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.