.

So for example the very recognition of 2 and 3 and 5 as prime numbers implies a natural number ordinal ranking among the primes of 1st 2nd and 3rd respectively. And if we assume that the ordinal is implied by its cardinal aspect, then this means that we must already assume the pre-existent identity of the cardinal numbers before we can even begin to explain their derivation from the primes!

This observation in fact implies a related circular notion of number (the significance of which is effectively unrecognised in conventional terms) that represents the appropriate (unreduced) appreciation of ordinal meaning.

Again even a little reflection on the matter will reveal how ambiguous is the ordinal notion of number! Clearly the meaning of 1st, 2nd, 3rd etc is of a relative nature depending on the size of the number group to which it refers.

So for example the 2nd of a group of 2 has a relatively distinct meaning from the second of a group of 3, 4, 5 etc. with potentially an unlimited range applying.

Therefore all natural numbers (in ordinal terms) have an unlimited range of potential meanings (depending on the size of the finite group to which they belong).

A little further reflection would clearly indicate that since cardinal and ordinal meaning are thereby in dynamic terms interconnected, that indirectly the cardinal number system is also of a merely relative nature!

In other words, in dynamic terms, all numbers have two complementary aspects that are - relatively - independent and also - relatively - interdependent with each other.

The static notion of numbers as abstract entities is therefore just an illusion (based on a reduced - merely quantitative – interpretation).

In my own dynamic treatment of number, I am therefore at pains to demonstrate that the number system is composed of two interacting aspects - which I refer to as Type 1 and Type 2 respectively.

The Type 1 is directly associated with the quantitative aspect (though indirectly through interdependence equally possessing a qualitative aspect).

The Type 2 is then directly associated with the qualitative aspect (though again through interdependence) indirectly possessing a quantitative aspect.

Then I refer to the full combined interaction of both aspects (Type 1 and Type 2) as Type 3.

Basically the Type 1 aspect treats numbers as homogeneous collective units in quantitative terms (thus allowing for no qualitative distinction as between individual units). So 3 as a cardinal number from this perspective = 1 + 1 + 1 (in quantitative terms).

From the Type 1 cardinal aspect the prime numbers represent the basic building blocks of the natural number system. All natural numbers (except 1) from this perspective represent a unique combination of prime factors.

The Type 2 aspect by contrast is based the notion of a number group as composed of unique individual members (in qualitative terms). Thus from this perspective 3 relates to its distinctive 1st, 2nd and 3rd individual members as ordinally defined.

From the Type 2 aspect the natural numbers represent the basic building blocks of each prime number (in ordinal terms). So every prime number (p) from an ordinal perspective is composed of a unique set of natural numbers 1st, 2nd, 3rd,...pth!

Then using these distinctions, I go on to define both Zeta 1 and Zeta 2 Functions with respect to uncovering the true nature of the Riemann Hypothesis. This relates to the ultimate identity of both the quantitative and qualitative aspects of number.

From this enlarged perspective we now have two complementary sets of (non-trivial) zeta zeros.

Indeed perhaps the best explanation of these incredibly significant sets of numbers can be given as follows.

The Type 1 zeta zeros indirectly represent the holistic aspect of the cardinal number system, whereas the Type 2 zeta zeros directly represent its corresponding holistic ordinal aspect.

Ultimately - which can be approximated in Type 3 terms where both systems simultaneously interact - both Type 1 and Type 2 zeros can be seen as fully complementary and indeed identical in an ineffable manner with the natural number system (in both cardinal and ordinal terms).

The very manner by which the quantitative and qualitative aspects of the number system interact is though the two-way interaction of the primes with the natural numbers (and the natural numbers with the primes), both of which are mediated through the two sets of zeta zeros respectively.

This leads to the remarkable conclusion that the hidden holistic activity of both sets of zeta zeros necessarily underlies all human experience in an innate fashion in the seemingly obvious identification of cardinal with ordinal meaning (i.e. 1st with 1, 2nd with 2, 3rd with 3 etc.)

Indeed even more remarkably this innate activity necessarily underlies all natural processes as the very means by which they acquire a phenomenal identity!

So what seems most accessible at a conscious level (appearing totally obvious) therefore remains least accessible in corresponding unconscious terms.

Mathematics is now in urgent need of addressing its great shadow i.e. the unrecognised holistic aspect of interpretation. As such understanding intimately affects all of the sciences - now greatly in need of a more integrated understanding - our very civilisation may now well depend on such holistic realisation occurring rapidly in the very near future.

## Thursday, March 28, 2013

### Two Complementary Zeta Functions (7)

We can also connect Euler's famous product formula to the Zeta 2 Function.

Indeed this can provide a new perspective on this formula (which I have not seen developed elsewhere).

For example when s = 2 in the Zeta 1 expression,

ζ1(2) = (1/1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + .......

= 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...

So the product terms on the right hand side involve the expression p^2/(p^2 - 1) where p ranges over all the prime numbers!

However this same expression can be shown to be intimately related to the Zeta 2 Function.

Now in illustrating this it will perhaps be easier initially to start with the case corresponding in the Zeta 1 where s = 1.

Here according to Euler's formula

ζ1(1) = 1 + 1/2 + 1/3 + 1/4 = 2/1 * 3/2 * 5/4 * 7/6 * ...

So the summation series on the LHS represents the well known harmonic series while the terms in the product series on the RHS conform to p/(p - 1) where again the value of p ranges over all the prime numbers! However because Euler's formula holds only for convergent series (and these diverge) we have an exception in this case. In fact in truth where s = 1, the R.H.S. conforms to p/(p + 1),

so that

ζ1(1) = 1 + 1/2 + 1/3 + 1/4 ~ 3/2 * 4/3 * 6/5 * 8/7 i.e. (1 + 1/2) * (1 + 1/3) * (1 + 1/5) * (1 + 1/7) *....

However to illustrate our procedures we will illustrate initially by proceeding as if the Euler formula is true for the case s = 1!

As we have already seen the value of the infinite Zeta 2 Function (where s = 1/2)i.e. ζ2(1/2)

= 1 + 1/2 + 1/4 + 1/8 +..... = 2

Therefore the 1st term in the harmonic series = ζ2(1/2) - 1

Likewise the 2nd term in the harmonic series = ζ2(1/3) - 1, the 3rd term ζ2(1/4) - 1, the 4th term ζ2(1/5) - 1 etc.

Now the corresponding terms in the related product formula can be derived in a consistently simple manner with respect to the Zeta 2 expression.

So in each case where s represents the reciprocal of a prime value, we simply divide the value for the Function (representing the sum of an infinite series of terms) by the corresponding first term of the series.

So when s = 1/2 we divide 2 (the sum of the series) by 1 (the 1st term) to obtain 2/1 i.e. the 1st term in the product series

Then when s = 1/3 we divide 3/2 (the sum for the series by a) to obtain 3/2.

Then when s = 1/5 we divide 5/4 (the sum for the series by 1) to obtain 5/4.

Finally to illustrate when s = 1/7 we divide 7/6 (the sum of the series by 1) to obtain 7/6 and so on.

Now because the 1st term is always 1, the process seems somewhat trivial (and as we have seen does not actually apply in this case).

However the procedure assumes much more relevance when we deal with the product values corresponding to higher integer values of s (with respect to the Zeta 1).

So once again

ζ1(2) = 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...

Now with respect to the Zeta 2 the additive terms on the LHS correspond now

to {ζ2(1/2) - 1}^2, {ζ2(1/3) - 1}^2, {ζ2(1/4) - 1}^2,{ζ2(1/5) - 1}^2 and so on!

Now this time because each terms is raised to the power of 2, to obtain the corresponding terms in the product formula we divide the sum of the Function (for each value in question) by the corresponding sum of the first two terms in the Function.

So once again where s = 1/2 the value of the Function = 2.

The sum of the corresponding 1st two terms of the Function = 1 + 1/2 = 3/2.

Then when we divide 2 by 3/2 (= 2 * 2/3) we get 4/3 i.e. the 1st term in the product formula!

The other terms in the product formula can be obtained in like manner

So for example when s = 1/3 the sum of the first two terms = 1 + 1/3 = 4/3.

When we divide the sum of the Function 3/2 by 4/3 we obtain 3/2 * 3/4 = 9/8 i.e. the 2nd value in the product formula.

The next relevant value (representing the reciprocal of a prime number) = 1/5.

Thus sum of the first two terms = 1 + 1/5 = 6/5 and the sum of the Function 5/4.

Then dividing 5/4 by 6/5 we get 5/4 * 5/6 = 25/24 i.e. the 3rd term in the product formula.

Finally to illustrate, the next relevant value for the Zeta 2 Function is 1/7

The value of the Function = 7/6 and the sum of first two terms = 1 + 1/7 = 8/7.

So 7/6 divided by 8/7 = 7/6 * 7/8 = 49/48 i.e. the next value in the product formula.

Now this process can be extended indefinitely for higher values of s.

For example when s = 3 (with respect to the Zeta 1 Function), the corresponding terms in additive series of terms in terms of the Zeta 2 will be

ζ2(1/2) - 1}^3, {ζ2(1/3) - 1}^3, {ζ2(1/4) - 1}^3,{ζ2(1/5) - 1}^3 and so on!

Then to get the corresponding terms in the product formula we divide the value of the Zeta 2 Function (ranging again over the reciprocals of all prime numbered values)by the sum of the first 3 terms of the Function in each case.

In this way both the additive and product parts of the Euler Formula can be intimately demonstrated in terms of the Zeta 2 Function (thus indicating its complementary nature to the Zeta 1).

In fact in many ways a more natural fit exists (especially for the multiplication part) in terms of the Zeta 2 expression.

Better notational representation and further illustrations of approach can be found under Section 2 at Two Zeta Functions.

Indeed this can provide a new perspective on this formula (which I have not seen developed elsewhere).

For example when s = 2 in the Zeta 1 expression,

ζ1(2) = (1/1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + .......

= 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...

So the product terms on the right hand side involve the expression p^2/(p^2 - 1) where p ranges over all the prime numbers!

However this same expression can be shown to be intimately related to the Zeta 2 Function.

Now in illustrating this it will perhaps be easier initially to start with the case corresponding in the Zeta 1 where s = 1.

Here according to Euler's formula

ζ1(1) = 1 + 1/2 + 1/3 + 1/4 = 2/1 * 3/2 * 5/4 * 7/6 * ...

So the summation series on the LHS represents the well known harmonic series while the terms in the product series on the RHS conform to p/(p - 1) where again the value of p ranges over all the prime numbers! However because Euler's formula holds only for convergent series (and these diverge) we have an exception in this case. In fact in truth where s = 1, the R.H.S. conforms to p/(p + 1),

so that

ζ1(1) = 1 + 1/2 + 1/3 + 1/4 ~ 3/2 * 4/3 * 6/5 * 8/7 i.e. (1 + 1/2) * (1 + 1/3) * (1 + 1/5) * (1 + 1/7) *....

However to illustrate our procedures we will illustrate initially by proceeding as if the Euler formula is true for the case s = 1!

As we have already seen the value of the infinite Zeta 2 Function (where s = 1/2)i.e. ζ2(1/2)

= 1 + 1/2 + 1/4 + 1/8 +..... = 2

Therefore the 1st term in the harmonic series = ζ2(1/2) - 1

Likewise the 2nd term in the harmonic series = ζ2(1/3) - 1, the 3rd term ζ2(1/4) - 1, the 4th term ζ2(1/5) - 1 etc.

Now the corresponding terms in the related product formula can be derived in a consistently simple manner with respect to the Zeta 2 expression.

So in each case where s represents the reciprocal of a prime value, we simply divide the value for the Function (representing the sum of an infinite series of terms) by the corresponding first term of the series.

So when s = 1/2 we divide 2 (the sum of the series) by 1 (the 1st term) to obtain 2/1 i.e. the 1st term in the product series

Then when s = 1/3 we divide 3/2 (the sum for the series by a) to obtain 3/2.

Then when s = 1/5 we divide 5/4 (the sum for the series by 1) to obtain 5/4.

Finally to illustrate when s = 1/7 we divide 7/6 (the sum of the series by 1) to obtain 7/6 and so on.

Now because the 1st term is always 1, the process seems somewhat trivial (and as we have seen does not actually apply in this case).

However the procedure assumes much more relevance when we deal with the product values corresponding to higher integer values of s (with respect to the Zeta 1).

So once again

ζ1(2) = 1 + 1/4 + 1/9 + 1/16 + .... = 4/3 * 9/8 * 25/24 * 49/48 *...

Now with respect to the Zeta 2 the additive terms on the LHS correspond now

to {ζ2(1/2) - 1}^2, {ζ2(1/3) - 1}^2, {ζ2(1/4) - 1}^2,{ζ2(1/5) - 1}^2 and so on!

Now this time because each terms is raised to the power of 2, to obtain the corresponding terms in the product formula we divide the sum of the Function (for each value in question) by the corresponding sum of the first two terms in the Function.

So once again where s = 1/2 the value of the Function = 2.

The sum of the corresponding 1st two terms of the Function = 1 + 1/2 = 3/2.

Then when we divide 2 by 3/2 (= 2 * 2/3) we get 4/3 i.e. the 1st term in the product formula!

The other terms in the product formula can be obtained in like manner

So for example when s = 1/3 the sum of the first two terms = 1 + 1/3 = 4/3.

When we divide the sum of the Function 3/2 by 4/3 we obtain 3/2 * 3/4 = 9/8 i.e. the 2nd value in the product formula.

The next relevant value (representing the reciprocal of a prime number) = 1/5.

Thus sum of the first two terms = 1 + 1/5 = 6/5 and the sum of the Function 5/4.

Then dividing 5/4 by 6/5 we get 5/4 * 5/6 = 25/24 i.e. the 3rd term in the product formula.

Finally to illustrate, the next relevant value for the Zeta 2 Function is 1/7

The value of the Function = 7/6 and the sum of first two terms = 1 + 1/7 = 8/7.

So 7/6 divided by 8/7 = 7/6 * 7/8 = 49/48 i.e. the next value in the product formula.

Now this process can be extended indefinitely for higher values of s.

For example when s = 3 (with respect to the Zeta 1 Function), the corresponding terms in additive series of terms in terms of the Zeta 2 will be

ζ2(1/2) - 1}^3, {ζ2(1/3) - 1}^3, {ζ2(1/4) - 1}^3,{ζ2(1/5) - 1}^3 and so on!

Then to get the corresponding terms in the product formula we divide the value of the Zeta 2 Function (ranging again over the reciprocals of all prime numbered values)by the sum of the first 3 terms of the Function in each case.

In this way both the additive and product parts of the Euler Formula can be intimately demonstrated in terms of the Zeta 2 Function (thus indicating its complementary nature to the Zeta 1).

In fact in many ways a more natural fit exists (especially for the multiplication part) in terms of the Zeta 2 expression.

Better notational representation and further illustrations of approach can be found under Section 2 at Two Zeta Functions.

## Wednesday, March 27, 2013

### Two Complementary Zeta Functions (6)

Yesterday I illustrated just one important example of the intimate relationship exists as between the Zeta 1 and Zeta 2 Functions.

Once again - though this web-page does not ideally lend itself to sophisticated mathematical notation - I denote the Zeta 1 as ζ1(s) and the Zeta 2 as ζ2(s)respectively.

We saw then that that ζ1(1)i.e. the harmonic series = 1 + 1/2 + 1/3 + 1/4 + .... can be expressed in terms of a sum of Zeta 2 series where the value of s (for the Zeta 2) ranges over the reciprocals of all the natural numbers (except 1).

So the complementarity here relates to the fact that the value of s in the Zeta 1 (= 1) is related to all other natural numbers (except 1) with respect to the Zeta 2.

Thus ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, ....

Now a fascinating reverse form of complementarity also exists as between the Zeta 2 and the Zeta 1 Functions.

When s = 1

ζ2(1) = 1 + 1 + 1 + 1 +.....

Therefore for a finite range of values (i.e. for s = 1, up to n)

ζ2(1) = n

Now with respect to the corresponding Zeta 1 Function

ζ1(2) = (π^2)/6 = 1.6449340668...

ζ1(3) = 1.2020569032...

ζ1(4) = (π^4)/90 = 1.0823232337...

ζ1(5) = 1.0369277551...

ζ1(6) = (π^6)/945 = 1.0173430619...

ζ1(7) = 1.0083492774...

ζ1(8) = (π^8)/9450 = 1.0040773561...

ζ1(9) = 1.0020083928...

ζ1(10)= (π^10)/93555 = 1.0009945751...

and so on

Now if we subtract 1 from each value of the Zeta 1 and sum up the remaining values the total sum (for all natural number values of s > 1) = 1.

Indeed, when we sum up the first 10 values (with again 1 subtracted from each value) the total already converges very closely on 1 i.e. .9990146221...

Therefore when we obtain the total for all these Zeta 1 Functions over the finite range (this time from s = 2 up to n)

the answer converges on (n - 1) + 1 = n

So just as we have shown that,

ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, .... ,

Now in like reverse manner where the value of s with respect to Zeta 2 can range from 2 to n (with no finite upper limit on n)

ζ2(1) ~ ∑{ζ1(s)

Now in the former expression we subtracted 1 from the sum of each of the Zeta 2 values.

We do not do the same in the case of each Zeta 1 value as there is a lagged nature between both Functions differing by 1.

For example in the case where s = 0 with respect the the Zeta 1,

ζ1(0) = 1 + 1 + 1 + 1 +...

However this corresponds exactly with the Zeta 2 where now s = 1!

i.e. ζ2(1) = 1 + 1 + 1 + 1 +...

In fact other fascinating aspects with respect to Zeta 1 values can be briefly illustrated here!

Again when we subtract 1 from the sum of the Zeta 1 Function (ranging over the natural numbers from 2 upwards),

the sum of even numbered values converges to .75!

Again using the values listed above for the first 5 even values (up to 10), the relevant sum is

.6449340668 + ..08232332337 +.01734306198 + .00407735619 + .0009945751 = .74967229717

So the answer here has already converged very closely on .75!

This implies that the sum of the odd numbered values (excluding s = 1) converges on .25!

It equally implies that the alternating series (where each even is balanced by its succeeding odd term)

i.e. ζ1(2) - ζ1(3) + ζ1(4) - ζ1(5) +..... = .5

It is well known that with respect to the Zeta 1, for the sum of the harmonic series,

ζ1(1)~ log n + γ (where γ is the Euler-Mascheroni Constant = .5772156649..)

Fascinatingly γ is directly connected with the Zeta 1 Function (for all natural number values starting with 2) in the following manner

i.e. γ = ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....

Therefore

ζ1(1)~ log n + ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....

So,

log n ~ ζ1(1) - ζ1(2)/2 + ζ1(3)/3 - ζ1(4)/4 + ζ1(5)/5 - .... where the Zeta Functions are summed over a finite range of terms.

Alternatively,

γ ~ ζ1(1) – {ζ1(2)/2 + ζ(13)/3 + ζ1(4)/4 + ζ1(5)/5 +…..} again when summed to a finite n with approximation improving as n increases.

Thus,

log n ~ ζ1(2)/2 + ζ1(3)/3 + ζ1(4)/4 + ζ1(5)/5 +….. when summed to a finite n with approximation improving as n improves.

Once again - though this web-page does not ideally lend itself to sophisticated mathematical notation - I denote the Zeta 1 as ζ1(s) and the Zeta 2 as ζ2(s)respectively.

We saw then that that ζ1(1)i.e. the harmonic series = 1 + 1/2 + 1/3 + 1/4 + .... can be expressed in terms of a sum of Zeta 2 series where the value of s (for the Zeta 2) ranges over the reciprocals of all the natural numbers (except 1).

So the complementarity here relates to the fact that the value of s in the Zeta 1 (= 1) is related to all other natural numbers (except 1) with respect to the Zeta 2.

Thus ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, ....

Now a fascinating reverse form of complementarity also exists as between the Zeta 2 and the Zeta 1 Functions.

When s = 1

ζ2(1) = 1 + 1 + 1 + 1 +.....

Therefore for a finite range of values (i.e. for s = 1, up to n)

ζ2(1) = n

Now with respect to the corresponding Zeta 1 Function

ζ1(2) = (π^2)/6 = 1.6449340668...

ζ1(3) = 1.2020569032...

ζ1(4) = (π^4)/90 = 1.0823232337...

ζ1(5) = 1.0369277551...

ζ1(6) = (π^6)/945 = 1.0173430619...

ζ1(7) = 1.0083492774...

ζ1(8) = (π^8)/9450 = 1.0040773561...

ζ1(9) = 1.0020083928...

ζ1(10)= (π^10)/93555 = 1.0009945751...

and so on

Now if we subtract 1 from each value of the Zeta 1 and sum up the remaining values the total sum (for all natural number values of s > 1) = 1.

Indeed, when we sum up the first 10 values (with again 1 subtracted from each value) the total already converges very closely on 1 i.e. .9990146221...

Therefore when we obtain the total for all these Zeta 1 Functions over the finite range (this time from s = 2 up to n)

the answer converges on (n - 1) + 1 = n

So just as we have shown that,

ζ1(1) = ∑{ζ2(s)- 1} where s represents the reciprocal of 2, 3, 4, .... ,

Now in like reverse manner where the value of s with respect to Zeta 2 can range from 2 to n (with no finite upper limit on n)

ζ2(1) ~ ∑{ζ1(s)

Now in the former expression we subtracted 1 from the sum of each of the Zeta 2 values.

We do not do the same in the case of each Zeta 1 value as there is a lagged nature between both Functions differing by 1.

For example in the case where s = 0 with respect the the Zeta 1,

ζ1(0) = 1 + 1 + 1 + 1 +...

However this corresponds exactly with the Zeta 2 where now s = 1!

i.e. ζ2(1) = 1 + 1 + 1 + 1 +...

In fact other fascinating aspects with respect to Zeta 1 values can be briefly illustrated here!

Again when we subtract 1 from the sum of the Zeta 1 Function (ranging over the natural numbers from 2 upwards),

the sum of even numbered values converges to .75!

Again using the values listed above for the first 5 even values (up to 10), the relevant sum is

.6449340668 + ..08232332337 +.01734306198 + .00407735619 + .0009945751 = .74967229717

So the answer here has already converged very closely on .75!

This implies that the sum of the odd numbered values (excluding s = 1) converges on .25!

It equally implies that the alternating series (where each even is balanced by its succeeding odd term)

i.e. ζ1(2) - ζ1(3) + ζ1(4) - ζ1(5) +..... = .5

It is well known that with respect to the Zeta 1, for the sum of the harmonic series,

ζ1(1)~ log n + γ (where γ is the Euler-Mascheroni Constant = .5772156649..)

Fascinatingly γ is directly connected with the Zeta 1 Function (for all natural number values starting with 2) in the following manner

i.e. γ = ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....

Therefore

ζ1(1)~ log n + ζ1(2)/2 - ζ1(3)/3 + ζ1(4)/4 - ζ1(5)/5 + ....

So,

log n ~ ζ1(1) - ζ1(2)/2 + ζ1(3)/3 - ζ1(4)/4 + ζ1(5)/5 - .... where the Zeta Functions are summed over a finite range of terms.

Alternatively,

γ ~ ζ1(1) – {ζ1(2)/2 + ζ(13)/3 + ζ1(4)/4 + ζ1(5)/5 +…..} again when summed to a finite n with approximation improving as n increases.

Thus,

log n ~ ζ1(2)/2 + ζ1(3)/3 + ζ1(4)/4 + ζ1(5)/5 +….. when summed to a finite n with approximation improving as n improves.

## Tuesday, March 26, 2013

### Two Complementary Zeta Functions (5)

Once again it is important to bear in mind that I define two complementary Zeta Functions.

The first relates to the recognised Riemann Zeta Function (Zeta 1) which is defined as an infinite series:

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….

Now if with respect to a number raised to a power we refer to the initial number as the base quantity and the exponent or power as the dimensional number,

for the Zeta 1 Function the natural numbers serve as the base quantities and s as the dimensional power - negative in this case - to which the base quantities are raised.

The Zeta 2 Function - the role of which is largely unrecognised - is defined as a finite series:

1 + s^1 + s^2 + ….+ s^(n – 1).

Here the role of base quantities and dimensional powers is reversed with s now serving as the base quantity which is defined with respect to the natural numbers (from 1 to n) where n has no finite limit.

In a qualified sense, the Zeta 2 can be defined in infinite terms as:

1 + s^1 + s^2 + s^3 + s^4 + ...

It is fascinating to probe the close connections between both Functions.

As is well known with respect to the recognised Riemann Zeta Function (i.e. Zeta 1). (I am inserting 1 and 2 after the ζ sign to distinguish the Zeta 1 and Zeta 2 expressions respectively).

ζ1(1) = 1 + 1/2 + 1/3 + 1/4 + .......

Now each of these terms can be directly related to the Zeta 2 Function.

In terms of this latter Function (using the infinite expression),

ζ2(s) = 1 + s^1 + s^2 + s^3 + s^4 + ...

Now when s = 1/2 with respect to this series,

ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ... = 2

Therefore, s^1 + s^2 + s^3 + s^4 + ... = ζ2(s)- 1 = 1

Then when s = 1/3

ζ2(1/3) = 1 + 1/3 + 1/9 + 1/27 +... = 3/2

Therefore, s^1 + s^2 + s^3 + s^4 + ... = 3/2 - 1 = 1/2

In general s^1 + s^2 + s^3 + s^4 + ... = 1(1 - s)

So ζ2(1/4) = 1/3, ζ2(1/5)= 1/4 and so on!

Again,

ζ(1)1 = 1 + 1/2 + 1/3 + 1/4 + .......

= ∑(s^1 + s^2 + s^3 + s^4 + ... ) where the value of s ranges over the reciprocals of all natural numbers except 1).

and s^1 + s^2 + s^3 + s^4 + ... = ζ2(s) - 1.

Now with respect to the Zeta 1,

ζ1(k) = (1)^k + (1/2)^k + (1/3)^k + (1/4)^k + ...

So for example when k = 2.

ζ1(2) = (1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ...

= 1 + 1/4 + 1/9 + 1/16 + ....

Therefore in more general terms.

ζ1(k) = ∑(s^1k + s^2k + s^3k + s^4k + ... ),

where s^1k + s^2k + s^3k + s^4k + ... = {ζ2(s)^k} - 1

This relationship holds where the Zeta 1 ranges over real positive dimensional values (≥ 1), and the Zeta 2 over real fractional values for s < 1/2). So each term in the Zeta 1, is directly related to an entire Zeta 2 expression. I have mentioned before how each cardinal number is related (ordinally) to its natural number members. So we have an equivalent relationship here where the Zeta 1 expression is composed of a number of Zeta 2 members. Because of notational difficulties when writing on a web-page, I have produced some accompanying material in text! See Two Zeta Functions (Section 1).

The first relates to the recognised Riemann Zeta Function (Zeta 1) which is defined as an infinite series:

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….

Now if with respect to a number raised to a power we refer to the initial number as the base quantity and the exponent or power as the dimensional number,

for the Zeta 1 Function the natural numbers serve as the base quantities and s as the dimensional power - negative in this case - to which the base quantities are raised.

The Zeta 2 Function - the role of which is largely unrecognised - is defined as a finite series:

1 + s^1 + s^2 + ….+ s^(n – 1).

Here the role of base quantities and dimensional powers is reversed with s now serving as the base quantity which is defined with respect to the natural numbers (from 1 to n) where n has no finite limit.

In a qualified sense, the Zeta 2 can be defined in infinite terms as:

1 + s^1 + s^2 + s^3 + s^4 + ...

It is fascinating to probe the close connections between both Functions.

As is well known with respect to the recognised Riemann Zeta Function (i.e. Zeta 1). (I am inserting 1 and 2 after the ζ sign to distinguish the Zeta 1 and Zeta 2 expressions respectively).

ζ1(1) = 1 + 1/2 + 1/3 + 1/4 + .......

Now each of these terms can be directly related to the Zeta 2 Function.

In terms of this latter Function (using the infinite expression),

ζ2(s) = 1 + s^1 + s^2 + s^3 + s^4 + ...

Now when s = 1/2 with respect to this series,

ζ2(1/2) = 1 + 1/2 + 1/4 + 1/8 + ... = 2

Therefore, s^1 + s^2 + s^3 + s^4 + ... = ζ2(s)- 1 = 1

Then when s = 1/3

ζ2(1/3) = 1 + 1/3 + 1/9 + 1/27 +... = 3/2

Therefore, s^1 + s^2 + s^3 + s^4 + ... = 3/2 - 1 = 1/2

In general s^1 + s^2 + s^3 + s^4 + ... = 1(1 - s)

So ζ2(1/4) = 1/3, ζ2(1/5)= 1/4 and so on!

Again,

ζ(1)1 = 1 + 1/2 + 1/3 + 1/4 + .......

= ∑(s^1 + s^2 + s^3 + s^4 + ... ) where the value of s ranges over the reciprocals of all natural numbers except 1).

and s^1 + s^2 + s^3 + s^4 + ... = ζ2(s) - 1.

Now with respect to the Zeta 1,

ζ1(k) = (1)^k + (1/2)^k + (1/3)^k + (1/4)^k + ...

So for example when k = 2.

ζ1(2) = (1)^2 + (1/2)^2 + (1/3)^2 + (1/4)^2 + ...

= 1 + 1/4 + 1/9 + 1/16 + ....

Therefore in more general terms.

ζ1(k) = ∑(s^1k + s^2k + s^3k + s^4k + ... ),

where s^1k + s^2k + s^3k + s^4k + ... = {ζ2(s)^k} - 1

This relationship holds where the Zeta 1 ranges over real positive dimensional values (≥ 1), and the Zeta 2 over real fractional values for s < 1/2). So each term in the Zeta 1, is directly related to an entire Zeta 2 expression. I have mentioned before how each cardinal number is related (ordinally) to its natural number members. So we have an equivalent relationship here where the Zeta 1 expression is composed of a number of Zeta 2 members. Because of notational difficulties when writing on a web-page, I have produced some accompanying material in text! See Two Zeta Functions (Section 1).

## Saturday, March 23, 2013

### The Shadow of Mathematics

I have already suggested that one revealing way of appreciating the nature of the non-trivial zeros (both for the Zeta 1 and Zeta 2 Functions) is as the shadow to the conventional natural system of number (in both cardinal and ordinal terms).

In Jungian psychological terms this would thereby imply that these two sets of non-trivial zeros represent the unconscious complement of what we conventionally understand as number (in conscious terms).

Likewise in physical terms it equally implies that the two sets of non-trivial zeros represent the holistic complement of what is conventionally interpreted in analytic terms with respect to the natural world.

Again from a Jungian perspective, true psychological integration requires the combined integration of both the (revealed) conscious and (hidden) unconscious aspects of personality.

Complementarity exists in the relationship between the natural number system (cardinal and ordinal) and the two sets of non-trivial zeros.

So from the conscious extreme the natural counting numbers seem fully accessible to understanding (even from a very early age).

However from the corresponding unconscious extreme, the two sets of non-trivial zeros – certainly in terms of what they inherently represent – seem almost entirely inaccessible to conventional understanding (at any age of development).

So what appears as completely obvious in terms of (revealed) conscious appears completely inaccessible (in terms of (hidden) unconscious behaviour. So identification with the conscious thereby blots out entire recognition with respect to the opposite unconscious extreme (relating to the hidden nature of the number system).

This thereby entails that ultimately true psychological integration requires appropriating this enormous shadow with respect to the nature of both sets of non-trivial zeros, so that eventually they can become fully accessible in intuitive terms to understanding.

The deeper significance is that Conventional Mathematics thereby is greatly lacking a true integral dimension with respect to overall understanding.

Once again this is due to a total failure in formal terms to recognise its hidden unconscious dimension.

Alternatively it represents the complete domination (in formal terms) of quantitative over qualitative type interpretation of mathematical symbols.

Expressed in yet another way it represents sole recognition of the analytic dimension of mathematical interpretation through the exclusion of its equally important holistic aspect.

Yet again it entails the remarkable fact that Mathematics as it currently stands is entirely lacking in any genuine notion of the nature of interdependent type relationships.

And the saddest part – though his is easily appreciated in Jungian terms – is that because of its extreme one-sided emphasis, it thereby is greatly lacking any insight into the nature of its own considerable shadow.

Thus what are understood as rigorous procedures (within its limited terms of reference) can equally be seen from an outside perspective as representing highly reduced understanding at every turn.

And by definition when one accepts such reductionism – as a recognised member of the mathematical profession - without question, one thereby becomes unable to tolerate or even appreciate any outside criticism of its inherent nature.

Though the Riemann Hypothesis is not necessary to identify the fundamental problem with present mathematical interpretation, it does however – when appropriately interpreted – provide a wonderful illustration of the precise nature of this problem.

So let me once more state it simply! Properly understood Mathematics is inherently dynamic in nature representing the two-way relative interaction of aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.

However Conventional Mathematics is based on the special case where such interaction is effectively completely ignored. So the qualitative – though of equal importance - is thereby reduced to the quantitative.

Mathematics is then misleadingly viewed as relating merely to quantitative relationships in absolute terms.

The Riemann Hypothesis relates intrinsically to the fundamental nature of the number system and cannot thereby be properly understood – not alone solved – in conventional mathematical terms.

It thus reduces and thereby gravely distorts its inherent dynamic nature as the interaction of twin elements that are quantitative and qualitative (and equally physical and psychological) with respect to each other.

So number inherently represents the fundamental encoding of all physical and psychological phenomena in quantitative and qualitative terms.

Conventional Mathematics is thereby gravely in need of addressing its quite enormous shadow.

In physical terms this requires the recognition that all mathematical processes entail a qualitative holistic dimension that cannot be reduced in mere quantitative terms.

In complementary psychological terms it implies that that the understanding of such processes entails an unconscious (intuitive) aspect that cannot be reduced in a rational (linear) manner!

As I have repeatedly stated, because of the very nature of dynamic interaction, all interpretation that is directly analytic (i.e. quantitative) has a corresponding shadow – initially unrecognised - that is indirectly holistic and qualitative in nature.

Likewise all interpretation that is directly holistic (i.e. qualitative) has a corresponding unrecognised shadow that is indirectly of an analytic quantitative nature.

When we initially look at the number system it appears directly in a linear rational manner revealing its quantitative aspect. So here we literally view all numbers in quantitative terms as lying on a straight line!

However this number system has a corresponding shadow that is indirectly holistic and qualitative in nature.

Bringing this aspect to light depends much more on intuitive (unconscious) development rather than on linear rational understanding.

In fact indirectly it can be then given a paradoxical rational interpretation in a circular manner.

I have frequently over the past year or so in my blogs drawn direct attention to the nature of this holistic aspect identifying firstly the Type 2 aspect (of the number system) and then the corresponding Zeta 2 Function.

Not surprisingly, as Conventional Mathematics totally lacks in formal terms a holistic dimension, no recognition exists as yet of the enormous importance of what – I refer to as – as the Zeta 2 non-trivial zeros. These in fact are fully complementary with the Type 1 zeros.

The Type 2 non-trivial zeros essentially relate to the holistic nature of numbers (as dimensions).

As understanding becomes increasingly dynamic the nature of higher dimensions (the structure of which is related to the corresponding roots of the number in question) unfold in understanding.

Ultimately however as holistic appreciation deepens, remaining rigid elements of a linear rational level gradually are completely dissolved.

In psycho-spiritual terms this is identified with the continual deepening of a contemplative (intuitive) state that ultimately transcends all analytic interpretation (of a rigid linear kind). Increasingly however, dynamic rational structures of a circular (paradoxical) kind are required to mediate such understanding.

This represents the unconscious – as opposed to the conscious – extreme of interpretation.

So we start in development with the understanding of number with the direct linear rational understanding of number (i.e. its quantitative aspect).

Then when development proceeds to higher stages we gradually can uncover the shadow of this aspect of number (i.e. its initially unrecognised holistic qualitative dimension).

The full unfolding of this alternative holistic aspect leads to an increasingly dynamic appreciating of number that is directly intuitive but indirectly is conveyed in a refined circular rational fashion.

This understanding ultimately culminates in a spiritually contemplative appreciation of the transcendent nature of reality (i.e. the purely holistic appreciation of the infinite nature of the number system).

And the Type 2 zeros relate directly to this circular holistic aspect!

However the holistic also has a shadow aspect (in its unrecognised analytic aspect).

What this entails from a psycho spiritual perspective is that the pure holistic appreciation of the infinite nature of reality (as transcending all finite notions) must now be made immanent in all finite phenomena.

This means in effect that one now gradually learns to integrate the new qualitative aspect of appreciation with all scientific type phenomena (formerly understood in mere quantitative terms).

Let me briefly illustrate. From a conventional scientific perspective (i.e. quantitative) the recognition of object phenomena e.g. roses is of a reduced collective nature where we can impersonally classify all individual members as belonging to the same general class of rose!

Now the corresponding qualitative recognition requires that each rose be understood uniquely with respect to its distinctive attributes.

Remarkably it is the same with the number system.

The quantitative recognition of number is impersonal in nature where the qualitative distinction as between numbers is not taken into account.

Indeed the failure to appreciate the true nature of multiplication in Conventional Mathematics relates to this lack of qualitative distinction.

So 2^2 (represents a number expressed with respect to the second dimension). However the quantitative result = 4 (i.e. 4^1) represents a number expressed with respect to the (default) 1st dimension.

Thus the qualitative change in the nature of the number (which always results through multiplication) is thereby ignored. In other words the qualitative aspect of interpretation is reduced to the quantitative!

The very point about the Riemann Hypothesis is that it relates to this key problem of reconciling the quantitative with the qualitative nature of number.

The non-trivial zeros (i.e. Type 1) therefore relate to the holistic aspect of the number system now interpreted in an immanent manner, whereby the qualitative aspect can be properly incorporated with each finite member of the number system.

Put another way just as the conventional linear number system represents the interpretation of number with respect to its quantitative aspect, the non-trivial zeros (Type 1) represent the corresponding interpretation with respect to the qualitative aspect.

Now the conventional system (esp. primes and natural nos.) are understood as points on a real line. In qualitative terms “real” is synonymous with rational interpretation of a quantitative kind!

From a holistic perspective, “imaginary” relates to the indirect rational way of conveying meaning that is of an unconscious intuitive nature. So the recognition of “qualities” as associated with consciously recognised objects, always comes directly from the unconscious mind (in its interaction with the conscious).

In this sense such qualitative appreciation is of an imaginary nature. So whereas the quantitative number system is of a real, the corresponding qualitative appreciation of its nature is imaginary! So therefore the (Type 1) non-trivial zeros lie on a straight line that is imaginary.

If the unconscious remains undeveloped we thereby cannot properly identify true quality in any context!

Unfortunately this is a damning criticism that can be made of Conventional Mathematics (which fails to recognise its qualitative dimension)!

This therefore poses considerable difficulties in appreciating the nature of the non-trivial zeros (Type 1) which relate to the qualitative aspect of number (with respect to their finite nature). By contrast, the Type 2 non-trivial zeros relate to the qualitative aspect of number (with respect to their infinite nature).

Now I have already stated that all phenomena fundamentally represent the original dynamic interaction of the number system with respect to its quantitative and qualitative characteristics.

What is truly remarkable is that though remaining inaccessible to conventional understanding, the non-trivial zeros (Type 1 and Type 2) must necessarily implicitly exist in the unconscious mind of all human beings as the very means through which we are enabled to make qualitative distinctions with respect to objects.

What is even more remarkable is that these same zeros must innately exist with respect to all natural phenomena providing their very capacity to become manifest in nature (exhibiting both quantitative and qualitative characteristics).

In an important sense one key goal of evolution requires fully uncovering in understanding these hidden – though currently largely inaccessible - original aspects of our number system!

Indeed the very nature of both types of zeros is pointing clearly to the present inadequate nature of Conventional Mathematics.

Mathematics is not strictly about the quantitative but rather the dynamic interaction as between its quantitative and qualitative aspects. Therefore to properly understand this interaction we must face its persistent shadow in a continual failure to recognise the qualitative dimension.

In Jungian psychological terms this would thereby imply that these two sets of non-trivial zeros represent the unconscious complement of what we conventionally understand as number (in conscious terms).

Likewise in physical terms it equally implies that the two sets of non-trivial zeros represent the holistic complement of what is conventionally interpreted in analytic terms with respect to the natural world.

Again from a Jungian perspective, true psychological integration requires the combined integration of both the (revealed) conscious and (hidden) unconscious aspects of personality.

Complementarity exists in the relationship between the natural number system (cardinal and ordinal) and the two sets of non-trivial zeros.

So from the conscious extreme the natural counting numbers seem fully accessible to understanding (even from a very early age).

However from the corresponding unconscious extreme, the two sets of non-trivial zeros – certainly in terms of what they inherently represent – seem almost entirely inaccessible to conventional understanding (at any age of development).

So what appears as completely obvious in terms of (revealed) conscious appears completely inaccessible (in terms of (hidden) unconscious behaviour. So identification with the conscious thereby blots out entire recognition with respect to the opposite unconscious extreme (relating to the hidden nature of the number system).

This thereby entails that ultimately true psychological integration requires appropriating this enormous shadow with respect to the nature of both sets of non-trivial zeros, so that eventually they can become fully accessible in intuitive terms to understanding.

The deeper significance is that Conventional Mathematics thereby is greatly lacking a true integral dimension with respect to overall understanding.

Once again this is due to a total failure in formal terms to recognise its hidden unconscious dimension.

Alternatively it represents the complete domination (in formal terms) of quantitative over qualitative type interpretation of mathematical symbols.

Expressed in yet another way it represents sole recognition of the analytic dimension of mathematical interpretation through the exclusion of its equally important holistic aspect.

Yet again it entails the remarkable fact that Mathematics as it currently stands is entirely lacking in any genuine notion of the nature of interdependent type relationships.

And the saddest part – though his is easily appreciated in Jungian terms – is that because of its extreme one-sided emphasis, it thereby is greatly lacking any insight into the nature of its own considerable shadow.

Thus what are understood as rigorous procedures (within its limited terms of reference) can equally be seen from an outside perspective as representing highly reduced understanding at every turn.

And by definition when one accepts such reductionism – as a recognised member of the mathematical profession - without question, one thereby becomes unable to tolerate or even appreciate any outside criticism of its inherent nature.

Though the Riemann Hypothesis is not necessary to identify the fundamental problem with present mathematical interpretation, it does however – when appropriately interpreted – provide a wonderful illustration of the precise nature of this problem.

So let me once more state it simply! Properly understood Mathematics is inherently dynamic in nature representing the two-way relative interaction of aspects that are quantitative (analytic) and qualitative (holistic) with respect to each other.

However Conventional Mathematics is based on the special case where such interaction is effectively completely ignored. So the qualitative – though of equal importance - is thereby reduced to the quantitative.

Mathematics is then misleadingly viewed as relating merely to quantitative relationships in absolute terms.

The Riemann Hypothesis relates intrinsically to the fundamental nature of the number system and cannot thereby be properly understood – not alone solved – in conventional mathematical terms.

It thus reduces and thereby gravely distorts its inherent dynamic nature as the interaction of twin elements that are quantitative and qualitative (and equally physical and psychological) with respect to each other.

So number inherently represents the fundamental encoding of all physical and psychological phenomena in quantitative and qualitative terms.

Conventional Mathematics is thereby gravely in need of addressing its quite enormous shadow.

In physical terms this requires the recognition that all mathematical processes entail a qualitative holistic dimension that cannot be reduced in mere quantitative terms.

In complementary psychological terms it implies that that the understanding of such processes entails an unconscious (intuitive) aspect that cannot be reduced in a rational (linear) manner!

As I have repeatedly stated, because of the very nature of dynamic interaction, all interpretation that is directly analytic (i.e. quantitative) has a corresponding shadow – initially unrecognised - that is indirectly holistic and qualitative in nature.

Likewise all interpretation that is directly holistic (i.e. qualitative) has a corresponding unrecognised shadow that is indirectly of an analytic quantitative nature.

When we initially look at the number system it appears directly in a linear rational manner revealing its quantitative aspect. So here we literally view all numbers in quantitative terms as lying on a straight line!

However this number system has a corresponding shadow that is indirectly holistic and qualitative in nature.

Bringing this aspect to light depends much more on intuitive (unconscious) development rather than on linear rational understanding.

In fact indirectly it can be then given a paradoxical rational interpretation in a circular manner.

I have frequently over the past year or so in my blogs drawn direct attention to the nature of this holistic aspect identifying firstly the Type 2 aspect (of the number system) and then the corresponding Zeta 2 Function.

Not surprisingly, as Conventional Mathematics totally lacks in formal terms a holistic dimension, no recognition exists as yet of the enormous importance of what – I refer to as – as the Zeta 2 non-trivial zeros. These in fact are fully complementary with the Type 1 zeros.

The Type 2 non-trivial zeros essentially relate to the holistic nature of numbers (as dimensions).

As understanding becomes increasingly dynamic the nature of higher dimensions (the structure of which is related to the corresponding roots of the number in question) unfold in understanding.

Ultimately however as holistic appreciation deepens, remaining rigid elements of a linear rational level gradually are completely dissolved.

In psycho-spiritual terms this is identified with the continual deepening of a contemplative (intuitive) state that ultimately transcends all analytic interpretation (of a rigid linear kind). Increasingly however, dynamic rational structures of a circular (paradoxical) kind are required to mediate such understanding.

This represents the unconscious – as opposed to the conscious – extreme of interpretation.

So we start in development with the understanding of number with the direct linear rational understanding of number (i.e. its quantitative aspect).

Then when development proceeds to higher stages we gradually can uncover the shadow of this aspect of number (i.e. its initially unrecognised holistic qualitative dimension).

The full unfolding of this alternative holistic aspect leads to an increasingly dynamic appreciating of number that is directly intuitive but indirectly is conveyed in a refined circular rational fashion.

This understanding ultimately culminates in a spiritually contemplative appreciation of the transcendent nature of reality (i.e. the purely holistic appreciation of the infinite nature of the number system).

And the Type 2 zeros relate directly to this circular holistic aspect!

However the holistic also has a shadow aspect (in its unrecognised analytic aspect).

What this entails from a psycho spiritual perspective is that the pure holistic appreciation of the infinite nature of reality (as transcending all finite notions) must now be made immanent in all finite phenomena.

This means in effect that one now gradually learns to integrate the new qualitative aspect of appreciation with all scientific type phenomena (formerly understood in mere quantitative terms).

Let me briefly illustrate. From a conventional scientific perspective (i.e. quantitative) the recognition of object phenomena e.g. roses is of a reduced collective nature where we can impersonally classify all individual members as belonging to the same general class of rose!

Now the corresponding qualitative recognition requires that each rose be understood uniquely with respect to its distinctive attributes.

Remarkably it is the same with the number system.

The quantitative recognition of number is impersonal in nature where the qualitative distinction as between numbers is not taken into account.

Indeed the failure to appreciate the true nature of multiplication in Conventional Mathematics relates to this lack of qualitative distinction.

So 2^2 (represents a number expressed with respect to the second dimension). However the quantitative result = 4 (i.e. 4^1) represents a number expressed with respect to the (default) 1st dimension.

Thus the qualitative change in the nature of the number (which always results through multiplication) is thereby ignored. In other words the qualitative aspect of interpretation is reduced to the quantitative!

The very point about the Riemann Hypothesis is that it relates to this key problem of reconciling the quantitative with the qualitative nature of number.

The non-trivial zeros (i.e. Type 1) therefore relate to the holistic aspect of the number system now interpreted in an immanent manner, whereby the qualitative aspect can be properly incorporated with each finite member of the number system.

Put another way just as the conventional linear number system represents the interpretation of number with respect to its quantitative aspect, the non-trivial zeros (Type 1) represent the corresponding interpretation with respect to the qualitative aspect.

Now the conventional system (esp. primes and natural nos.) are understood as points on a real line. In qualitative terms “real” is synonymous with rational interpretation of a quantitative kind!

From a holistic perspective, “imaginary” relates to the indirect rational way of conveying meaning that is of an unconscious intuitive nature. So the recognition of “qualities” as associated with consciously recognised objects, always comes directly from the unconscious mind (in its interaction with the conscious).

In this sense such qualitative appreciation is of an imaginary nature. So whereas the quantitative number system is of a real, the corresponding qualitative appreciation of its nature is imaginary! So therefore the (Type 1) non-trivial zeros lie on a straight line that is imaginary.

If the unconscious remains undeveloped we thereby cannot properly identify true quality in any context!

Unfortunately this is a damning criticism that can be made of Conventional Mathematics (which fails to recognise its qualitative dimension)!

This therefore poses considerable difficulties in appreciating the nature of the non-trivial zeros (Type 1) which relate to the qualitative aspect of number (with respect to their finite nature). By contrast, the Type 2 non-trivial zeros relate to the qualitative aspect of number (with respect to their infinite nature).

Now I have already stated that all phenomena fundamentally represent the original dynamic interaction of the number system with respect to its quantitative and qualitative characteristics.

What is truly remarkable is that though remaining inaccessible to conventional understanding, the non-trivial zeros (Type 1 and Type 2) must necessarily implicitly exist in the unconscious mind of all human beings as the very means through which we are enabled to make qualitative distinctions with respect to objects.

What is even more remarkable is that these same zeros must innately exist with respect to all natural phenomena providing their very capacity to become manifest in nature (exhibiting both quantitative and qualitative characteristics).

In an important sense one key goal of evolution requires fully uncovering in understanding these hidden – though currently largely inaccessible - original aspects of our number system!

Indeed the very nature of both types of zeros is pointing clearly to the present inadequate nature of Conventional Mathematics.

Mathematics is not strictly about the quantitative but rather the dynamic interaction as between its quantitative and qualitative aspects. Therefore to properly understand this interaction we must face its persistent shadow in a continual failure to recognise the qualitative dimension.

## Wednesday, March 20, 2013

### Two Complementary Zeta Functions (4)

It is now recognised that the non-trivial zeros (i.e. the Type 1 zeros) in all probability have a spectral interpretation relating to the frequencies of some dynamic system of a physical nature.

Now this comes as no surprise to me at all, as I would have long suspected that this was necessarily the case. Indeed I would go considerably further in emphasising that this spectral interpretation relates to a dynamic system with twin physical and psychological aspects that are complementary.

However my own interest in the Riemann Hypothesis initially arose from a prolonged interest in the nature of the (unrecognised) non-trivial zeros (Type 2).

And in this context I had already made use of a spectral interpretation as just one illuminating pathway towards appreciating their true nature.

Now the electromagnetic spectrum is perhaps the best-known physical spectrum.

So natural light comprises just one narrow band on this overall spectrum. Below this band lies electromagnetic energy of higher frequency and shorter wavelength than light (e.g. x-rays and gamma rays). Above this band lie corresponding energies of lower frequency and longer wavelength (e.g. microwave and radio).

Because in dynamic terms of complementarity, a psychological aspect necessarily exists to such a spectrum with fascinating implications for the spiritual contemplative journey.

The term “light” of course is equally used in a spiritual as well as physical sense where it refers to the holistic aspect of intuition or illumination which necessarily interacts with (analytic) reason.

Just as natural light inhabits the middle band of the physical spectrum likewise the intuition that characterises conventional (linear) reason likewise inhabits the middle of the psychological spectrum.

However it has been long known that through authentic contemplative development that higher bands on this spectrum are attainable where the intuitive light becomes ever more refined and thereby capable of interacting with paradoxical (i.e. circular) rational structures.

Indeed ultimately these structures become so unrestricted and transparent that they no longer appear to possess any remaining rigid structure becoming fully integrated with the very light through which they are mediated.

Now associated with the deepest level of contemplation is a quality of peace and utter stillness. This illumination can therefore be readily characterised as (intuitive) light of exceptionally long wavelength (and corresponding low frequency).

By contrast in the spiritual literature the other extreme (of short wavelength and high frequency) is often related to periods of purgation where an incessant inner bombardment takes place exposing one’s hidden faults and failings!

One sobering point however to consider is this!

Just as energies outside the natural light band are not visible to the naked eye, likewise all these energies (and the understanding associated with them) cannot be appropriated through linear reason (informed by common intuition).

This therefore creates insuperable difficulties for Conventional Mathematics in trying to appropriate such understanding within its accepted axioms and definitions!

Now again in the spiritual esoteric traditions, these higher bands of understanding are often referred to as higher “dimensions”.

Now the key here is to recognise that number is used to represent dimensions in Mathematics.

So therefore associated with each number – now understood in a qualitative holistic sense – is a unique dynamic form of understanding where a more refined quality of intuition interacts with the ever more circular (paradoxical) use of reason!

I have often mentioned in this context the fact that Conventional Mathematics is based on linear reason i.e. literally in a 1-dimensional manner.

Therefore though the quantitative expression of other dimensional numbers (as powers or exponents) is indeed accommodated, this all takes place within a default qualitative interpretation (= 1).

And the very nature of such interpretation is that distinctive holistic meaning (in any relevant context) is necessarily reduced in a mere quantitative manner.

Using the language of string theory, we could say therefore that Conventional Mathematics represents the lowest possible energy state (with respect to its understanding) where entities such as numbers thereby assume a (misleading) absolute identity.

More accurately we could say that Conventional Mathematics is exclusively defined in 1-dimensional terms (where 1 is defined in a qualitative holistic manner).

However the remarkable truth that needs to be embraced is that mathematical relationships can be validly defined in an unlimited number of possible ways (where each number as dimension represents a unique overall manner of interpretation).

Now in all cases (except 1) these dimensional interpretations imply unique configurations with respect to the dynamic interaction of both the analytic (particle) and holistic (wave) aspects of mathematical understanding.

So in making this very point, I am illustrating how the well-known fact that the Zeta Function remains uniquely undefined on the complex plane for s = 1, can be given both a quantitative (analytic) and qualitative (holistic) interpretation.

If you can appreciate this then you can understand immediately why the Riemann Zeta Function (and its associated Riemann Hypothesis) cannot be properly interpreted from within the standpoint of Conventional Mathematics.

As – when correctly interpreted - it fundamental relates to the dynamic interaction of the quantitative (analytic) and qualitative (holistic) aspects of the number system, this remains uniquely undefined in linear (1-dimensional) terms where the qualitative aspect formally is not recognised.

And of course from this enlarged perspective, as the Riemann Hypothesis relates to the fundamental condition for the ultimate identity of both quantitative (analytic) and qualitative (holistic) aspects, it is futile trying to prove it through the axioms of Conventional Mathematics (which give no formal recognition to the qualitative aspect).

From what we have said it is but a short journey to appreciating that the number system itself represents the ultimate spectrum (in both physical and psychological terms).

So the vibrations of all dynamic systems (physical and psychological) can thereby be ultimately understood as the original vibrations of this number system with respect to both its quantitative and qualitative aspects.

Therefore all natural phenomena thereby represent but a secondary expression of an original system where number - when dynamically interpreted with respect to both analytic and holistic aspects – inherently exists as their deepest possible means of encoding.

Once again the recognised non-trivial zeros (of the Zeta 1 Function) are - when properly understood - fully complementary with the Zeta 2 zeros (which I have been explaining).

Whereas the Zeta 2 can be identified initially with each natural number except 1 (that constitutes each prime) in ordinal terms, The Zeta 1 inversely can be identified initially overall with the entire group of prime numbers (that comprise the natural number system except 1) in cardinal terms.

The solutions from both perspectives establish the identity of the quantitative and qualitative aspects of the number system. Ultimately through simultaneous appreciation of Zeta 1 and Zeta 2, we can perfectly see (in Zeta 3 terms) the primes and natural numbers (and indeed both sets of non-trivial zeros) as perfect mirrors of each other with an ineffable identity.

Now this comes as no surprise to me at all, as I would have long suspected that this was necessarily the case. Indeed I would go considerably further in emphasising that this spectral interpretation relates to a dynamic system with twin physical and psychological aspects that are complementary.

However my own interest in the Riemann Hypothesis initially arose from a prolonged interest in the nature of the (unrecognised) non-trivial zeros (Type 2).

And in this context I had already made use of a spectral interpretation as just one illuminating pathway towards appreciating their true nature.

Now the electromagnetic spectrum is perhaps the best-known physical spectrum.

So natural light comprises just one narrow band on this overall spectrum. Below this band lies electromagnetic energy of higher frequency and shorter wavelength than light (e.g. x-rays and gamma rays). Above this band lie corresponding energies of lower frequency and longer wavelength (e.g. microwave and radio).

Because in dynamic terms of complementarity, a psychological aspect necessarily exists to such a spectrum with fascinating implications for the spiritual contemplative journey.

The term “light” of course is equally used in a spiritual as well as physical sense where it refers to the holistic aspect of intuition or illumination which necessarily interacts with (analytic) reason.

Just as natural light inhabits the middle band of the physical spectrum likewise the intuition that characterises conventional (linear) reason likewise inhabits the middle of the psychological spectrum.

However it has been long known that through authentic contemplative development that higher bands on this spectrum are attainable where the intuitive light becomes ever more refined and thereby capable of interacting with paradoxical (i.e. circular) rational structures.

Indeed ultimately these structures become so unrestricted and transparent that they no longer appear to possess any remaining rigid structure becoming fully integrated with the very light through which they are mediated.

Now associated with the deepest level of contemplation is a quality of peace and utter stillness. This illumination can therefore be readily characterised as (intuitive) light of exceptionally long wavelength (and corresponding low frequency).

By contrast in the spiritual literature the other extreme (of short wavelength and high frequency) is often related to periods of purgation where an incessant inner bombardment takes place exposing one’s hidden faults and failings!

One sobering point however to consider is this!

Just as energies outside the natural light band are not visible to the naked eye, likewise all these energies (and the understanding associated with them) cannot be appropriated through linear reason (informed by common intuition).

This therefore creates insuperable difficulties for Conventional Mathematics in trying to appropriate such understanding within its accepted axioms and definitions!

Now again in the spiritual esoteric traditions, these higher bands of understanding are often referred to as higher “dimensions”.

Now the key here is to recognise that number is used to represent dimensions in Mathematics.

So therefore associated with each number – now understood in a qualitative holistic sense – is a unique dynamic form of understanding where a more refined quality of intuition interacts with the ever more circular (paradoxical) use of reason!

I have often mentioned in this context the fact that Conventional Mathematics is based on linear reason i.e. literally in a 1-dimensional manner.

Therefore though the quantitative expression of other dimensional numbers (as powers or exponents) is indeed accommodated, this all takes place within a default qualitative interpretation (= 1).

And the very nature of such interpretation is that distinctive holistic meaning (in any relevant context) is necessarily reduced in a mere quantitative manner.

Using the language of string theory, we could say therefore that Conventional Mathematics represents the lowest possible energy state (with respect to its understanding) where entities such as numbers thereby assume a (misleading) absolute identity.

More accurately we could say that Conventional Mathematics is exclusively defined in 1-dimensional terms (where 1 is defined in a qualitative holistic manner).

However the remarkable truth that needs to be embraced is that mathematical relationships can be validly defined in an unlimited number of possible ways (where each number as dimension represents a unique overall manner of interpretation).

Now in all cases (except 1) these dimensional interpretations imply unique configurations with respect to the dynamic interaction of both the analytic (particle) and holistic (wave) aspects of mathematical understanding.

So in making this very point, I am illustrating how the well-known fact that the Zeta Function remains uniquely undefined on the complex plane for s = 1, can be given both a quantitative (analytic) and qualitative (holistic) interpretation.

If you can appreciate this then you can understand immediately why the Riemann Zeta Function (and its associated Riemann Hypothesis) cannot be properly interpreted from within the standpoint of Conventional Mathematics.

As – when correctly interpreted - it fundamental relates to the dynamic interaction of the quantitative (analytic) and qualitative (holistic) aspects of the number system, this remains uniquely undefined in linear (1-dimensional) terms where the qualitative aspect formally is not recognised.

And of course from this enlarged perspective, as the Riemann Hypothesis relates to the fundamental condition for the ultimate identity of both quantitative (analytic) and qualitative (holistic) aspects, it is futile trying to prove it through the axioms of Conventional Mathematics (which give no formal recognition to the qualitative aspect).

From what we have said it is but a short journey to appreciating that the number system itself represents the ultimate spectrum (in both physical and psychological terms).

So the vibrations of all dynamic systems (physical and psychological) can thereby be ultimately understood as the original vibrations of this number system with respect to both its quantitative and qualitative aspects.

Therefore all natural phenomena thereby represent but a secondary expression of an original system where number - when dynamically interpreted with respect to both analytic and holistic aspects – inherently exists as their deepest possible means of encoding.

Once again the recognised non-trivial zeros (of the Zeta 1 Function) are - when properly understood - fully complementary with the Zeta 2 zeros (which I have been explaining).

Whereas the Zeta 2 can be identified initially with each natural number except 1 (that constitutes each prime) in ordinal terms, The Zeta 1 inversely can be identified initially overall with the entire group of prime numbers (that comprise the natural number system except 1) in cardinal terms.

The solutions from both perspectives establish the identity of the quantitative and qualitative aspects of the number system. Ultimately through simultaneous appreciation of Zeta 1 and Zeta 2, we can perfectly see (in Zeta 3 terms) the primes and natural numbers (and indeed both sets of non-trivial zeros) as perfect mirrors of each other with an ineffable identity.

## Sunday, March 17, 2013

### Two Complementary Zeta Functions (3)

In the latter part of my last blog entry I emphasised the inherent holistic nature of the Type 2 aspect of the number system. Ultimately this relates to the notion of number with respect to its qualitative - rather than quantitative - identity.

Alternatively, it relates to the understanding of number as interdependent (which conflicts strongly with the cultural dominance of numbers as representing solely independent entities).

The clue again with respect to this latter holistic emphasis is to view a number i.e. natural, with respect to the relationship as between its individual ordinal members.

So for example the analytic linear notion of 2 as cardinal relates to this number in static absolute terms as representing a collective whole (whose individuals units are necessarily of a homogenous - and thereby - indistinguishable nature).

However the corresponding holistic circular notion of 2 as ordinal relates to this same number, now understood in a dynamic relative fashion, as representing the interaction of its individual members (i.e. 1st and 2nd) which can be uniquely distinguished in qualitative terms.

Once again, this interaction entails two polar directions (i.e. 1st and 2nd dimensions which are inversely associated with the 1st and 2nd roots of 1) that are positive and negative with respect to each other represented as + 1 and – 1 respectively.

So the combined fusion of both as interdependent is represented as (+) 1 – 1 = 0.

Therefore from this holistic perspective every number can be given a qualitative meaning as a pure energy state (= 0 from a quantitative perspective). Of course in actual experience, this holistic energy state interacts with the customary rational understanding of number (as discrete form).

Then in formal interpretation of a conventional kind, the holistic aspect (which is of a uniquely distinct nature) is reduced in a mere rational fashion! We are thereby conditioned to view numbers as representing distinct quantitative entities with no recognition therefore of their corresponding holistic qualitative aspect as energy states (in both physical and psychological terms).

Thus, to emphasise once more, the holistic notion of any number relates to the dynamic interaction of its individual ordinal members (which are uniquely defined for the number).

Through reference to the corresponding roots of 1 for each number, this does indeed enable one to demonstrate the holistic aspect of all individual numbers. However, in itself it does not directly lend itself to quantitative type analysis, as by definition the sum of all these roots (demonstrating the qualitative relationship aspect of ordinal members) = 0.

However just as the Type 1 (analytic) aspect indirectly can be given a holistic interpretation (with respect to the nature of Zeta 1 trivial zeros) equally the Type 2 (holistic) aspect indirectly can be given an analytic expression (with respect to the Zeta 2 trivial zeros)!

Once again a fascinating form of complementarity is at work. The Zeta 1 non-trivial zeros are expressed in quantitative terms in a linear (1-dimensional) manner. The key to the holistic understanding is then to provide a “higher” dimensional interpretation for these zeros in a qualitative manner.

In reverse fashion, the Zeta 2 non-trivial zeros are expressed - representing their qualitative meaning - directly in a circular (higher dimensional) manner.

So again for example, I have been at pains to express the precise qualitative meaning of 2 as a dimensional number (i.e. with respect to its 1st and 2nd ordinal members).

Therefore the key here is to represent higher dimensional roots in a reduced linear (1-dimensional) fashion from a quantitative perspective.

In effect this implies treating negative signs as positive and imaginary numbers as real!

So when we convert the various roots of 1 in this quantitative manner, a fascinating new set of analytic type relationships emerge, leading both to an alternative Type 2 Prime Number Theorem and Riemann Hypothesis respectively.

I have expressed the nature of this alternative formulation on many occasions previously in my blog entries.

Once again we initially treat both the cos and sin parts of the roots of 1 (now expressed in this reduced linear quantitative fashion) obtaining the average for each prime number grouping (representing all of its ordinal number members)

So to illustrate where p = 3 the three roots (1st, 2nd and 3rd) are 1, .5 + .866 and .5 + .866.

So the sum of the 3 cos parts = 2 and three sin parts (the first of which is zero) = 1.732.

The average of the cos parts therefore = .666.. and the sin parts = .5773.

Now when we obtain the average of cos and sin parts for higher prime numbered root,s the answer quickly converges for both parts (with a small remaining deviation) on the value 2/π = .6366….

The alternative Prime Number Theorem relates to the fact that 2/π = i/log i.

So i/log i with respect to the Type 2 aspect (as the mean reduced average for both parts of the ordinal number roots of p, as p increases without limit) plays a complementary role to that which n/log n plays with respect to the Type 1 (representing the average number of primes among the natural numbers as n increases without limit).

Also it quickly becomes apparent that the average for the cos part always exceeds 2/π (= i/log i) while the corresponding sin part always is less than 2/π (= i/log i).

Indeed the absolute value of the ratio of the deviation of the cos part to the deviation of the sin part from 2/π (= i/log i) quickly converges towards .5 (which operates as the alternative Riemann Hypothesis with respect to the Type 2 aspect).

Even here further complementarity is in evidence: From the Type 1 perspective, n/log n is an approximate varying quantity, whereas .5 according to the Riemann Hypothesis is the fixed point on the real axis through which the imaginary line on which all the non-trivial zeros lie; however from the Type 2 perspective i/log i is a fixed value with .5 as the absolute ratio of cos and sin deviations by contrast an approximate varying quantity.

Now here is the important point!

As we saw in the last blog entry, in the Zeta 1 function, both the prime numbers (with respect to the natural numbers) and trivial zeros each possess extreme analytic properties (representing independence) and extreme holistic properties (representing interdependence) respectively depending on the frame of reference from which they are viewed.

So from one perspective we can use the non-trivial zeros to eliminate all remaining (independent) deviations of the prime numbers with respect to their overall collective (interdependent) relationship with the real natural numbers.

Equally from the other perspective we can use the prime numbers (as a group) to eliminate all (independent) deviations of non-trivial zeros with respect to their overall collective (interdependent) relationship with the imaginary number system.

Again a complementary type connection exists with respect to the Zeta 2 function. Here the two aspects (independence and interdependence respectively) are reconciled within the same distribution

I have already illustrated how the prime numbers (and by extension all the natural numbers) can be used to explain to a very high degree of accuracy the deviations of the average of both cos and sin parts from 2/π (= i/log i).

However what is perhaps even more interesting is that these deviations can in principle be used in such a manner that the frequency of primes within the natural numbers can be approximated to any degree of accuracy.

It would work something like this. In the normal general calculation of prime number frequency (among the primes) each natural number is given a weighting of 1.

However using the (reduced) Zeta 2 deviations a slightly modified weighting would be given with earliest natural numbers differing most. So for example when predicting the frequency of primes relating to the first 100 natural numbers (i.e. n = 100) we would substitute a replacement number based on the total representing the sum of each slightly modified weighted natural number (up to 100) which could then correctly predict from the general formula the actual frequency associated with 100!

So whereas with the Zeta 1 function, the role of deviations associated with the non-trivial zeros must be understood in the context of the number system as a whole, here in the Zeta 2 approach we would have a one-to one relationship of a new weighted number (differing slightly from 1) with each natural number (based on the default weighting of 1).

Once again the greatest deviations would occur - with respect to these newly weighted members - amongst the earliest natural numbers. Then as we ascend up the natural number scale the weightings would approach ever closer to the default natural number weightings of 1.

Alternatively, it relates to the understanding of number as interdependent (which conflicts strongly with the cultural dominance of numbers as representing solely independent entities).

The clue again with respect to this latter holistic emphasis is to view a number i.e. natural, with respect to the relationship as between its individual ordinal members.

So for example the analytic linear notion of 2 as cardinal relates to this number in static absolute terms as representing a collective whole (whose individuals units are necessarily of a homogenous - and thereby - indistinguishable nature).

However the corresponding holistic circular notion of 2 as ordinal relates to this same number, now understood in a dynamic relative fashion, as representing the interaction of its individual members (i.e. 1st and 2nd) which can be uniquely distinguished in qualitative terms.

Once again, this interaction entails two polar directions (i.e. 1st and 2nd dimensions which are inversely associated with the 1st and 2nd roots of 1) that are positive and negative with respect to each other represented as + 1 and – 1 respectively.

So the combined fusion of both as interdependent is represented as (+) 1 – 1 = 0.

Therefore from this holistic perspective every number can be given a qualitative meaning as a pure energy state (= 0 from a quantitative perspective). Of course in actual experience, this holistic energy state interacts with the customary rational understanding of number (as discrete form).

Then in formal interpretation of a conventional kind, the holistic aspect (which is of a uniquely distinct nature) is reduced in a mere rational fashion! We are thereby conditioned to view numbers as representing distinct quantitative entities with no recognition therefore of their corresponding holistic qualitative aspect as energy states (in both physical and psychological terms).

Thus, to emphasise once more, the holistic notion of any number relates to the dynamic interaction of its individual ordinal members (which are uniquely defined for the number).

Through reference to the corresponding roots of 1 for each number, this does indeed enable one to demonstrate the holistic aspect of all individual numbers. However, in itself it does not directly lend itself to quantitative type analysis, as by definition the sum of all these roots (demonstrating the qualitative relationship aspect of ordinal members) = 0.

However just as the Type 1 (analytic) aspect indirectly can be given a holistic interpretation (with respect to the nature of Zeta 1 trivial zeros) equally the Type 2 (holistic) aspect indirectly can be given an analytic expression (with respect to the Zeta 2 trivial zeros)!

Once again a fascinating form of complementarity is at work. The Zeta 1 non-trivial zeros are expressed in quantitative terms in a linear (1-dimensional) manner. The key to the holistic understanding is then to provide a “higher” dimensional interpretation for these zeros in a qualitative manner.

In reverse fashion, the Zeta 2 non-trivial zeros are expressed - representing their qualitative meaning - directly in a circular (higher dimensional) manner.

So again for example, I have been at pains to express the precise qualitative meaning of 2 as a dimensional number (i.e. with respect to its 1st and 2nd ordinal members).

Therefore the key here is to represent higher dimensional roots in a reduced linear (1-dimensional) fashion from a quantitative perspective.

In effect this implies treating negative signs as positive and imaginary numbers as real!

So when we convert the various roots of 1 in this quantitative manner, a fascinating new set of analytic type relationships emerge, leading both to an alternative Type 2 Prime Number Theorem and Riemann Hypothesis respectively.

I have expressed the nature of this alternative formulation on many occasions previously in my blog entries.

Once again we initially treat both the cos and sin parts of the roots of 1 (now expressed in this reduced linear quantitative fashion) obtaining the average for each prime number grouping (representing all of its ordinal number members)

So to illustrate where p = 3 the three roots (1st, 2nd and 3rd) are 1, .5 + .866 and .5 + .866.

So the sum of the 3 cos parts = 2 and three sin parts (the first of which is zero) = 1.732.

The average of the cos parts therefore = .666.. and the sin parts = .5773.

Now when we obtain the average of cos and sin parts for higher prime numbered root,s the answer quickly converges for both parts (with a small remaining deviation) on the value 2/π = .6366….

The alternative Prime Number Theorem relates to the fact that 2/π = i/log i.

So i/log i with respect to the Type 2 aspect (as the mean reduced average for both parts of the ordinal number roots of p, as p increases without limit) plays a complementary role to that which n/log n plays with respect to the Type 1 (representing the average number of primes among the natural numbers as n increases without limit).

Also it quickly becomes apparent that the average for the cos part always exceeds 2/π (= i/log i) while the corresponding sin part always is less than 2/π (= i/log i).

Indeed the absolute value of the ratio of the deviation of the cos part to the deviation of the sin part from 2/π (= i/log i) quickly converges towards .5 (which operates as the alternative Riemann Hypothesis with respect to the Type 2 aspect).

Even here further complementarity is in evidence: From the Type 1 perspective, n/log n is an approximate varying quantity, whereas .5 according to the Riemann Hypothesis is the fixed point on the real axis through which the imaginary line on which all the non-trivial zeros lie; however from the Type 2 perspective i/log i is a fixed value with .5 as the absolute ratio of cos and sin deviations by contrast an approximate varying quantity.

Now here is the important point!

As we saw in the last blog entry, in the Zeta 1 function, both the prime numbers (with respect to the natural numbers) and trivial zeros each possess extreme analytic properties (representing independence) and extreme holistic properties (representing interdependence) respectively depending on the frame of reference from which they are viewed.

So from one perspective we can use the non-trivial zeros to eliminate all remaining (independent) deviations of the prime numbers with respect to their overall collective (interdependent) relationship with the real natural numbers.

Equally from the other perspective we can use the prime numbers (as a group) to eliminate all (independent) deviations of non-trivial zeros with respect to their overall collective (interdependent) relationship with the imaginary number system.

Again a complementary type connection exists with respect to the Zeta 2 function. Here the two aspects (independence and interdependence respectively) are reconciled within the same distribution

I have already illustrated how the prime numbers (and by extension all the natural numbers) can be used to explain to a very high degree of accuracy the deviations of the average of both cos and sin parts from 2/π (= i/log i).

However what is perhaps even more interesting is that these deviations can in principle be used in such a manner that the frequency of primes within the natural numbers can be approximated to any degree of accuracy.

It would work something like this. In the normal general calculation of prime number frequency (among the primes) each natural number is given a weighting of 1.

However using the (reduced) Zeta 2 deviations a slightly modified weighting would be given with earliest natural numbers differing most. So for example when predicting the frequency of primes relating to the first 100 natural numbers (i.e. n = 100) we would substitute a replacement number based on the total representing the sum of each slightly modified weighted natural number (up to 100) which could then correctly predict from the general formula the actual frequency associated with 100!

So whereas with the Zeta 1 function, the role of deviations associated with the non-trivial zeros must be understood in the context of the number system as a whole, here in the Zeta 2 approach we would have a one-to one relationship of a new weighted number (differing slightly from 1) with each natural number (based on the default weighting of 1).

Once again the greatest deviations would occur - with respect to these newly weighted members - amongst the earliest natural numbers. Then as we ascend up the natural number scale the weightings would approach ever closer to the default natural number weightings of 1.

## Thursday, March 14, 2013

### Two Complementary Zeta Functions (2)

As we have seen, all mathematical symbols can be given two complementary interpretations (analytic and holistic) respectively.

This is then reflected in the number system which itself has Type 1 (analytic) and Type 2 (holistic) aspects which are complementary. Then indirectly the Type 1 aspect can be given a Type 2 interpretation while then Type 2 indirectly can be given a Type 1 formulation!

Then with respect to the famed complex zeta function, it too has too complementary aspects.

So corresponding to Type 1 we have the recognised Riemann Zeta Function (which I refer to as Zeta 1).

Then corresponding to Type 2 we have a - largely unrecognised – complementary function (which I refer to as Zeta 2). Strictly, though the Zeta 2 Function is in fact well known; its true holistic significance is not at all appreciated.

Once again the Zeta 1 is defined as the infinite series:

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….

And the non-trivial (pair) solutions of the form s = a + it and a – it (where a according to the Riemann Hypothesis = ½) occur when

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + …. = 0

The Zeta 2 is then in inverse complementary terms (i.e. where the natural numbers now represent the powers and s the base quantities) as the finite series:

1 + s^1 + s^2 + ….+ s^(n – 1).

And the non-trivial solutions of the form s = a + it arise from the solutions of the equation,

1 + s^1 + s^2 + ….+ s^(n – 1) = 0

Now in a qualified sense we can equally define this second function in an infinite manner!

So if we take terms in in a strict cyclical order in accordance with the value of n, then the infinite equation will hold.

Thus for example in the simplest case where n = 2, the non-trivial 2nd root as solution to

1 + s^1 = 0 gives s = – 1.

So when we take terms two at a time the value of the infinite series

1 + s^1 + s^2 + s^3 + ….. = 0.

Now in this blog entry I will try to highlight some of the key complementary relations existing between both series and their implications.

Indeed when one begins to appreciate the very nature of the number system from this dynamic complementary perspective it leads to a completely new type of insight that itself directly fosters appreciation of its holistic nature.

Customarily we tend to view the number system in (real) linear terms with the prime numbers viewed as its independent building blocks! So using the terminology that I customarily employ this reflects an analytic (quantitative) interpretation.

Then parallel with this finding we find that the system of non-trivial zeros relating to the Zeta 1 in turn - assuming the truth of the Riemann Hypothesis – represents a corresponding linear number system (this time however in imaginary terms).

Now properly understood this indicates a direct complementary relationship.

So the numbers on the real number line can represented as separate (independent) points i.e. in analytic terms within this frame of reference.

Likewise the non-trivial zeros on the imaginary number line equally can be represented as separate (independent) points i.e. in analytic terms within this different frame of reference.

However the clear implication here is that when we seek to switch reference frames so as to interpret the imaginary points (from a real perspective) they now assume a holistic – rather than an analytic – identity.

In other words the significance of the non-trivial zeros, from a real number perspective, is that they can be used as an entire group to reconcile the unique individual identity of each prime (which is random with respect to the natural numbers) with an overall collective nature (where they are perfectly synchronised with the natural number system).

In other words, from the perspective of the imaginary number line, each non-trivial zero has an (extreme) analytic identity; however from the perspective of the real number line the non-trivial zeros as an entire group (which is always ultimately indeterminate in finite terms) have a complementary (extreme) holistic identity!

The same in fact applies to the prime numbers.

From the perspective of the real number line, each prime number has an (extreme) analytic identity i.e. as a unique building block of the natural number system!

However from the complementary perspective of the imaginary number line, the prime numbers as an entire group (the ultimate nature of which is necessarily indeterminate in finite terms) play a remarkable holistic role in serving to completely reconcile the random individual identity of each non-trivial zero with their overall collective identity (i.e. through being perfectly synchronised with the imaginary number system).

So we see that when we adopt both perspectives that:

1) Each individual prime number has an extreme analytic identity as separate (in real number terms).

2) That the prime numbers as an entire collective group have a corresponding extreme holistic identity (in imaginary number terms).

3) That each individual non-trivial zero likewise has an extreme analytic identity (in imaginary number terms).

4) That the non-trivial zeros as an entire collective group have a corresponding extreme holistic identity (from the real number perspective).

However to properly appreciate such relationships we must preserve both analytic (Type 1) and holistic (Type 2) perspectives with respect to the number system.

Clearly this cannot be achieved in conventional mathematical terms, as it is based formally on mere Type 1 interpretation.

So each number type contains have both particle and wave identities with an individual (analytic) and collective (holistic) identity respectively.

We can then approach the same number relationship issues in an inverted fashion through the Type 2 aspect of the number system.

Now whereas the Type 1 is directly geared to analytic type appreciation (and indirectly holistic), Type 2 is - by contrast - directly geared to holistic type appreciation (and indirectly analytic).

With Type 2, rather than a linear, we adopt a circular notion of number.

Therefore, from a real perspective instead of representing the natural numbers as equidistant points on the number line, we represent them in alternative fashion as equidistant points on the unit circle (where now they strictly represent an ordinal - rather than cardinal - identity)!

So in this approach the prime number 3 - for example – can be represented by the 3 equidistant points labelled 1, 2 and 3 respectively relating to the three corresponding ordinal members of 3 i.e. 1st, 2nd and 3rd respectively.

Thus once again whereas in the cardinal (Type 1) approach the natural numbers, (as an entire collective group) are all uniquely derived from individual prime constituents, in the complementary ordinal (Type 2) approach each prime number (as an individual group) is uniquely defined by its individual natural number members.

This in turn leads to a fascinating alternative explanation of prime number identity.

From the Type 1 perspective, a prime number has no factors (other than itself and 1).

From the Type 2 perspective a prime number - by definition - will always be a factor of the sum of its individual members.

For example 3 as prime number is necessarily a factor of 1 + 2 + 3 = 6 (and this relationship will always hold where the number is prime). However this is not unique for prime numbers and in fact is shared by all odd numbers.

Put another way, expressing this in modular (clock arithmetic) if the modulus is prime, then the sum of the individual natural number members of this prime = 0.

So with the Type 1 a prime number is defined by having no non-trivial factor in natural number terms (i.e. other than itself and 1).

In Type 2 terms, in inverse complementary fashion, a prime number is defined as always constituting a non-trivial factor of the sum of its natural number members!

This in fact therefore provides an alternative means in practice for testing for primeness (which I have already illustrated in other contexts). See "Interesting Prime Result"

However this circular (Type 2) system of (real) numbers has a counterpart system of (complex) numbers as the non-trivial zeros arising as solutions to the equation,

1 + s^1 + s^2 + ….+ s^(n – 1) = 0, where n is prime.

In contrast to Type 1, appreciation of the nature of these zeros is directly of a holistic nature.

Though 1 is always included as one of the prime roots of 1, in a certain sense it is trivial in that it is thereby not unique!

So the non-trivial strictly refer to the remaining n – 1 roots!

We could perhaps illustrate the true holistic nature of such solutions with respect to the simplest case of the 2 roots of 1.

So once again these are 1 and – 1 respectively.

The holistic interdependence of these roots can be illustrated by the fact that the sum of these roots i.e. 1 – 1 = 0.

Now this interdependence strictly relates to an energy state!

For example at the sub-atomic level, a particle and anti-particle are + 1 and – 1 with respect to each other. Then when combined their separate identities are annihilated resulting in a pure energy state.

So in a very literal sense such a pure energy state thereby characterises the holistic nature of 2.

Indeed if we were to equally consider virtual (imaginary) as well as real particles the combined annihilation with anti-particles at both levels would characterise the true holistic nature of 4!

Though other numbers are indeed more difficult to illustrate, the basic principle is clear in that the holistic nature of number in fact relates directly to a physical energy state!

And as physical and psychological aspects are complementary, this equally implies that associated with each number in holistic terms is a corresponding psychological energy state relating to the (unconscious) intuitive aspect of number experience.

For example, in the dynamics of experience external and internal polarities of understanding are positive and negative with respect to each other.

As the very process of understanding necessarily entails integrating both of these polarities + 1 and – 1 as directions (i.e. dimensions) the holistic understanding of 2 from a psychological perspective necessarily relates to a psycho-spiritual energy state! So intuitive recognition (as a psychological eneregy state) always reflects the holistic aspect of understanding. Unfortunately such intuitive recognition is then reduced to merely rational (analytic) interpretation in formal conventional mathematical terms.

So when one looks at the issue from a number perspective it should come as no surprise that likewise the holistic appreciation of the non-trivial zeros (with respect to Type 1) equally relate to energy states.

Though a degree of recognition of this fact has admittedly been obtained in recent years (with reference to the behaviour of certain quantum chaotic physical systems) Conventional Mathematics still lacks the means to intuitively explain why this is so (due to the complete absence in formal terms of a holistic aspect to its means of interpretation).

In my own case, as I had been specialising for some years in the holistic appreciation of number, I had long reached the conclusion that the Riemann zeros necessarily had quantum mechanical implications.

However even more, I had firmly reached the conclusion that these same zeros necessarily possess extremely important psycho-spiritual implications (which is completely missing as yet from conventional type understanding).

Furthermore the appropriate way to view both these aspects - physical and psychological - is in a dynamic complementary manner!

Moreover, correctly understood, we have in fact two sets of zeros (in accordance with Type 1 and Type 2 appreciation respectively).

Whereas Type 1 directly relates to the understanding of prime numbers as discrete phenomenal entities, by contrast Type 2 directly relates to the same prime numbers as (formless) energy states in both physical and psychological terms.

The deeper implications of all this is that - correctly understood - number itself must be viewed as inherent in all physical and psychological processes as the most fundamental way in which their quantitative and qualitative attributes are encoded.

In this sense, we can truly say that all living phenomenal forms at their most fundamental level represent but the dynamic interaction of number processes with respect to both their quantitative and qualitative aspects!

This is then reflected in the number system which itself has Type 1 (analytic) and Type 2 (holistic) aspects which are complementary. Then indirectly the Type 1 aspect can be given a Type 2 interpretation while then Type 2 indirectly can be given a Type 1 formulation!

Then with respect to the famed complex zeta function, it too has too complementary aspects.

So corresponding to Type 1 we have the recognised Riemann Zeta Function (which I refer to as Zeta 1).

Then corresponding to Type 2 we have a - largely unrecognised – complementary function (which I refer to as Zeta 2). Strictly, though the Zeta 2 Function is in fact well known; its true holistic significance is not at all appreciated.

Once again the Zeta 1 is defined as the infinite series:

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + ….

And the non-trivial (pair) solutions of the form s = a + it and a – it (where a according to the Riemann Hypothesis = ½) occur when

1^(–s) + 2^(– s) + 3^(– s) + 4^(– s) + …. = 0

The Zeta 2 is then in inverse complementary terms (i.e. where the natural numbers now represent the powers and s the base quantities) as the finite series:

1 + s^1 + s^2 + ….+ s^(n – 1).

And the non-trivial solutions of the form s = a + it arise from the solutions of the equation,

1 + s^1 + s^2 + ….+ s^(n – 1) = 0

Now in a qualified sense we can equally define this second function in an infinite manner!

So if we take terms in in a strict cyclical order in accordance with the value of n, then the infinite equation will hold.

Thus for example in the simplest case where n = 2, the non-trivial 2nd root as solution to

1 + s^1 = 0 gives s = – 1.

So when we take terms two at a time the value of the infinite series

1 + s^1 + s^2 + s^3 + ….. = 0.

Now in this blog entry I will try to highlight some of the key complementary relations existing between both series and their implications.

Indeed when one begins to appreciate the very nature of the number system from this dynamic complementary perspective it leads to a completely new type of insight that itself directly fosters appreciation of its holistic nature.

Customarily we tend to view the number system in (real) linear terms with the prime numbers viewed as its independent building blocks! So using the terminology that I customarily employ this reflects an analytic (quantitative) interpretation.

Then parallel with this finding we find that the system of non-trivial zeros relating to the Zeta 1 in turn - assuming the truth of the Riemann Hypothesis – represents a corresponding linear number system (this time however in imaginary terms).

Now properly understood this indicates a direct complementary relationship.

So the numbers on the real number line can represented as separate (independent) points i.e. in analytic terms within this frame of reference.

Likewise the non-trivial zeros on the imaginary number line equally can be represented as separate (independent) points i.e. in analytic terms within this different frame of reference.

However the clear implication here is that when we seek to switch reference frames so as to interpret the imaginary points (from a real perspective) they now assume a holistic – rather than an analytic – identity.

In other words the significance of the non-trivial zeros, from a real number perspective, is that they can be used as an entire group to reconcile the unique individual identity of each prime (which is random with respect to the natural numbers) with an overall collective nature (where they are perfectly synchronised with the natural number system).

In other words, from the perspective of the imaginary number line, each non-trivial zero has an (extreme) analytic identity; however from the perspective of the real number line the non-trivial zeros as an entire group (which is always ultimately indeterminate in finite terms) have a complementary (extreme) holistic identity!

The same in fact applies to the prime numbers.

From the perspective of the real number line, each prime number has an (extreme) analytic identity i.e. as a unique building block of the natural number system!

However from the complementary perspective of the imaginary number line, the prime numbers as an entire group (the ultimate nature of which is necessarily indeterminate in finite terms) play a remarkable holistic role in serving to completely reconcile the random individual identity of each non-trivial zero with their overall collective identity (i.e. through being perfectly synchronised with the imaginary number system).

So we see that when we adopt both perspectives that:

1) Each individual prime number has an extreme analytic identity as separate (in real number terms).

2) That the prime numbers as an entire collective group have a corresponding extreme holistic identity (in imaginary number terms).

3) That each individual non-trivial zero likewise has an extreme analytic identity (in imaginary number terms).

4) That the non-trivial zeros as an entire collective group have a corresponding extreme holistic identity (from the real number perspective).

However to properly appreciate such relationships we must preserve both analytic (Type 1) and holistic (Type 2) perspectives with respect to the number system.

Clearly this cannot be achieved in conventional mathematical terms, as it is based formally on mere Type 1 interpretation.

So each number type contains have both particle and wave identities with an individual (analytic) and collective (holistic) identity respectively.

We can then approach the same number relationship issues in an inverted fashion through the Type 2 aspect of the number system.

Now whereas the Type 1 is directly geared to analytic type appreciation (and indirectly holistic), Type 2 is - by contrast - directly geared to holistic type appreciation (and indirectly analytic).

With Type 2, rather than a linear, we adopt a circular notion of number.

Therefore, from a real perspective instead of representing the natural numbers as equidistant points on the number line, we represent them in alternative fashion as equidistant points on the unit circle (where now they strictly represent an ordinal - rather than cardinal - identity)!

So in this approach the prime number 3 - for example – can be represented by the 3 equidistant points labelled 1, 2 and 3 respectively relating to the three corresponding ordinal members of 3 i.e. 1st, 2nd and 3rd respectively.

Thus once again whereas in the cardinal (Type 1) approach the natural numbers, (as an entire collective group) are all uniquely derived from individual prime constituents, in the complementary ordinal (Type 2) approach each prime number (as an individual group) is uniquely defined by its individual natural number members.

This in turn leads to a fascinating alternative explanation of prime number identity.

From the Type 1 perspective, a prime number has no factors (other than itself and 1).

From the Type 2 perspective a prime number - by definition - will always be a factor of the sum of its individual members.

For example 3 as prime number is necessarily a factor of 1 + 2 + 3 = 6 (and this relationship will always hold where the number is prime). However this is not unique for prime numbers and in fact is shared by all odd numbers.

Put another way, expressing this in modular (clock arithmetic) if the modulus is prime, then the sum of the individual natural number members of this prime = 0.

So with the Type 1 a prime number is defined by having no non-trivial factor in natural number terms (i.e. other than itself and 1).

In Type 2 terms, in inverse complementary fashion, a prime number is defined as always constituting a non-trivial factor of the sum of its natural number members!

This in fact therefore provides an alternative means in practice for testing for primeness (which I have already illustrated in other contexts). See "Interesting Prime Result"

However this circular (Type 2) system of (real) numbers has a counterpart system of (complex) numbers as the non-trivial zeros arising as solutions to the equation,

1 + s^1 + s^2 + ….+ s^(n – 1) = 0, where n is prime.

In contrast to Type 1, appreciation of the nature of these zeros is directly of a holistic nature.

Though 1 is always included as one of the prime roots of 1, in a certain sense it is trivial in that it is thereby not unique!

So the non-trivial strictly refer to the remaining n – 1 roots!

We could perhaps illustrate the true holistic nature of such solutions with respect to the simplest case of the 2 roots of 1.

So once again these are 1 and – 1 respectively.

The holistic interdependence of these roots can be illustrated by the fact that the sum of these roots i.e. 1 – 1 = 0.

Now this interdependence strictly relates to an energy state!

For example at the sub-atomic level, a particle and anti-particle are + 1 and – 1 with respect to each other. Then when combined their separate identities are annihilated resulting in a pure energy state.

So in a very literal sense such a pure energy state thereby characterises the holistic nature of 2.

Indeed if we were to equally consider virtual (imaginary) as well as real particles the combined annihilation with anti-particles at both levels would characterise the true holistic nature of 4!

Though other numbers are indeed more difficult to illustrate, the basic principle is clear in that the holistic nature of number in fact relates directly to a physical energy state!

And as physical and psychological aspects are complementary, this equally implies that associated with each number in holistic terms is a corresponding psychological energy state relating to the (unconscious) intuitive aspect of number experience.

For example, in the dynamics of experience external and internal polarities of understanding are positive and negative with respect to each other.

As the very process of understanding necessarily entails integrating both of these polarities + 1 and – 1 as directions (i.e. dimensions) the holistic understanding of 2 from a psychological perspective necessarily relates to a psycho-spiritual energy state! So intuitive recognition (as a psychological eneregy state) always reflects the holistic aspect of understanding. Unfortunately such intuitive recognition is then reduced to merely rational (analytic) interpretation in formal conventional mathematical terms.

So when one looks at the issue from a number perspective it should come as no surprise that likewise the holistic appreciation of the non-trivial zeros (with respect to Type 1) equally relate to energy states.

Though a degree of recognition of this fact has admittedly been obtained in recent years (with reference to the behaviour of certain quantum chaotic physical systems) Conventional Mathematics still lacks the means to intuitively explain why this is so (due to the complete absence in formal terms of a holistic aspect to its means of interpretation).

In my own case, as I had been specialising for some years in the holistic appreciation of number, I had long reached the conclusion that the Riemann zeros necessarily had quantum mechanical implications.

However even more, I had firmly reached the conclusion that these same zeros necessarily possess extremely important psycho-spiritual implications (which is completely missing as yet from conventional type understanding).

Furthermore the appropriate way to view both these aspects - physical and psychological - is in a dynamic complementary manner!

Moreover, correctly understood, we have in fact two sets of zeros (in accordance with Type 1 and Type 2 appreciation respectively).

Whereas Type 1 directly relates to the understanding of prime numbers as discrete phenomenal entities, by contrast Type 2 directly relates to the same prime numbers as (formless) energy states in both physical and psychological terms.

The deeper implications of all this is that - correctly understood - number itself must be viewed as inherent in all physical and psychological processes as the most fundamental way in which their quantitative and qualitative attributes are encoded.

In this sense, we can truly say that all living phenomenal forms at their most fundamental level represent but the dynamic interaction of number processes with respect to both their quantitative and qualitative aspects!

## Monday, March 11, 2013

### Two Complementary Zeta Functions (1)

Once again my basic contention is that – properly understood – there are two distinct aspects to the number system.

1) An analytic aspect where the fundamental poles of understanding (i.e. internal and internal and quantitative and qualitative are clearly separated). This indeed is what makes such Mathematics linear (i.e. 1-dimensional) in nature as it literally interprets its symbols exclusively – in any context - in terms of the dominance of just one pole of understanding.

So relationships are considered absolutely in terms of the objective nature of symbols (where the external pole dominates); equally relationships are considered absolutely in terms of the quantitative nature of symbols (where the qualitative is thereby reduced to the quantitative).

(It is important to appreciate that when I use the word analytic, typically I use it is this general context of absolute uni-polar interpretation of symbols and not in the narrower more specialised conventional mathematical sense relating to infinite series).

I refer to this aspect of Mathematics as Type 1.

2) A holistic aspect where the same fundamental poles of understanding are now considered as dynamically related. In this new distinctive interpretation the objective nature of mathematical symbols strictly has no meaning in the absence of their corresponding means of (subjective) mental interpretation.

And as a wide variety of partially valid interpretations can be employed the objective nature of such symbols is thereby necessarily of a merely relative nature.

Likewise the quantitative interpretation of symbols strictly has no means in the absence of qualitative appreciation (of a distinctive holistic variety).

To be more accurate the holistic aspect properly refers to the dynamic interaction of poles that are (a) external and internal and (b) quantitative and qualitative with respect to each other.

Likewise the analytic aspect refers to the extreme situation where such interaction is ignored through reducing one entirely in terms of the other. And again in Conventional Mathematics the internal aspect is reduced in terms of the external and the qualitative in terms of the quantitative!

However for convenience I continually contrast analytic and holistic (as shorthand for there respectively different ways of treating such polar interaction)!

I refer to this aspect of Mathematics as Type 2.

3) The actual experience of Mathematics entails the mutual dynamic interaction of both Type 1 and Type 2 understanding. Though customary objective quantitative distinctions of a valid nature can indeed be made, as the polar reference frames to which they relate continually change they are now understood in a merely relative manner.

I refer to this most comprehensive aspect of Mathematics as Type 3.

So properly understood (i.e. in a comprehensive manner) all mathematical relationships are necessarily understood in merely relative terms. The customary fixation with absolute type relationships in Conventional Mathematics reflects its reduced – and thereby limited – (1-dimensional) manner of interpretation.

The distinction as between the Type 1 and Type 2 aspects Mathematics leads to a corresponding distinction with respect to the number system.

For example the natural numbers in Type 1 terms are understood as representing cardinal whole units. So 2 in cardinal terms represent a collective whole unit (i.e. an integer).

The problem with this quantitative approach however is that when one attempts to describe the individual members of 2, one must treat them in homogenous terms as without qualitative distinction. So 2 = 1 + 1.

However this leaves one with the considerable problem of having no means of making a meaningful distinction as between the 1st and 2nd members of 2 (in ordinal terms).

The fact that this fundamental issue is completely glossed over in Conventional Mathematics clearly reflects its reduced nature i.e. where the qualitative (ordinal) nature of 1st and 2nd is misleadingly assumed to be implied by the quantitative (cardinal) understanding of 1 and 2!

So the Type 2 understanding of number resembles – as it were - a sub-atomic approach where each of the individual members of a number group are now given a unique individual identity in relative terms.

So once again from a Type 1 approach, 2 is treated from a cardinal perspective as a collective whole unit (in quantitative terms).

From the complementary Type 2 approach, 2 is treated from an ordinal perspective as comprising unique individual units i.e. 1st and 2nd (in qualitative terms).

Now Type 1 is – literally - defined in 1-dimensional terms, where each natural number is implicitly defined with respect to 1 as exponent (i.e. dimensional number).

So the natural numbers 1, 2, 3, 4,…. are defined in Type 1 terms as

1^1, 2^1, 3^ 1. 4^1,….

The 1-dimensional nature of this approach is clearly illustrated by the fact that any number initially raised to another exponent (dimensional number) will be given a reduced linear interpretation i.e. where its ultimate value is expressed in reduced quantitative terms.

So 2^2 in Type 1 = 4^1!

The Type 1 approach is properly geared to explain the fundamental nature of addition where quantitative change occurs (without qualitative transformation)!

So 2 + 3 in this additive system = 2^ 1 + 3^ 1 = 5^1.

In the Type 2 aspect of the number system, 2 is by contrast is defined with respect to 1 (representing a default base number quantity).

So the natural number 1, 2, 3, 4,….. are defined from a Type 2 perspective as

1^1, 1^2, 1^3, 1^4,…..

Just as the Type 1 approach is geared to the pure nature of addition (where no qualitative transformation in the variables takes place), The Type 2 approach is then geared to the pure nature of multiplication (where no quantitative transformation in the variables takes place).

So 2 + 3 in this system represents 1^2 * 1^ 3 = 1 ^ (2 + 3) = 1^5.

So what appears as multiplication from the Type 1 perspective represents addition from the corresponding Type 2 perspective.

However to properly express the nature of the Type 2 aspect we need to move to a circular number system (based on successive roots of unity).

The key to understanding the nature of this system is that a two-way interaction is necessarily entailed as between variables given both an analytic and holistic interpretation respectively.

Thus for example the number 2 in type 2 terms = 1^2.

2 here is identified with the two individual unique dimensions of 2 i.e. its 1st and 2nd dimensions (with a holistic interpretation as interdependent).

Now in analytic terms these two dimensions can be given expression as the 1st and 2nd roots of 1 respectively or more correctly the roots of 1^2 and 1^1 respectively.

So the first root of 1 is therefore (1^2) raised to the power ½ = + 1 (i.e. 1^1).

The second root of 1 is (1^1) raised to the power of ½ = – 1.

However in dynamic terms these two separate values (as analytically understood) are in continual relationship with the corresponding interdependent value (as holistically understood).

So the quantitative independent interpretation of 1 and – 1 as separate strictly have no meaning in dynamic relative terms apart from the combined holistic interpretation of 1 and – 1 as interdependent.

Once again I have repeatedly used the example of a crossroads to illustrate this type of understanding. When understood (within independent reference frames) left and right turns at a crossroads can be given a – relatively – separate interpretation which are + 1 and – 1 with respect to each other. (1 refers here to the dimensional unit i.e. representing a direction).

However the appreciation that both left and right ultimately have a purely relative i.e. paradoxical meaning (when both frames are simultaneously combined) represents the true qualitative holistic nature of 2!

And this could be represented quantitatively as 1 – 1 = 0, i.e. as strictly without quantitative significance!

So in the actual understanding of the nature of a direction at a crossroads, both types of understanding (analytic and holistic) are necessarily combined.

Indeed this 2-dimensional type of dynamic understanding is well-recognised in spiritual literature (e.g. Taoism and Buddhism) in philosophy (e.g. Heraclitus and Hegel) in psychology (e.g. Jung) and indeed even indirectly in quantum physics (e.g. wave-particle duality).

What is vital to appreciate however is that it necessarily combines both analytic and holistic aspects of understanding in a complementary fashion.

Now clearly this cannot be appreciated in Type 1 terms (where holistic notions are reduced to analytic in a static absolute manner).

As Conventional Mathematics is fundamentally based on such absolute notions it is severely limited in terms of dealing with the very notion of interdependence.

And as the nature of prime numbers likewise involves such interdependence e.g. in the relationship of the primes - constituting the natural number system in cardinal terms - to the non-trivial zeta zeros, it is fatally flawed in terms of appreciating the true nature of the number system.

Another key point is that this interpretation of the number 2 from a Type 2 perspective represents but the simplest version of Type 2 understanding.

Indeed associated with every number as an exponent (i.e. dimensional number) is a unique means of configuring the dynamic interaction as between its analytic and holistic elements! Or using the shorthand means that I have been employing every number has a unique holistic significance serving as a distinct means of interpretation of the dynamic interaction of polarities (underlying all mathematical understanding).

In this holistic context 1 (as 1-dimensional) has an extreme limiting interpretation where the holistic aspect is directly reduced to analytic interpretation.

And this is what precisely defines the nature of Conventional Mathematics which is based on absolute objective interpretation of its symbols in a merely (reduced) quantitative manner!

However in truth in a more comprehensive vision of Mathematics, an unlimited number of possible interpretations exist (all with a partial relative validity). And in all these other systems (based on a dimensional number ≠ 1).

And if you understand this you will immediately understand why from a holistic perspective, the Riemann zeta function is uniquely undefined for s = 1 (as this is the one value where the inherent dynamic interaction as between the analytic and holistic aspects of the number system is broken).

So once again, putting it bluntly the Riemann zeta function cannot be properly interpreted in conventional mathematical terms. And of course neither can the Riemann Hypothesis (which is fundamentally based on true appropriate interpretation of the Riemann Function).

Finally it is futile therefore attempting to prove the Riemann Hypothesis from a conventional (1-dimensional) perspective!

However fortunately, the key to appreciating the nature of the Riemann Hypothesis can be related to the simplest case of Type 2 understanding (with respect to the number 2).

Once again the most comprehensive form of mathematical understanding involves the interaction of both Type 1 and Type 2 aspects where both analytic (as separate) and holistic appreciation can reach a high level of refinement.

In fact whenever a non-unitary base number is raised to a non-unitary power, both Type 1 and Type 2 aspects of interpretation are both required.

So in the simplest case of 2^2, strictly both a quantitative (analytic) and qualitative (holistic) transformation is entailed the mutual interaction of which entails Type 3 understanding.

Next we find that associated with the Type 1 and Type 2 aspects of the number system are corresponding Type 1 and Type 2 zeta functions.

Now the Riemann zeta function refers strictly to the Type 1 aspect; however proper appreciation of the Riemann zeta function and its associated Riemann Hypothesis requires incorporation of a complementary Type 2 zeta function.

Though the conventional Riemann zeta function can indeed be given an analytic Type 1 formulation (as the Zeta 1 function), strictly this has no proper meaning in the absence of corresponding holistic Type 2 appreciation.

In like manner through the Riemann zeta function can be given an – unrecognised – Type 2 formulation (as the Zeta 2 function), strictly again this has no proper meaning in the absence of corresponding analytic Type 1 formulation.

Finally the most comprehensive understanding – in what I refer to as the Type 3 zeta function - entails the simultaneous interaction of both Type 1 and Type 2 formulations in a manner ultimately approaching purely ineffable understanding!

1) An analytic aspect where the fundamental poles of understanding (i.e. internal and internal and quantitative and qualitative are clearly separated). This indeed is what makes such Mathematics linear (i.e. 1-dimensional) in nature as it literally interprets its symbols exclusively – in any context - in terms of the dominance of just one pole of understanding.

So relationships are considered absolutely in terms of the objective nature of symbols (where the external pole dominates); equally relationships are considered absolutely in terms of the quantitative nature of symbols (where the qualitative is thereby reduced to the quantitative).

(It is important to appreciate that when I use the word analytic, typically I use it is this general context of absolute uni-polar interpretation of symbols and not in the narrower more specialised conventional mathematical sense relating to infinite series).

I refer to this aspect of Mathematics as Type 1.

2) A holistic aspect where the same fundamental poles of understanding are now considered as dynamically related. In this new distinctive interpretation the objective nature of mathematical symbols strictly has no meaning in the absence of their corresponding means of (subjective) mental interpretation.

And as a wide variety of partially valid interpretations can be employed the objective nature of such symbols is thereby necessarily of a merely relative nature.

Likewise the quantitative interpretation of symbols strictly has no means in the absence of qualitative appreciation (of a distinctive holistic variety).

To be more accurate the holistic aspect properly refers to the dynamic interaction of poles that are (a) external and internal and (b) quantitative and qualitative with respect to each other.

Likewise the analytic aspect refers to the extreme situation where such interaction is ignored through reducing one entirely in terms of the other. And again in Conventional Mathematics the internal aspect is reduced in terms of the external and the qualitative in terms of the quantitative!

However for convenience I continually contrast analytic and holistic (as shorthand for there respectively different ways of treating such polar interaction)!

I refer to this aspect of Mathematics as Type 2.

3) The actual experience of Mathematics entails the mutual dynamic interaction of both Type 1 and Type 2 understanding. Though customary objective quantitative distinctions of a valid nature can indeed be made, as the polar reference frames to which they relate continually change they are now understood in a merely relative manner.

I refer to this most comprehensive aspect of Mathematics as Type 3.

So properly understood (i.e. in a comprehensive manner) all mathematical relationships are necessarily understood in merely relative terms. The customary fixation with absolute type relationships in Conventional Mathematics reflects its reduced – and thereby limited – (1-dimensional) manner of interpretation.

The distinction as between the Type 1 and Type 2 aspects Mathematics leads to a corresponding distinction with respect to the number system.

For example the natural numbers in Type 1 terms are understood as representing cardinal whole units. So 2 in cardinal terms represent a collective whole unit (i.e. an integer).

The problem with this quantitative approach however is that when one attempts to describe the individual members of 2, one must treat them in homogenous terms as without qualitative distinction. So 2 = 1 + 1.

However this leaves one with the considerable problem of having no means of making a meaningful distinction as between the 1st and 2nd members of 2 (in ordinal terms).

The fact that this fundamental issue is completely glossed over in Conventional Mathematics clearly reflects its reduced nature i.e. where the qualitative (ordinal) nature of 1st and 2nd is misleadingly assumed to be implied by the quantitative (cardinal) understanding of 1 and 2!

So the Type 2 understanding of number resembles – as it were - a sub-atomic approach where each of the individual members of a number group are now given a unique individual identity in relative terms.

So once again from a Type 1 approach, 2 is treated from a cardinal perspective as a collective whole unit (in quantitative terms).

From the complementary Type 2 approach, 2 is treated from an ordinal perspective as comprising unique individual units i.e. 1st and 2nd (in qualitative terms).

Now Type 1 is – literally - defined in 1-dimensional terms, where each natural number is implicitly defined with respect to 1 as exponent (i.e. dimensional number).

So the natural numbers 1, 2, 3, 4,…. are defined in Type 1 terms as

1^1, 2^1, 3^ 1. 4^1,….

The 1-dimensional nature of this approach is clearly illustrated by the fact that any number initially raised to another exponent (dimensional number) will be given a reduced linear interpretation i.e. where its ultimate value is expressed in reduced quantitative terms.

So 2^2 in Type 1 = 4^1!

The Type 1 approach is properly geared to explain the fundamental nature of addition where quantitative change occurs (without qualitative transformation)!

So 2 + 3 in this additive system = 2^ 1 + 3^ 1 = 5^1.

In the Type 2 aspect of the number system, 2 is by contrast is defined with respect to 1 (representing a default base number quantity).

So the natural number 1, 2, 3, 4,….. are defined from a Type 2 perspective as

1^1, 1^2, 1^3, 1^4,…..

Just as the Type 1 approach is geared to the pure nature of addition (where no qualitative transformation in the variables takes place), The Type 2 approach is then geared to the pure nature of multiplication (where no quantitative transformation in the variables takes place).

So 2 + 3 in this system represents 1^2 * 1^ 3 = 1 ^ (2 + 3) = 1^5.

So what appears as multiplication from the Type 1 perspective represents addition from the corresponding Type 2 perspective.

However to properly express the nature of the Type 2 aspect we need to move to a circular number system (based on successive roots of unity).

The key to understanding the nature of this system is that a two-way interaction is necessarily entailed as between variables given both an analytic and holistic interpretation respectively.

Thus for example the number 2 in type 2 terms = 1^2.

2 here is identified with the two individual unique dimensions of 2 i.e. its 1st and 2nd dimensions (with a holistic interpretation as interdependent).

Now in analytic terms these two dimensions can be given expression as the 1st and 2nd roots of 1 respectively or more correctly the roots of 1^2 and 1^1 respectively.

So the first root of 1 is therefore (1^2) raised to the power ½ = + 1 (i.e. 1^1).

The second root of 1 is (1^1) raised to the power of ½ = – 1.

However in dynamic terms these two separate values (as analytically understood) are in continual relationship with the corresponding interdependent value (as holistically understood).

So the quantitative independent interpretation of 1 and – 1 as separate strictly have no meaning in dynamic relative terms apart from the combined holistic interpretation of 1 and – 1 as interdependent.

Once again I have repeatedly used the example of a crossroads to illustrate this type of understanding. When understood (within independent reference frames) left and right turns at a crossroads can be given a – relatively – separate interpretation which are + 1 and – 1 with respect to each other. (1 refers here to the dimensional unit i.e. representing a direction).

However the appreciation that both left and right ultimately have a purely relative i.e. paradoxical meaning (when both frames are simultaneously combined) represents the true qualitative holistic nature of 2!

And this could be represented quantitatively as 1 – 1 = 0, i.e. as strictly without quantitative significance!

So in the actual understanding of the nature of a direction at a crossroads, both types of understanding (analytic and holistic) are necessarily combined.

Indeed this 2-dimensional type of dynamic understanding is well-recognised in spiritual literature (e.g. Taoism and Buddhism) in philosophy (e.g. Heraclitus and Hegel) in psychology (e.g. Jung) and indeed even indirectly in quantum physics (e.g. wave-particle duality).

What is vital to appreciate however is that it necessarily combines both analytic and holistic aspects of understanding in a complementary fashion.

Now clearly this cannot be appreciated in Type 1 terms (where holistic notions are reduced to analytic in a static absolute manner).

As Conventional Mathematics is fundamentally based on such absolute notions it is severely limited in terms of dealing with the very notion of interdependence.

And as the nature of prime numbers likewise involves such interdependence e.g. in the relationship of the primes - constituting the natural number system in cardinal terms - to the non-trivial zeta zeros, it is fatally flawed in terms of appreciating the true nature of the number system.

Another key point is that this interpretation of the number 2 from a Type 2 perspective represents but the simplest version of Type 2 understanding.

Indeed associated with every number as an exponent (i.e. dimensional number) is a unique means of configuring the dynamic interaction as between its analytic and holistic elements! Or using the shorthand means that I have been employing every number has a unique holistic significance serving as a distinct means of interpretation of the dynamic interaction of polarities (underlying all mathematical understanding).

In this holistic context 1 (as 1-dimensional) has an extreme limiting interpretation where the holistic aspect is directly reduced to analytic interpretation.

And this is what precisely defines the nature of Conventional Mathematics which is based on absolute objective interpretation of its symbols in a merely (reduced) quantitative manner!

However in truth in a more comprehensive vision of Mathematics, an unlimited number of possible interpretations exist (all with a partial relative validity). And in all these other systems (based on a dimensional number ≠ 1).

And if you understand this you will immediately understand why from a holistic perspective, the Riemann zeta function is uniquely undefined for s = 1 (as this is the one value where the inherent dynamic interaction as between the analytic and holistic aspects of the number system is broken).

So once again, putting it bluntly the Riemann zeta function cannot be properly interpreted in conventional mathematical terms. And of course neither can the Riemann Hypothesis (which is fundamentally based on true appropriate interpretation of the Riemann Function).

Finally it is futile therefore attempting to prove the Riemann Hypothesis from a conventional (1-dimensional) perspective!

However fortunately, the key to appreciating the nature of the Riemann Hypothesis can be related to the simplest case of Type 2 understanding (with respect to the number 2).

Once again the most comprehensive form of mathematical understanding involves the interaction of both Type 1 and Type 2 aspects where both analytic (as separate) and holistic appreciation can reach a high level of refinement.

In fact whenever a non-unitary base number is raised to a non-unitary power, both Type 1 and Type 2 aspects of interpretation are both required.

So in the simplest case of 2^2, strictly both a quantitative (analytic) and qualitative (holistic) transformation is entailed the mutual interaction of which entails Type 3 understanding.

Next we find that associated with the Type 1 and Type 2 aspects of the number system are corresponding Type 1 and Type 2 zeta functions.

Now the Riemann zeta function refers strictly to the Type 1 aspect; however proper appreciation of the Riemann zeta function and its associated Riemann Hypothesis requires incorporation of a complementary Type 2 zeta function.

Though the conventional Riemann zeta function can indeed be given an analytic Type 1 formulation (as the Zeta 1 function), strictly this has no proper meaning in the absence of corresponding holistic Type 2 appreciation.

In like manner through the Riemann zeta function can be given an – unrecognised – Type 2 formulation (as the Zeta 2 function), strictly again this has no proper meaning in the absence of corresponding analytic Type 1 formulation.

Finally the most comprehensive understanding – in what I refer to as the Type 3 zeta function - entails the simultaneous interaction of both Type 1 and Type 2 formulations in a manner ultimately approaching purely ineffable understanding!

## Thursday, March 7, 2013

### In a Nutshell!

As we have seen natural numbers can be given two distinct definitions (in relative isolation from each other):

1. In cardinal terms as a collective homogeneous group (i.e. where individual units have no unique identity). This relates to the analytic (quantitative) aspect of number which I refer to as the Type 1 interpretation.

2. In ordinal terms as the relationship between unique individual members of a number group (where the collective group - from this perspective - has no distinct identity). This relates to the holistic (qualitative) aspect of number which I refer to as the Type 2 interpretation.

In actual experience, both of these aspects of number continually interact. Strictly speaking therefore, we cannot form an analytic appreciation of the nature of number without its corresponding holistic aspect; likewise we cannot form a holistic appreciation of number without its corresponding analytic aspect.

Ultimately both analytic and holistic aspects are of a purely relative nature.

Though such pure relativity relates to an absolute ineffable state, in the phenomenal experience of number it can only be approximated.

And properly understood this is what the Riemann Hypothesis relates to as the closest approximation in phenomenal terms to the ultimate ineffable state where both the analytic (quantitative) and holistic (qualitative) aspects of number are fully identical.

I have already explained in an earlier blog how the non-trivial zeros - corresponding to the Type 1 interpretation - represent in fact the perfect shadow number system to the primes (that comprise the natural number system - except 1 - in cardinal terms).

Equally the non-trivial zeros – corresponding to the Type 2 interpretation – represent the perfect shadow number system to the natural numbers (that comprise each prime group - except 1 - in ordinal terms).

Properly understood therefore, both the cardinal and ordinal numbers have no meaning in the absence of their corresponding shadow (wave-like) number systems as represented by the non-trivial zeros in both cases.

So both these particle-like features of number (as independent) and their corresponding wave-like properties (as interdependent) are simultaneously co-determined in a manner that is ultimately ineffable.

Put another way both the Type 1 and Type 2 aspects of the number system have a particle and wave-like identity reflecting their analytic and holistic aspects respectively.

From a psychological perspective this implies that all numbers have both conscious aspects (as analytic) and unconscious aspects (as holistic) respectively.

Now just as in quantum physics through interaction, every particle possesses wave-like properties and every wave particle-like properties, likewise with respect to number, the analytic (Type 1) interpretation of number possesses holistic (Type 2) properties and the holistic (Type 2) interpretation analytic properties respectively.

This means - as I have repeatedly stated – that we cannot possibly hope to understand the nature of the Type 1 (quantitative) non-trivial zeros – to which the Riemann Hypothesis directly relates - in the absence of Type 2 (qualitative) interpretation.

Equally, we cannot possibly hope to understand the nature of the Type 2 (qualitative) non-trivial zeros - to which an unrecognised counterpart Riemann Type Hypothesis relates - in the absence of Type 1 (quantitative) interpretation.

And I have indicated in several of my blogs what such a quantitative interpretation entails.

The basic message is that the inherent nature of the number system (and indeed of all mathematical activity) is of a dynamic relative nature entailing the interaction of both its quantitative (analytic) and qualitative (holistic) aspects.

It therefore cannot possibly be understood within the current mathematical paradigm, which is geared formally to - mere - quantitative interpretation!

In a nutshell this is the essential message underlying all my blog entries and urgently needs to be grasped!

So every mathematical symbol with a standard (Type 1) quantitative interpretation in conventional mathematical terms possesses an equally important (Type 2) qualitative interpretation from a holistic perspective.

Properly understood therefore mathematical meaning necessarily represents the combined interaction of both perspectives.

Thus once again to illustrate the number 2 has a standard quantitative interpretation in linear mathematical terms (representing the separate collective whole identity of a homogeneous set of units in a cardinal manner).

However the number 2 equally has a qualitative interpretation in circular terms (representing the interdependence of the two unique individual members of a group in an ordinal manner).

Likewise the operations + and – have a recognised quantitative meaning in standard mathematical terms; however they equally have a - largely unrecognised - qualitative meaning! So + in this context relates to the direct positing of meaning in a (conscious) rational fashion. – relates to the corresponding negation of such rational meaning in an (unconscious) intuitive manner.

So deeply implicit in the holistic notion of 2 is the interaction of two polar directions of experience that are positive and negative with respect to each other.

Indeed deeply implicit in the holistic notion of 1 is the existence of just one direction of experience that is positive (in an absolute manner).

So linear (1-dimensional) rational understanding with just one direction, necessarily leads to the reduction of holistic to analytic (and unconscious to conscious) interpretation respectively.

Once again though indeed it represents an important special case, in holistic terms, 1 represents the only dimension that interprets mathematical meaning in an absolute manner.

For all other numbers (as dimensions) a dynamic relationship as between analytic and holistic (conscious and unconscious) is implied.

Once again this is the Type 2 explanation as to why the Riemann Zeta Function remains uniquely undefined for s = 1!

As the Function - when properly interpreted - relates to the relationship as between analytic and holistic type meaning, it thereby remains undefined in conventional (1-dimensional) terms, where analytic type meaning is solely recognised in an absolute manner!

Therefore it is ultimately futile - not only in trying to prove - but more importantly in even trying to understand the Riemann Hypothesis from this limited perspective.

1. In cardinal terms as a collective homogeneous group (i.e. where individual units have no unique identity). This relates to the analytic (quantitative) aspect of number which I refer to as the Type 1 interpretation.

2. In ordinal terms as the relationship between unique individual members of a number group (where the collective group - from this perspective - has no distinct identity). This relates to the holistic (qualitative) aspect of number which I refer to as the Type 2 interpretation.

In actual experience, both of these aspects of number continually interact. Strictly speaking therefore, we cannot form an analytic appreciation of the nature of number without its corresponding holistic aspect; likewise we cannot form a holistic appreciation of number without its corresponding analytic aspect.

Ultimately both analytic and holistic aspects are of a purely relative nature.

Though such pure relativity relates to an absolute ineffable state, in the phenomenal experience of number it can only be approximated.

And properly understood this is what the Riemann Hypothesis relates to as the closest approximation in phenomenal terms to the ultimate ineffable state where both the analytic (quantitative) and holistic (qualitative) aspects of number are fully identical.

I have already explained in an earlier blog how the non-trivial zeros - corresponding to the Type 1 interpretation - represent in fact the perfect shadow number system to the primes (that comprise the natural number system - except 1 - in cardinal terms).

Equally the non-trivial zeros – corresponding to the Type 2 interpretation – represent the perfect shadow number system to the natural numbers (that comprise each prime group - except 1 - in ordinal terms).

Properly understood therefore, both the cardinal and ordinal numbers have no meaning in the absence of their corresponding shadow (wave-like) number systems as represented by the non-trivial zeros in both cases.

So both these particle-like features of number (as independent) and their corresponding wave-like properties (as interdependent) are simultaneously co-determined in a manner that is ultimately ineffable.

Put another way both the Type 1 and Type 2 aspects of the number system have a particle and wave-like identity reflecting their analytic and holistic aspects respectively.

From a psychological perspective this implies that all numbers have both conscious aspects (as analytic) and unconscious aspects (as holistic) respectively.

Now just as in quantum physics through interaction, every particle possesses wave-like properties and every wave particle-like properties, likewise with respect to number, the analytic (Type 1) interpretation of number possesses holistic (Type 2) properties and the holistic (Type 2) interpretation analytic properties respectively.

This means - as I have repeatedly stated – that we cannot possibly hope to understand the nature of the Type 1 (quantitative) non-trivial zeros – to which the Riemann Hypothesis directly relates - in the absence of Type 2 (qualitative) interpretation.

Equally, we cannot possibly hope to understand the nature of the Type 2 (qualitative) non-trivial zeros - to which an unrecognised counterpart Riemann Type Hypothesis relates - in the absence of Type 1 (quantitative) interpretation.

And I have indicated in several of my blogs what such a quantitative interpretation entails.

The basic message is that the inherent nature of the number system (and indeed of all mathematical activity) is of a dynamic relative nature entailing the interaction of both its quantitative (analytic) and qualitative (holistic) aspects.

It therefore cannot possibly be understood within the current mathematical paradigm, which is geared formally to - mere - quantitative interpretation!

In a nutshell this is the essential message underlying all my blog entries and urgently needs to be grasped!

So every mathematical symbol with a standard (Type 1) quantitative interpretation in conventional mathematical terms possesses an equally important (Type 2) qualitative interpretation from a holistic perspective.

Properly understood therefore mathematical meaning necessarily represents the combined interaction of both perspectives.

Thus once again to illustrate the number 2 has a standard quantitative interpretation in linear mathematical terms (representing the separate collective whole identity of a homogeneous set of units in a cardinal manner).

However the number 2 equally has a qualitative interpretation in circular terms (representing the interdependence of the two unique individual members of a group in an ordinal manner).

Likewise the operations + and – have a recognised quantitative meaning in standard mathematical terms; however they equally have a - largely unrecognised - qualitative meaning! So + in this context relates to the direct positing of meaning in a (conscious) rational fashion. – relates to the corresponding negation of such rational meaning in an (unconscious) intuitive manner.

So deeply implicit in the holistic notion of 2 is the interaction of two polar directions of experience that are positive and negative with respect to each other.

Indeed deeply implicit in the holistic notion of 1 is the existence of just one direction of experience that is positive (in an absolute manner).

So linear (1-dimensional) rational understanding with just one direction, necessarily leads to the reduction of holistic to analytic (and unconscious to conscious) interpretation respectively.

Once again though indeed it represents an important special case, in holistic terms, 1 represents the only dimension that interprets mathematical meaning in an absolute manner.

For all other numbers (as dimensions) a dynamic relationship as between analytic and holistic (conscious and unconscious) is implied.

Once again this is the Type 2 explanation as to why the Riemann Zeta Function remains uniquely undefined for s = 1!

As the Function - when properly interpreted - relates to the relationship as between analytic and holistic type meaning, it thereby remains undefined in conventional (1-dimensional) terms, where analytic type meaning is solely recognised in an absolute manner!

Therefore it is ultimately futile - not only in trying to prove - but more importantly in even trying to understand the Riemann Hypothesis from this limited perspective.

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