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!