Riemann Zeta Function: Important Number Relationships (10)

We earlier looked at how the Riemann zeta function (for positive integer values of s) has a role to play with respect to the maximum frequency that any prime can occur (with respect to the unique factor compositions of the natural numbers).

We concluded then the probability that a number chosen at random will have a factor composition with at least one prime repeating n or more times  = 1 – 1/ζ(n).
Alternatively we can state that the probability that a number chosen at random will not contain a factor composition where at least one prime repeats n or more times = 1/ζ(n).

Therefore, for example, the probability that such a random number will have at least one prime factor repeating 2 or more times = 1 – 1/ζ(2) = 1 – .6079 = .3921 (approx).

Alternatively we could express this by stating that that the probability that such a number will have no prime factor repeating more than once =  1/ζ(2) = .6079 (approx).

We also derived expressions - again based on the Riemann zeta function for the positive integer values - that express the probability that a number chosen at random will contain at most n repeating prime factors.

So again this general probability is given as 1/ζ(n + 1) – 1/ζ(n).

So therefore to illustrate, the probability that a number chosen at random will contain at most 2 repeating prime factors = 1/ζ(3) – 1/ζ(2)  = .8319 - .6079 = .224 (approx).

However these measurements are confined to the external aspect of the number system. So basically in this case, we are trying to ascertain the relative frequency (with respect to the overall system) that certain numbers occur with such stipulated prime factor compositions.

Therefore we can state - using our most recent example - that 22.4% of all numbers will contain factor combinations, where at most one prime repeats twice!

But as I was anxious to emphasise in my previous entries, based on recognition of both the Type 1 and Type 2 aspects of the number system, that a complementary interpretation (entailing the Riemann zeta function for the positive integers) will also necessarily exist in an internal number fashion.

So therefore in this context, we are looking at the frequency with respect to the overall occurrence of factors associated with such stipulated prime combinations.

Now, there is little point in trying to count in absolute terms the total number of factors associated  (as this total will necessarily increase without limit for numbers associated with all factor compositions).

However we can make meaningful comparisons by comparing the average number of factors for each prime combination (at a given appropriately high level of the number system).

We can then for example compare the average number of factors belonging to those numbers which contain - say - at most a prime that occurs just once (i.e. no repeating prime) with the average count of factors belonging to those numbers where one (or more) primes repeat at most 2 times.

Now one would reasonably expect to find, on average, more factors belonging to those numbers where one or more primes can repeat twice compared to those where no prime factor can repeat more than once!

And this ratio is given it would appear (as based on preliminary sample evidence) as ζ(2).

Alternatively in reverse, the average ratio of the total frequency of factors associated with those numbers, where no prime repeats more than once, in relation to the total frequency of factors associated with numbers where one (or more) primes repeat at most twice = 1/ζ(2).

And in like fashion, the average ratio of the total frequency of factors associated with numbers where  one (or more) primes repeat at most 2 times in relation to the total frequency of factors for numbers where one (or more) primes repeat at most 3 times = 1/ ζ(3).

And in general terms, the average ratio of the total frequency of factors associated with numbers where one or more primes repeat at most (n – 1) times in relation to the total frequency of factors for numbers where one (or more) primes repeat at most n times =  1/ ζ(n).

And as ζ(n) → 1 (for large n), this means that the corresponding ratio in relation to the total frequency of factors would likewise → 1. So for example is we were to count up the total frequency of factors in a certain  region (appropriately high up the number scale) for all those numbers, where at most one or more primes repeat - say - 7 times and then count up the corresponding total of factors for the same amount of numbers where at most one or more primes repeat 8 times, there would be little difference with respect to the overall total of factors in each case! 

We can also get equivalent expressions where the stipulation - with relation to the factor composition - is that a prime (or primes) repeats more than n times.

Then from this perspective, the average ratio of the total frequency of factors for numbers where a prime (or primes) repeats 1 time or more (which represents 100% of numbers) in relation to the total frequency of factors where a prime (or primes) repeats 2 times or more = {1/ζ(2)}/{1/ζ(3)} = ζ(3)/ζ2).

And in general terms the average ratio of the total frequency of factors for numbers where a prime (or primes) repeats (n – 1) times or more, in relation to the total frequency of factors where a prime (or primes) repeats n times or more = ζ(n + 1)/ζ(n).  

Riemann Zeta Function: Important Number Relationships (9)

We were discussing yesterday the Riemann zeta function for ζ (s) where s = – 1.

I was at pains then to point out that appreciation of what is involved, requires incorporation of a distinctive holistic notion (literally of the qualitative meaning of "whole") which unfortunately is completely edited out from conventional mathematical understanding.

In other words due to its reduced quantitative nature, "the whole" in any numerical context is viewed as merely the sum of its constituent parts.

It might be possible here to provide some insight into the true holistic meaning of s = – 1, with reference to - what initially might appear - a very simple example.

As we have seen in the conventional (Type 1) approach to Mathematics number is defined in linear terms i.e. with respect to the default dimensional number (as power or index) = 1.

So 3 for example in this context can be more fully expressed as 3¹.
 
Now something fascinating happens when we now negate this dimension of 1 i.e. by raising 3 to – 1.

So 3^{– 1} = 1/3.

Therefore enshrined in this simple relationship is the central mystery regarding the relationship between whole and part.

To make this more accessible, let us consider the example of a cake that is divided into 3 (equal) slices.
We will look initially at interpretation of the relationship between the (part) slices and (whole) cake from the conventional mathematical perspective.

As we have seen, this conventional approach is based on the notion of quantitative reductionism.

Therefore in this context the (whole) cake is equated merely as the sum of its 3 (part) slices.

Therefore in this context the whole cake (as the quantitative sum of its component slices) = 3.

Each part in turn (as a quantitative fraction of the cake) = 1/3.

However the true relationship here as between whole and part is of a much subtler nature (entailing both quantitative and qualitative notions).

The key here is the recognition that the whole has a distinctive qualitative - as well as quantitative - meaning.

The quantitative notion is related directly to the recognition of independent identity with respect to its component parts. So we little difficulty in recognising in this context that each of the slices assumes an individual quantitative identity, so that for example we can then serve each of the slices to 3 different people.

However each of the parts also necessarily enjoys a common qualitative identity, which is based on their interdependent relationship with respect to the whole cake.

For example it is this distinctive qualitative identity that enables one to identify a 1st, 2nd and 3rd slice, which depends purely on relative context. In other words, each of the 3 slices can be identified as 1st, 2nd or 3rd depending on the arbitrary method of choosing the slices.

Now, when one recognises this distinction as between the quantitative identity of the slices (as relatively independent of each other in an individual manner) and the qualitative identity of the slices (through their common shared interdependence with the whole cake), the relationship in turn as between parts and whole (and whole and parts) is crucially transformed.

So basically now in this dynamic interactive mode of understanding, the parts and the whole are understood as in complementary fashion as quantitative and qualitative with respect to each other.

Remarkably what this entails is that number itself - using the language of quantum physics - is now understood to have complementary particle and wave aspects.

Thus we can identify the cake in quantitative terms as identical with its 3 component slices.

So therefore the cake = 3 (in this particle context).

However equally we can identify the whole cake as expressive of the collective interdependence of its component slices.

So therefore the cake = 1 (in this wavelength context).

Therefore when we understand this simple example in appropriate fashion, we realise that a continual transition takes place as between the use of number with respect to both quantitative and qualitative interpretation.

Or to put it in somewhat equivalent terms, one keeps switching as between an analytic appreciation of number (in quantitative terms) and a corresponding holistic appreciation of number (in a qualitative manner).

Therefore in the dynamics of understanding, if we start with recognition of
3 quantitative slices (i.e. 3^{– 1}, a crucial qualitative recognition is required before one can now switch to the corresponding recognition of each slice representing a fraction of the (whole) cake. And this entails implicit understanding of the qualitative meaning of s = – 1.

So in psycho spiritual terms the negation of 1 as dimension implies an unconscious intuitive recognition of the notion of the whole (in a qualitative manner). And implicit recognition of this qualitative aspect of understanding is vital in enabling one to switch from quantitative recognition of the combined 3 slices to corresponding quantitative recognition of each slice as existing as part of the combined whole.

However though implicitly this qualitative recognition (entailing unconscious intuitive insight) is involved, explicitly in conventional mathematical terms, a merely reduced interpretation is given.

So the great mystery of the true relationship as between whole and part is lost here with merely a quantitative reductionist interpretation remaining.

Of course it is equally true in reverse. When we are now aware in quantitative terms of each slice (as a fractional part of  the cake) it implicitly requires a corresponding qualitative recognition of the whole, so as to be able to switch from quantitative recognition of each individual part to corresponding quantitative recognition of the combined 3 slices.

So (1/3)^{– 1} = 3 ¹.

In fact all this opens up an entirely new holistic appreciation in physical (and psychological) terms of the true nature of space and time.

So quite remarkably, every number (and sign) with a recognised quantitative meaning in conventional mathematical terms, can equally be given a corresponding qualitative interpretation which intimately relates to the dynamic manner in which space and time are experienced (again in both physical and psychological terms).

Thus in this context, I have explained the holistic psychological significance of the negative 1st dimension as intimately involved in the - apparently - simple act of concrete number recognition.

Indeed one could sum up true appreciation of the holistic mathematical significance of the negative 1st dimension, as the keen realisation that the whole (in any context) has a distinctive qualitative meaning that cannot be reduced in terms of its quantitative parts.


As I stated in another context (which in fact is related), this is deeply relevant in terms of the spiritual life. Here with the onset of authentic contemplative progress, customary dualistic distinctions (through which quantitatively reduced notions are expressed) begin to break down in dramatic fashion. So through a greatly refined intuitive recognition, one begins to truly discover a new qualitative appreciation of  "the whole".

Again this is deeply relevant in terms of appreciating the Riemann zeta function for s = – 1, where we are confronted with a series (i.e. the sum of the natural numbers).

Now due to conventional training (in unquestioned acceptance of the reduced quantitative approach) one immediately tries to interpret this series in quantitative terms and this clearly appears (from this perspective) to diverge to infinity.

However the clue as to its finite interpretation = – 1/12, lies in holistic rather than strict analytic appreciation.

Here is yet another clue as to what dimensional interpretation according to the negative 1st dimension entails.

Now in conventional mathematical terms, there is strong belief in the abstract validity of mathematical "objects" such as numbers (and especially the primes).

And in holistic mathematical terms, this equates with the conscious (positive) direction of experience.

However strictly speaking mathematical "objects" can have no meaning independent of our mental interaction with them. And relative to these "objects", the subjective mental constructs we use in their understanding equate with the negative (unconscious) direction.

So the development in such "negative consciousness" leads to a growing appreciation of the holistic dimension underlying conventional analytic type understanding.

In particular one recognises that what is known objectively as "mathematical truth" necessarily reflects a particular limited interpretation of reality (in which such truth appears unambiguous).

And as we have seen Mathematics is characterised to an extraordinary degree by interpretation that corresponds qualitatively to the positive 1st dimension, in what perhaps more simply can be expressed as linear rational understanding. For example this is well represented through the very notion of a number line, on which all real numbers are supposed to lie!

However when one begins to understand with respect to the negative 1st dimension, one realises that this standard version of mathematical truth represents just one limited interpretation (i.e. according to the positive 1st dimension). In particular, one now recognises that what chiefly characterises such understanding is whole/part reductionism in quantitative terms i.e. where the whole in any context is viewed as the quantitative total of its various parts.

Thus when one starts experiencing in terms of the negative 1st dimension, an important inversion takes place whereby now truth becomes defined in true holistic (i.e. whole) terms as the relationship of interdependent parts to an overall qualitative whole.

And this is directly relevant to the Riemann zeta function ζ(s ) where s = – 1.

So the correct way of interpreting the sum of the resulting series i.e. 1 + 2 + 3 + 4 + ... result is not in terms of its quantitative sum, but rather in terms of the relationship in some unique context of constituent parts (of an interdependent nature) to an overall collective whole in qualitative terms.

Now the problem with concrete macro objects is that parts immediately assume a recognisable independence in their own right, thus encouraging our typical understanding of the relationship as between whole and parts in a reduced quantitative manner.

So when we divided in our example the whole cake into 3 constituent slices, each individual slice can be seen to possess a recognisable quantitative independence. Therefore we are enabled to overlook the fact that a corresponding qualitative interdependence must necessarily apply to the relationship between the three slices.

However at the quantum physical level, it is very different in that sub-atomic particles lose any distinct independent identity and therefore can only be properly understood in relation to an overall qualitative whole context.

Therefore, despite the great advances made at the analytic level in terms of the understanding of quantum mechanical relationships, a truly major problem remains a failure to properly introduce holistic qualitative notions into the interpretation of particle interactions.

Thus quantum mechanics continues to appear so non-intuitive, precisely because we insist on an interpretation that has been heavily conditioned by the conventional mathematical paradigm (that is of a limited 1-dimensional nature).


However indirectly, holistic ideas can shown to be at work.

For example the understanding of the behaviour of strings (in the case of bosonic strings) is strongly linked with the Riemann zeta function for s = – 1.

Now here a string can vibrate in 24 different vertical directions based on a 2-dimensional horizontal surface. And this is associated with the finding that this earlier version of strings was found to be only consistent in 26 dimensions.

So these dimensions here provide an overall holistic context in which the physical behaviour of the strings can operate in a consistent manner.

Thus, if one then attempted to break up this whole in terms of a single string, this would represent 1/24 of the total (which is just half of 1/12).
The negative sign here could then signify the inverted understanding where the part is now being derived from the overall whole (in qualitative dimensional terms) whereas conventionally in linear (1-dimensional terms) the whole is derived from the constituent parts (in a quantitative manner).

So 12 and 24 are "magic" numbers that arise in a variety of physical circumstances such as quantum physics and the monster group. Thus special properties of holistic symmetry (which are of a qualitative nature) are associated with these numbers.
And the Riemann zeta function (for s = – 1) is directly related to these holistic qualitative notions of symmetry. 

Some time ago I became aware of a fascinating alternative example with direct relevance to psychology.

I had been studying Jung and become especially interested in his treatment of personality types. 

Though Jung was not a mathematician in the conventional sense, I found that many of his ideas could be readily formulated in a holistic mathematical manner.

Now Jung defined 8 personality types which in the well known Myers-Briggs type Indicator was then enlarged to 16 (that was heavily based on his own work).

However, having studied it deeply for some time, I gradually became convinced that there were 8 more "missing" personality types.

So I then devised a new system of 24 Personality Types based directly on holistic mathematical notions with each offering a valid - though necessarily limited  - perspective on the understanding of reality.

Basically in this approach I start with 4 fundamental dimensions (in accordance with the holistic complex meaning of the 4 roots of 1). 

Then each personality type is viewed as a unique permutation with respect to the four original dimensions. And as 4P₄ = 24, this entails that 24 relatively distinct Personality Types can be defined.

In then struck me that each of these types represents a way of configuring the experience of space and time.

This then led to a new appreciation of dimension (in this context) as a unique configuration with respect to the original 4 dimensions. 

I then realised that this would necessarily have a complementary interpretation in physical terms. So the 24 vibrations of the bosonic string, would thereby likewise represent unique permutations with respect to the 4 original dimensions. Clearly at the string level, space and time would remain entangled with each other in an interdependent fashion. Only later would they become sufficiently separated to be viewed as 4 dimensions in an independent manner.

So here we have two domains physical and psychological connected through the same holistic notion of dimensions.

Interestingly in the Jungian approach and the later Myers- Briggs approach, an even balance is maintained as between the positive (conscious) types (designated as S types) and the negative (unconscious) types (designated as N types).  So S types identify directly with actual reality in a conscious manner; N types relate to potential (holistic) reality in a more unconscious manner.

Now if we extend this to the 24 types then 12 would lean towards conscious and another 12 to unconscious orientation. And in holistic mathematical terms these would be + and – with respect to each other.

Thus if we start from standard appreciation of the world (especially in mathematical terms) as S based, true holistic appreciation would be N based.

So for the 12 types that identify with the conscious aspect (+ 12) there would be another 12 that are identify more with the corresponding unconscious aspect (– 12).

–1/12 would then arise in this context as the attempt to express one of the 12 personality types (geared to holistic type appreciation) as an analytic fraction of all 12 (which in truth are highly interrelated as one integral whole).

So the key point here is that the numerical result – 1/12 properly relates to a holistic type relationship of a qualitative nature (though indirectly employing the analytic notion of a fraction).

This is likewise the case with the bosonic string. When the string is postulated to vibrate in 24 directions, this relates directly to a holistic qualitative capacity (associated with the string).

Therefore the fractional relationship of the vibration of one string in terms of the total (where all are highly interrelated) represents directly a qualitative - rather than quantitative - relationship.

We have dealt here with the important case of ζ(s ) where s = – 1.

And what we have found is that a unique interpretation, entailing holistic qualitative and (indirect) analytic quantitative notions are associated with this relationship.

So a unique interpretation attaches to s = – 1 in the context of the Riemann zeta function.

Equally a unique holistic relationship attaches to s for every odd integer value. And in general, the resulting fractional results that emerge are associated with high levels of symmetry with respect to holistic qualitative type relationships (i.e. where the whole in any context can only be meaningfully understood in terms of the interdependence of its constituent parts).  

Though in some respect the values of ζ(s) for the negative odd integers appear somewhat arbitrary, this is not the case.

In fact I believe that a remarkable feature is associated with the denominators of all these results that directly relates to the very nature of the primes 

Once again in conventional terms, we can demonstrate that a number is prime, by attempting to break it into constituent factors. Then if no factors can be found other that the number itself and 1, then that number is prime.

However the Riemann zeta function for the negative odd integers turns this process on its head, whereby we can now - in principle - show that a number is prime if it is a factor of a certain number (which is given by the denominator of the corresponding values for the Riemann zeta function).

Now through the Riemann zeta function, the value of ζ(1 – s) is related to the corresponding value of ζ(s).

So for example the value of ζ(– 1) is related to ζ(2).

Then in general terms, I postulate this simple rule.

In the case of the odd integers if the absolute value of the denominator of ζ(1 – s) is divisible by s + 1, then s + 1 is prime.

So in the case of s = 2, ζ(1 – s) =  ζ(– 1) with  the absolute value of its denominator = 12. 

And as 12 is divisible by 3, therefore 3 is a prime!

Then in the case of s = 4, ζ(1 – s) =  ζ(– 3) with  the absolute value of its denominator = 120. 

And 120 is divisible by s + 1 (= 5). So 5 is prime

And continuing, in the case of s = 6, ζ(1 – s) =  ζ(– 5) with  the absolute value of its denominator = 252.

And as 252 is divisible by s + 1 (= 7), then 7 is a prime.

Finally to illustrate in the case of s = 8, ζ(1 – s) =  ζ(– 7) with  the absolute value of its denominator = 240.  

And as 240 is not divisible by s + 1 (= 9), then 9 is not prime!

I was able to check these values for values of s up to 200, with no exceptions arising. However as the numerators get truly enormous for higher odd integer values of s, it is not possible to keep checking at higher values.

So as a practical way of finding primes, this approach would be of little value.

However in principle, it remains very important as it seems that the unique nature of denominator values of the Riemann zeta function (for odd integers) is determined by this inverse qualitative nature of prime identity.

In other words, as I have pointed out before from the quantitative analytic perspective, we view the primes as the quantitative building blocks of the natural numbers.

However from the corresponding holistic perspective, this relationship is reversed with the qualitative nature of the primes (through their unique relationship as factors) depending on the natural numbers!

Riemann Zeta Function: Important Number Relationships (8)

We now will attempt to probe the deeper meaning of ζ(– 1).

Thus when s = – 1,

ζ(– 1) = 1/1^{– 1} + 1/2^{– 1} + 1/3^{– 1} + 1/4^{– 1} + ........

= 1 + 2 + 3 + 4 + ......

Now in conventional linear terms, this series necessarily diverges to infinity.

However in terms of the Riemann zeta function it has a finite value =  – 1/12.

Therefore the issue that has baffled generations of mathematicians is the provision of a satisfactory explanation as why this series (representing the sum of the natural numbers) can have what appears to be a nonsensical result!

Once more the key to obtaining some insight into the matter is the recognition that one must now step beyond mere analytic interpretation of number relationships.

I have persistently stated that Conventional Mathematics is decidedly linear (i.e. 1-dimensional) in qualitative terms. 

What this means again is that all its relationships are studied within isolated (independent) polar reference frames. So as we have seen, mathematical objects are formally studied in an abstract manner (without reference to our subjective mental interaction with them); likewise number relationships are studied in a quantitative fashion (without reference to the necessary qualitative type interdependence between them).

So numbers are therefore for example understood in a static absolute - rather than a dynamic interactive - manner.

However the remarkable fact remains - which is yet to be properly recognised - is that all mathematical symbols, operations and relationships can be given a coherent holistic (i.e. qualitative) as well as standard analytic meaning.

So for example when – 1 is used (as in this case) to represent a dimensional power (or index) this carries an important holistic meaning! And this then provides the clue as to appreciation of why such a non-intuitive numerical result for ζ(– 1) can arise (when considered from the standard analytic perspective).

In holistic terms,  – 1 carries the meaning of (unconscious) negation of consciously posited symbols, strictly negation of the former analytic meaning attached to linear interpretation of such symbols (i.e. as 1-dimensional in a positive manner).

This issue is faced in a very different spiritual context with the onset of authentic contemplative experience. This often entails an existential crisis where the customary dualistic distinctions that characterise everyday life start to break down dramatically .

Therefore for a considerable time, one can feel suspended as between two worlds. On the one hand one may still be attempting to hold on to the familiarity associated with the dualistic (i.e. linear) worldview. However authentic spiritual progress leads to the slow emergence of a new nondual (i.e. holistic intuitive) perspective. and initially these two standpoints can seem incompatible to a considerable degree with each other. 

Now this is all deeply relevant to the mathematical issue we are are now considering.

The very reason why 1 + 2 + 3 + 4 + seems clearly to diverge (from the conventional analytic viewpoint) is that we assign here an unambiguous meaning to the mathematical operation of  +.

However as we have seen frequently with reference to our crossroads example, the very essence of holistic intuition is that it creates paradox in terms of customary linear distinctions.

So once again when we approach the crossroads from just one direction (i.e. within a linear framework of understanding) left and right turns can be defined in an unambiguous manner. So + 1 (representing say the left turn) can be clearly distinguished from – 1 (the corresponding right turn).

However, when we appreciate the approach to the crossroads from two opposite directions simultaneously (i.e. from N and S directions), left and right turns are rendered paradoxical. So what is left from one perspective is right from the other (and vice versa). In other words, in this holistic context, from one relative perspective + 1 = – 1 (and  then from the opposite relative perspective, – 1 =  + 1).

Thus when we bear this in mind, it can transform our appreciation of the zeta series.

Now in this context it is easier to see what is precisely involved by starting with the Riemann zeta function for s = 0, i.e. ζ(0).

So ζ(0) = 1/1⁰ + 1/2⁰ + 1/3⁰ + 1/4⁰ + ........

= 1 + 1 + 1 + 1 +.........

In conventional terms, this represents the standard linear manner of mathematical interpretation (i.e. that is 1-dimensional in qualitative terms). 

Thus 1 + 1 + 1 + 1 +......... clearly diverges towards infinity.

However the fact that it can indeed be given a finite value, means that it must be interpreted now according to the qualitative dimensional meaning of 0 (rather than 1).

And just like in a shrinking circle, the central point = 0, represents the midpoint of both the line and the circle (so that both are ultimately inseparable) likewise interpretation according to the qualitative dimensional meaning of 0 requires that both linear and circular aspects of understanding are directly incorporated.  

To do this we must look initially at the corresponding eta function where s = 0, which is made up of alternating positive and negative terms

So η(0) = 1 – 1 + 1 – 1 + 1 .....

Now when we interpret this holistically, the pairing of (+) 1 and  – 1 represents the circular notion of the complementarity of opposites So correctly understood from this perspective 1 – 1 represents an interdependent energy state (i.e. as 0 in phenomenal terms). Then with the next term we have the linear quantitative addition of + 1 (in an analytic manner).

So therefore the series properly represents the continual transition as between both holistic (circular)  and analytic (linear) type meaning.

Thus when we have an even number of terms, so that all can be paired off with each other in complementary fashion) the sum of the series = 0 (which in this context has directly a holistic meaning).   

However when we have an odd number of terms (with the last positive term necessarily independent and free of complementary pairing) the sum of the series assumes the analytic value of 1.

Therefore when we take the average of these two values (based on the equal probability of the series ending in an even or odd number of terms), 

η(0) = 1/2.

However the startling observation to be made here is that this value properly represents a hybrid interpretation (entailing the combination of both analytic and holistic notions of number).

One can then in turn - based on this value for η(0) - obtain a corresponding hybrid value for ζ(0).

This is based on the general result that, 

ζ(s) = η(s)/{1 – 1/2^{s – 1}}.

So we start with ζ(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + ........

To get η(s), we then subtract 2 * ( 1/2^s + 1/4^s + 1/6^s + 1/8^s +...) i.e. 2ζ(s) from ζ(s)

= ζ(s)   –  {2 * 1/2^s (1/1^s + 1/2^s + 1/3^s + 1/4^s + ........)}ζ(s)

= ζ(s)  – (1/2^{s – 1})ζ(s 

So η(s)  = (1  – 1/2^{s – 1})ζ(s)

Therefore ζ(s) = η(s)/(1 – 1/2^{s – 1})

So when s = 0

ζ(0)  = (1/2)/ (– 1) =  – 1/2.

Thus because the value of η(s) itself represents a hybrid value, this likewise applies to the value for
ζ(0).

So the crucial point to bear in mind is that in the case of the seemingly non-intuitive numerical values that arise with respect to ζ(0), together with the other negative integer values for ζ(s), standard analytic interpretation cannot strictly be applied.  

Now once again, standard linear interpretation is based on the unambiguous type logic associated with 1-dimensional appreciation (i.e. where s in holistic terms = 1).

However for s ≤  0, we must now apply a distinctive interpretation based directly on the qualitative meaning uniquely associated with the dimension in question.

We saw yesterday that where s is a negative even integer that a pure holistic explanation can be given for the numerical value arising.

So ζ(– 2) = ζ(– 4) = ζ(– 6) = ζ(– 8) =.......  = 0

The reason again for this is that direct complementarity arises in all these cases where every positive aspect of number interpretation is dynamically balanced by a corresponding negative.

Because of the key importance of such holistic interpretation, I will explain again briefly with respect to the simplest case  ζ(– 2).

Now ζ(– 2) = 1/1^{– 2} + 1/2^{– 2} + 1/3^{– 2} + 1/4^{– 2} + .....

= 1² + 2² + 3² + 4² + ..... = 0.

Clearly from the standard analytic (which is 1-dimensional in qualitative terms), the sum of this series diverges to infinity.

However 2-dimensional interpretation (i.e. where s = 2 in qualitative terms) is now appropriate.

In psychological terms this is associated with a dynamic relative interpretation of number, where understanding represents the complementarity of two opposite poles which are positive and negative with respect to each other.

Thus one now intuitively realises that like the two turns at a crossroads that positive and negative directions with respect to understanding are merely relative, thus dynamically cancelling out in an intuitive holistic realisation.

So when each of the natural numbers with respect to this series is posited (in conscious manner) it is then quickly negated (in an unconscious fashion). Put another way each number now loses any separate independent identity to assume a common interdependent meaning in terms of the whole series (which is 0 in phenomenal terms).

Therefore though we are still using the symbol "0" to represent our result, in this context, it takes on a purely relative holistic (rather than absolute analytic) meaning.


Now the enormous problem for Conventional Mathematics is that it has become completely based in formal terms on a merely reduced analytic interpretation associated with the number 1.

In other words, Mathematics has greatly lost appreciation of what the qualitative notion of "the whole" in any context truly entails (with "the whole" in all cases being reduced to separate independent parts).

This is why ζ(– 2) = 1² + 2² + 3² + 4² + ..... = 0, appears such a non-intuitive result, as we are conditioned to look at the result of the "whole" series merely as the reduced sum of its component part members.

Indeed strictly speaking, the standard mathematical way of looking at divergent series as approaching infinity is just nonsense, for in finite terms no matter how large a series becomes it still necessarily remains of a finite nature!

So the true notion of the infinite relates to holistic - rather than analytic - appreciation.

This then explains why - when correctly interpreted in holistic terms - the various series that apparently diverge to infinity (from an analytic perspective) are now understood to have a finite value.

In other words their infinite nature is properly seen to emanate, as it were, from the finite in the intuitive realisation of their number interdependence. 

Indeed in this context it is fascinating to explain from a holistic perspective why the only dimensional value for s where a finite value cannot be obtained for the Riemann zeta function is where s = 1.

This is due to the fact that as conventional analytic interpretation is - by definition - based on 1-dimensional understanding (i.e. where s in holistic terms = 1), so that both the quantitative and qualitative interpretations of the series necessarily coincide for this value.

However for all other values of s (≠ 1), unique qualitative interpretations are associated with the dimensional numbers involved, enabling a finite value to be obtained in all those cases (where the series diverges in 1-dimensional terms).


This holistic explanation also explains clearly why the attempt to obtain proof (or disproof) of the Riemann Hypothesis is strictly speaking meaningless in conventional mathematical terms!

From a dynamic interactive perspective - that properly combines both analytic (quantitative) and holistic (qualitative) appreciation - the only number in analytic terms where the Riemann zeta function remains undefined is where s = 1.

However the holistic counterpart to this is that the only mathematical interpretation for which it likewise remains undefined is where s = 1 (in holistic terms). and this - as we have seen defines the conventional approach to Mathematics.

So, when properly understood, the true nature of the Riemann zeta function relates to the dynamic interactive relationship as between both analytic (quantitative) and holistic (qualitative) aspects of number.  

And this crucial holistic dimension is not even recognised in conventional mathematical terms!

So not alone can the Riemann Hypothesis be neither proved (nor disproved) from the standard mathematical perspective, much more importantly it cannot be properly interpreted in this manner!


Now the appreciation as to why ζ(– 1) = (– 1/12), represents a more convoluted version of what was involved in arriving at the value of ζ(0) = – 1/2.

Though Riemann achieved such results through the advanced technique of analytic continuation, it is possible to ascertain values for the more common negative values i.e. where s =  – 1 and – 3 respectively, through much simpler means.

In fact, I worked out these two values for myself some time ago, which can be found at "Calculating  ζ (- 1) and ζ (- 3)".

However the key to understanding these results is that they in fact represent a hybrid mix of both analytic (quantitative) and holistic (qualitative) type appreciation.

I will perhaps say a little more on the significance of  ζ(– 1) in the next entry. 

Riemann Zeta Function: Important Number Relationships (7)

We looked yesterday at the holistic explanation for  ζ (– 2), the first of the trivial zeros.

Indeed the holistic rationale used here is equally important with respect to appreciation of the non-trivial zeros.

We have seen for example that in holistic terms ζ (– 2) = 0 represents a psycho-spiritual energy state (i.e. as pure intuition).

And as in holistic terms, psychological and physical aspects are complementary, this of course also entails that ζ (– 2) = 0 has a corresponding meaning in terms of the appreciation of a physical energy state!

So what happens in this holistic generation of  energy is that two analytic quantitative values (having  unambiguous meanings within independent (1-dimensional) frames of reference are brought together and now holistically understood - in terms of each other - as directly complementary. So like matter and anti-matter particles in physics they negate or cancel each other out resulting in pure energy.

So the holistic ability to recognise - in any context - the complementarity of relationships is directly intuitive in nature though this can subsequently be given an indirect (circular) rational explanation.

Therefore in dynamic interactive terms, the direct qualitative - as opposed to quantitative -  appreciation of a mathematical relationship is holistic in nature. However this qualitative appreciation arises from the complementary pairing of quantitative type relationships (that have an unambiguous analytic meaning within fixed independent frames of reference).  

So the true mathematical appreciation of the holistic qualitative aspect of the relationship requires a more refined quantitative type interpretation in analytic terms (where one recognises that it takes place within frames of reference that strictly are always of an arbitrary nature).

So once again using our crossroads example we can unambiguously define a turn as left or right (assigning it a value of  + 1 or – 1 respectively if the crossroads is approached from just one direction). However when we recognise that the crossroads can be simultaneously approached from two opposite directions, then left and right (i.e. + 1 and – 1) now have a paradoxical meaning (where + 1 = – 1 and – 1 = + 1).

Thus it is in this moment of simultaneous recognition (from two complementary reference frames) that the holistic qualitative nature of the relationship is directly appreciated in intuitive fashion as 0 (i.e. nothing in dualistic phenomenal terms).

The huge significance of all this is that the fundamental nature of the relationship as between the primes and the natural numbers (and natural numbers and primes) is necessarily conditioned by polar opposites (such as external/internal and whole/part) that are complementary with each other in dynamic interactive terms.

So conventional analytic understanding always takes place within isolated independent frames of reference that are necessarily of an arbitrary contingent nature!

In fact at this point it might be of value to proceed a little further by now explaining the holistic mathematical significance of the second of the trivial zeros i.e. ζ (– 4).

Again ζ (– 4) = 0.

What this means is that appropriate holistic appreciation now entails two sets of complementary relationships (that indirectly can be represented by the four roots of 1).  

In other words at the 4-dimensional level of understanding one understands that all experiential relationships are conditioned by both real and imaginary polarities that are positive and negative with respect to each other.

At this level one would recognise that all mathematical relationships have both an objective and subjective aspect (i.e. as mental interpretation).
Therefore - though this is frequently overlooked - we can have no objective meaning (in mathematical terms) without corresponding mental interpretation. Thus the view - as so strongly elucidated for example by Hardy - that primes are objective in an absolute manner - is itself the product of a particular mental interpretation (that is of a crucially limited nature).

Likewise, as we have seen, we cannot - though again completely overlooked in conventional terms - strictly have quantitative mathematical meaning (in the absence of a corresponding qualitative aspect).

Indeed this entails the fundamental relationship of whole and parts, in dynamic interactive terms, is always quantitative and qualitative with respect to each other. Thus the key reductionism that defines conventional mathematical treatment of number is the attempt to define the whole (in any context) merely in terms of its quantitative parts!

However at the 4-dimensional level of appreciation, one now sees mathematical objects and corresponding mental interpretation in complementary terms as positive and negative with respect to each other.

Then one further understands the whole and part likewise in even more refined holistic mathematical terms as real and imaginary with respect to each other.

Thus from a qualitative perspective whereas the conventional analytic approach represents the "real" aspect of mathematical understanding, the unrecognised holistic aspect represents its corresponding "imaginary" aspect.

Therefore though both real and imaginary aspect (complex numbers) are now recognised in quantitative terms, there is as yet no corresponding recognition of an imaginary aspect (in a qualitative manner).

So the holistic aspect of mathematical understanding on which I am currently elaborating, represents this unrecognised imaginary aspect (in qualitative terms).
And future Mathematics will be required to move to a new complex rather than the current real rational approach (in qualitative terms) that characterises the conventional treatment of all relationships.


Therefore it requires a more refined intuitive state to properly recognise the truly complementary nature of both the real and imaginary aspects of mathematical relationships (in positive and negative terms).

However in principle it is very similar to 2-dimensional understanding (where now opposite reference frames cancel out in both real and imaginary terms).
And of course negative 4-dimensional understanding then entails the direct intuitive recognition of the coincidence of opposite poles (free of secondary rational interpretation). 

Now meaning can likewise be given to all the other negative even dimensions (representing trivial zeros) where complementarity can be established as between one set of reference frames and another set that are directly opposite in all cases.

All these "higher" even dimensions entail various configurations entailing real and imaginary values (as indirectly expressed through the corresponding roots of the number).

So therefore in holistic terms, these likewise represent varying configurations, in dynamic interactive terms, with respect to both objective recognition (and corresponding mental interpretation) and holistic (qualitative) and analytic (quantitative) recognition.

Therefore with respect to the prime and natural numbers the understanding corresponding to ζ (– 4) = 0, relates to the direct intuitive recognition that ultimately these numbers have no objective existence independent of mental interpretation. In other words their ultimate nature is ineffable!

Likewise it entails that both the prime and natural numbers ultimately can have no quantitative meaning independent of their corresponding qualitative aspect. This implies that a remarkable synchronistic interaction  dynamically characterises their relationship, which ultimately again is ineffable in nature. 

Riemann Zeta Function: Important Number Relationships (6)

So far we have looked at just the positive integer values for the Riemann zeta function ζ(s).

We will now probe a little into the corresponding negative integer values, attempting to properly explain their important role.

ζ(s) = 1/1^s + 1/2^s + 1/3^s + 1/4^s + .......


Therefore, when for example s = – 2,

ζ(– 2) = 1/1^{– 2} + 1/2^{– 2} + 1/3^{– 2} + 1/4^{– 2} +......

= 1 ² + 2 ²  + 3 ² + 4 ²  +........  =  1 + 4 + 9 + 16 + ......

From the standard linear interpretation, this infinite series clearly diverges (with no finite result).

However in terms of the Riemann zeta function ζ(– 2) = 0 (in what is referred to as the first of the trivial zeros).

Now this choice of expression is somewhat unfortunate. Whereas it is true that the trivial zeros do not play a significant role with respect to the quantitative nature of the primes, they do indeed have an extremely important qualitative function (which is entirely overlooked in the conventional approach).

In fact I have seen anything resembling a satisfactory explanation in conventional terms as to why the series 1 + 4 + 9 + 16 + ...... can have two diametrically opposing values!

Whereas there is highly technical literature available on the nature of analytic continuation using holomorphic functions  (on which the extension of the domain of the Riemann zeta function depends),  this only helps to obscure the fact that no satisfactory explanation has yet been given as to why two opposing interpretations can exist for the same series!

It was my own determination to properly understand the nature of this first "trivial" zero that transformed my whole understanding of the Riemann zeta function.

I then slowly began to understand that the apparent nonsensical values of the function for values of s < 0, related to the fact that they do not correspond to a standard analytic interpretation (that is merely quantitative), but rather to an unrecognised holistic interpretation (that strictly is of a qualitative nature). 

This in turn required that both of these aspects (analytic and holistic) must necessarily be viewed in a dynamic interactive context, where they are seen as directly complementary with each other.

So once again we cannot have number independence (in quantitative terms) without number interdependence (in a qualitative manner); likewise we cannot have number interdependence without number independence. so therefore in appropriate dynamic terms, both quantitative (analytic) and qualitative (holistic) aspects necessarily interact with each other in a bi-directional relative manner.

In particular this applies to interpretation of the Riemann zeta function which maps values for ζ(s) on the RHS of the functional domain with corresponding values for ζ(1 – s) on the LHS.  

So therefore when we apply the standard analytic interpretation in quantitative terms to ζ(s) for s > 1, this implies that the corresponding interpretation for ζ(1 – s) should be carried out - relatively - in a complementary holistic manner.

Thus when ζ(3) with a recognised quantitative value in analytic terms, is mapped with the first of the trivial zeros i.e. ζ( – 2) this latter interpretation relates to a holistic - rather than analytic - value.

The clue to this holistic interpretation lies in the fact that one should now consider the dimensional power involved (i.e. – 2) in a qualitative rather than quantitative manner.

In standard analytic terms, 2 relates to units that are independent (as befits the cardinal notion of number); however from the qualitative perspective it relates to units that are interdependent with each other (that befits the ordinal notion). So such interdependence of units implies that their positions can be interchanged, with each, depending on relative context, potentially existing as 1st or 2nd!

Now we have seen that we can indirectly express this qualitative view of number in quantitative terms, through obtaining the two roots of 1. In this way the two "units" can be expressed as + 1 and 
– 1 respectively (where the signs can interchange depending on context).

I have mentioned many times before that such 2-dimensional appreciation, in holistic terms, characterises our understanding that the two turns at a crossroads can be both left and right, or in mathematical terms, both + 1 and  – 1 (depending on the direction from which the crossroads is approached).

So with 1-dimensional interpretation only one polar reference frame is used. Thus when one approaches the crossroads from either a N or S direction (considered independently) one can unambiguously denote a turn at the crossroads as L or R. 

However, 2-dimensional interpretation requires the ability to simultaneously "see" from two opposite i.e. complementary polar reference frames. Therefore when one simultaneously views the approach to the crossroads from both N and S directions, then (circular) paradox is generated from a dualistic (i.e. 1-dimensional) perspective. So each turn is now interchangeable as both left and right (which seemingly confounds normal logic). In this sense, one implicitly recognises that left and right are purely relative (with no meaning independent of each other).


This is all deeply relevant in mathematical terms, which is likewise conditioned by fundamental polarities that necessarily interact with each other in dynamic fashion.

So for example we cannot have a mathematical object without a corresponding subjective interpretation. So numbers as independent objective entities strictly have no meaning apart from the subjective mental interpretations we place on them!

Likewise we cannot have a quantitative without a corresponding qualitative dimension to numbers. In other words, the independence of numbers in quantitative terms necessarily implies their corresponding interdependence in a qualitative manner (and vice versa).

So these are necessarily polar relative terms that ultimately have no meaning apart from each other.

Yet we have spent millennia now attempting to interpret numbers (especially the primes) as if they somehow possess an absolute objective identity. And quite simply this is an utterly mistaken approach!

Now with reference to the general expression a^b, both the base a and dimensional number b relate  to differing reference frames that are quantitative and qualitative with respect to each other.

Therefore, in dynamic interactive terms, when a is interpreted in an analytic (quantitative) manner, b  in complementary terms is thereby interpreted in a holistic (qualitative) fashion. Likewise in reverse when a is interpreted in holistic terms, b is then interpreted in a - relative - analytic fashion.

So all numbers, in base and dimensional terms have both (quantitative) analytic and (qualitative) holistic interpretations depending on relative context.

Thus when we look at 2 in this qualitative dimensional sense it implies - like with our crossroads - the simultaneous recognition of two complementary reference frames for number (i.e. Type 1 and Type 2). Though 2 in Type 1 and Type 2 terms represents a numerical value that seems unambiguous when considered separately, deep paradox arises when the frames are incorporated simultaneously with each other.

In holistic terms, + is always associated with the conscious notion of positing phenomena; however, – carries the holistic meaning of negation (i.e. of making what is conscious, unconscious). And the unconscious element of understanding then expresses itself as intuition (i.e. a psycho spiritual energy) which entails direct appreciation of the mutual interdependence of phenomena. 

So just like the combination of matter and anti-matter particles in physics leads to the direct generation of physical energy, likewise the combination of psychic matter and anti-matter objects, lead to the direct (unconscious) generation of psycho-spiritual energy, which is generally referred to as intuition!  

Now whereas 2 represents the conscious attempt to portray the complementary nature of + 1 and – 1 (as posited) so the dimension of – 2 expresses, in holistic terms, the direct intuitive recognition - through negation of what is phenomenally posited in experience - of the complementary nature of Type 1 and Type 2 aspects of the number system  And it is through these aspects that number keeps switching as between quantitative and qualitative (and in reverse terms qualitative and quantitative) recognition. 

So the holistic reason why ζ(– 2) = 0 is because understanding is now identified directly with the intuitive recognition of the complete interdependence of the polar opposites (entailing thereby two dimensions) that condition all number relationships.

Put another way the holistic  meaning as to why  ζ(– 2) = 0 is this direct intuitive realisation of the pure relativity of all number relationships in dynamic interactive terms. And this is of a qualitative psycho-spiritual nature that is thereby nothing (0) in a phenomenal quantitative manner!  

This represents therefore the complementary extreme to the conventional interpretation of number (in an absolute quantitative manner).


The deeper implications of this understanding are very revealing for true interpretation - even - of the quantitative nature of the number system.

As I have stated before we can study the behaviour of number both with respect to the external and internal nature of number.

Now in conventional terms, the emphasis is primarily on the external aspect. Even when some limited investigation takes place with respect to the corresponding internal aspect, its dynamic complementary relationship with the external is completely overlooked!

Great attention has been placed in external terms on the frequency of the primes with respect to the natural numbers.

For example, the simplest version of the prime number theorem states that the frequency is approximated by n/log n (with accuracy eventually approaching 100% for sufficiently large n).

However there is a equally important internal version of this prime number theorem, whereby, when n is sufficiently large, the average ratio of  the natural to the corresponding prime factors (or divisors) of a number approaches n₁/log n₁ (where n₁ = log n).

Therefore regarding the external aspect of number system, we can observe the relationship of prime to natural numbers; then with  respect to the internal aspect we can observe the relationship of  prime to natural number factors.

Now when we approach this issue from the standard rational (1-dimensional) mathematical  perspective, we treat both sets of relationships absolutely in a quantitative manner.

In terms of our crossroads analogy, this is equivalent to interpreting left and right turns in an unambiguous manner (i.e. when approached from just one direction either N or S). 

However when we interpret left and right (in a circular 2-dimensional manner) when approached from both N and S directions, left and right turns are rendered paradoxical with a purely relative meaning.


It is exactly similar in number terms! Therefore when we recognise that the behaviour of the number system can be simultaneously understood with respect to both its external and internal aspects, then the relationship of the primes to the natural numbers is rendered paradoxical.

Thus from the external perspective the natural numbers seem to depend on the primes;

Then from the internal perspective, the natural seem to likewise depend on the prime factors.

However these (internal) factors represent the dimensional aspect of number as opposed to their quantitative base behaviour (in external terms).

Thus simultaneous understanding with respect to both the external and internal behavior of the number system (which is of a holistic intuitive nature) leads to direct recognition of the paradoxical relationship of the primes to the natural numbers, which is merely of a relative nature.

Thus in quantitative (Type 1) terms the natural numbers appear to depend on the primes; also in quantitative (Type 2) terms the natural number factors appear to depend on corresponding prime factors.

However simultaneously in Type 3 terms (where both Type 1 and Type 2 are recognised as dynamically complementary) the prime and natural numbers are now understood as co-dependent on each other (in both quantitative and qualitative terms).

And this equates directly with recognition of the holistic synchronicity of the number system in an ultimately ineffable manner (where both the prime and natural numbers are ultimately seen as representing perfect mirrors of each other).


In the earlier contributions to this series, I concentrated on the relevance of ζ(s), where s is a real positive integer for the behaviour of factor composition externally with relation to the number system.

So for example we were able to derive the interesting finding that roughly 61%, i.e. 1/ζ(2) of numbers are comprised of factor combinations where no factor occurs more than once. 

However we can also study such behaviour in an internal manner with respect to deriving the average frequency with respect to the total cumulative factors belonging to numbers where at most any factor occurs 1, 2, 3, .....n times.

And from what we have said, such behaviour should complement the results in external terms that we have already found.

We will return to this later!