More on the Significance of ζ– 1) = – 1/12!

In my companion blog "Spectrum of Mathematics", I have shown how a fascinating formula of Ramanujan for 1/24 can be used to "convert" the existing series - in an intuitively accessible linear manner - for ζ(s), where s is a negative odd integer.

The standard "conversion" formula is given for the case of ζ(– 13).

Now in linear terms, 

ζ(– 13) = 1¹³ + 2¹³ + 3¹³ + ...., which diverges to infinity. However we know that the correct answer (in holistic terms) = – 1/12.

So the "converted" series, i.e.

1¹³/{(1 –  e^2π )/2} + 2¹³/{(1 –  e^4π )/2} + 3¹³/{(1 –  e^6π )/2} +...    = – 1/12,  now gives the correct answer from the standard linear perspective.

The remarkable fact is that this formula very much works in cyclical fashion in the manner of a 12 hr. clock for the denominator (with the hours referring to the common dimensional power to which each of the natural numbers in the series is raised).

Thus when for example we go back "12 hours" we find  that,

ζ(– 1) = 1¹ + 2¹ + 3¹ + ....,

And the correct "converted" series for ζ(– 1) is given by the same "converted" series for ζ(– 13), i.e.

1¹³/{(1 –  e^2π )/2} + 2¹³/{(1 –  e^4π )/2} + 3¹³/{(1 –  e^6π )/2} +...    = – 1/12.

When we now move forward "12 hours" we find that

ζ(– 25) = 1²⁵ + 2²⁵ + 3²⁵ + ....,  which  =  – 657931/12 (from the correct holistic perspective).

So the linear "conversion" for this series is given as,

657931[1¹³/{(1 –  e^2π )/2} + 2¹³/{(1 –  e^4π )/2} + 3¹³/{(1 –  e^6π )/2}] +...
= – 1/12.

Now the "12 hour" clock does not always work in this manner for the denominator. It does work again where s = – 37, but then breaks down for  s = – 49, with the denominator for ζ(– 49) = – 132.

However for all  "12 hour" cycles then up to (and including) s = – 97, the denominator of ζ(s) = – 12.

The importance of 12 is also in evidence with respect to the fact that it seems that in all cases (where s is a negative odd integer), that the denominator of ζ(s) is divisible by 12. I have also been aware for some time here of an apparent connection with twin primes. So apart from the first twin primes i.e. 3 and 5, the sum of all other twin prime pairings appears to be divisible by 12!

However it is another remarkable feature - with respect to the denominators of ζ(s) - where again s is  negative odd integer, that I wish to concentrate on here!

In conventional terms, we can determine that a number is prime if it has no factors (other than itself and 1). However, as is known so well, it becomes progressively difficult to test for primes in this manner (where the number is very large).

In fact, safe encryption systems - on which e-commerce so much depends - are based on the inherent difficulty of factorising very large numbers!

However the "Alice in Wonderland" world of the Riemann zeta function, where conventional expectations with respect to results are turned inside-out for negative values of s < 0, in principle offers a complementary opposite means for testing for number primality.

Once again in conventional linear terms, we demonstrate that a number is prime by showing that it has no other factors (other than the number itself and 1).

Then in complementary circular terms, we can demonstrate that a number is prime by showing in a certain unique manner - enabled by the Riemann zeta function - that it is a factor of a composite number!

In fact the "golden rule" for establishing such primality can be stated quite simply.

In Riemann's functional equation, a relationship is established as between values of ζ(s) and ζ(1 – s).

So when s is a positive even, 1 – s is thereby a negative odd integer.

Now the "golden rule" is as follows.

If the denominator of ζ(1 – s) is divisible by 1 + s, then 1 + s  is a prime number; also if the denominator of ζ(1 – s) is not divisible by (1 + s), then 1 + s  is not a prime number. 

When s = 2 the denominator of ζ(– 1) i.e. 12 is divisible by 3. Therefore 3 is prime.

When s = 4, the denominator of ζ(– 3) i.e. 120 is divisible by 5. Therefore 5 is prime.

When s = 6, the denominator of ζ(– 5) i.e. 252 is divisible by 7. Therefore 7 is prime.

Then when s = 8, the denominator of ζ(– 7) i.e. 240 is not divisible by 9. Therefore 9 is not prime!

When s = 10, the denominator of ζ(– 9) i.e. 132 is divisible by 11. Therefore 11 is prime.

When s = 12 the denominator of ζ(–11) i.e. 32760 is divisible by 13. Therefore 13 is prime.

Then when s = 14, the denominator of ζ(–11) i.e. 12 is not divisible by 15. Therefore 15 is not prime!

And my contention is that we can continue on indefinitely in this manner. I have found no exceptions in this procedure to s = 200, which is as far as the available tables enable me to test.

In fact the general rule also holds for the one case where a prime can be even (= 2).
For when s = 1, the denominator of ζ(0) i.e. 2 is divisible by 2!


Now of course, this does not provide a practical way of testing for large prime numbers as the calculation of the corresponding denominators of ζ(1 – s) becomes prohibitively cumbersome.

However what remains fascinating is that in principle it does provide a means of testing for primes which completely inverts the customary procedure. And this in turn is valuable to point out, as it demonstrates from yet another perspective the truly complementary nature of both ζ(s) and ζ(1 – s) which require analytic and holistic interpretation with respect to each other.

And this can only be properly understood in a dynamic interactive context, where both analytic and holistic aspects - that are relatively quantitative and qualitative with respect to each other - are clearly recognised.

In addition, it can be stated that when ζ(1 – s) is divisible by (1 + s), this then represents the largest prime by which ζ(1 – s) is divisible.

One can readily make comparisons with the denominator of ζ(s) where again s even. Here the denominator of ζ(s) entails the product of all primes (where repetition of the same prime is allowed) from 2 to 1 + s (where s is a power of 2) and from 3 to 1 + s in all other cases.
However we cannot use the denominator here to universally establish whether a number is prime!

Where ζ– 1) = – 1/12!

Again I will start by clarifying a little, some of the final comments in the previous entry.

In dynamic terms, wholes can only be understood as in relationship to parts and parts in turn as in relationship to wholes. So a relationship of qualitative to quantitative (and quantitative to qualitative) is always necessarily involved.

However in conventional mathematical terms, the attempt is made to deal with this relationship in an absolute quantitative manner whereby the whole (in any context) is understood as the sum of its differentiated parts.

However properly understood, we have an alternative qualitative interpretation of this relationship whereby (in any context) each part is understood in an integrated manner as fully containing each whole (and thereby as - loosely - representing the sum of wholes).

So in the first instance where the whole is seen as the sum of the parts, we are referring to a quantitative relationship entailing (relatively) independent parts; in the second instance, where each part is seen as fully containing the wholes, we are referring to a qualitative relationship of interdependent wholes. So when, for example, William Blake could see a "World in a grain of sand", he clearly was referring to the second type of qualitative relationship of high interdependence (where everything is seen as integrated with everything else).

Now whereas the first type of (quantitative) interpretation, directly depends on conscious recognition, the second type of (qualitative) interpretation directly depends on unconscious recognition, which indirectly however can be explained in a circular i.e. paradoxical) type manner.

And remarkably, to properly understood the number results that arise for both positive and negative integer values of s with respect to the Riemann zeta function, both types of recognition (conscious and unconscious) are required.

So whereas the results of ζ(s) where s > 1, correspond directly with intuition relating to the standard analytic type interpretation of number (in quantitative terms), the results of ζ(s) where s ≤ 0 correspond directly with intuition relating to the unrecognised holistic interpretation of number (in qualitative terms).

Thus once again, the Riemann zeta function must be interpreted in a dynamic interactive manner (with complementary quantitative and qualitative aspects).

So with respect to the Zeta 2 relationship when we concluded earlier that 1 + 2 + 4 + 8  + ... =  – 1, and then 1 + 4 + 16 + 64 + ...  =  – 1/3, these results concur directly with the holistic qualitative manner of interpretation!


We then showed how these Zeta 2 results can then be used to "convert" corresponding Eta 1 results in an appropriate Zeta 1 manner.

So ζ₁(0) = η₁(0) * ζ₂(s₂) where s₂ = 2, and ζ₁(– 1) = η₁(– 1) * ζ₂(s₂) where s₂ = 4.

In this way we can show the important case that ζ₁(– 1) =  1/4 * (– 1/3)  = – 1/12.

So 1 + 2 + 3 + 4 +...    = – 1/12.     

Now clearly this result does not intuitively correspond with standard analytic interpretation (in a quantitative manner).

Rather it intuitively corresponds directly with (unrecognised) holistic interpretation (in a qualitative fashion).

Now the essence of holistic type interpretation is that one looks at the overall dimensional structure of the relationship (rather than the individual terms) in interpreting the numerical results that arise.

So - quite literally - the various terms must be treated as an integrated whole with respect to derivation of the result.

Therefore, for example when we maintain (in a holistic sense) that 1 + 2 + 3 + 4 +...  – 1/12, it is meaningless in this context therefore to maintain that 2 + 3 + 4 + ...   = – 13/12.


My holistic understanding of this relationship has been deeply tied up for some time now with a seemingly unrelated task, which I first seriously addressed some 20 years ago.

I was very much influenced by Jungian psychology at the time and especially his theory of Personality Types (which I found directly amenable to holistic mathematical interpretation). See
"Personality Types and Superstrings"

Now the original Jungian profile of 8 fundamental Personality Types was later extended in the well-known Myers-Briggs classification to 16.

Though I found this classification very useful, I felt - on initial close examination of my own personality - that there were certain "missing" types.

So eventually I worked out a simple holistic mathematical approach for generating 24 fundamental personality types.

Now the basic starting point here is a new holistic mathematical understanding of the 4 dimensions of space and time (which are directly based on the unit circle as interpreted in a holistic mathematical manner).

So the four roots of 1 are 1, – 1, + i and – i respectively.

What this entails in holistic mathematical terms is that the relationship between all real phenomena entails external (objective) and internal (subjective) polarities that are relatively positive and negative with respect to each other.

Then secondly all phenomenal relationships further entail quantitative (part) and qualitative (whole) polarities that are relatively real and imaginary with respect to each other.

Thus from this dynamic holistic perspective, our very experience of space and time necessarily reflects the manner in which these two sets of polarities are configured.

In holistic terms, the four polarities (representing the 4 dimensions) are highly interdependent with each other in a unified manner. So in this holistic sense each of the four dimensions represents 1/4 of the unified whole.


I then realised by permutating each of these original four dimensions that 24 unique personality types would arise.

So in a very true holistic sense, each personality type therefore represents a certain distinctive manner in which the four original dimensions of space and time are configured.  

I then realised that there were intimate connections here with the physical world of strings.

Just as in psychological terms we can speak of 24 unique vibrations as it were (each corresponding to all the fundamental  personality types) likewise with respect to strings (in the heterotic theory) we can speak also speak of 24 unique vibrations corresponding to the fundamental "impersonality" types of matter. 

So this leads in fact to a new notion of "dimension" in this context as relating to a certain distinctive configuration with respect to the four original primary dimensions.

Now with respect to the 24 personality types, I identify three major groupings.

8 of these types fall into the "real" mode where orientation is primarily conscious in origin geared to actual events.

Mathematics as a rational discipline is entirely identified with this one grouping, which I customarily refer to as the Type 1 aspect.

8 other types fall entirely into the "imaginary" mode, where orientation is primarily unconscious in origin geared to the intuitive potential inherent in events.  

So recognition of the distinctive holistic intuitive nature of mathematical relationships - that indirectly can be conveyed in a circular logical fashion - is at present entirely missing from accepted mathematical interpretation.
For many years therefore I found myself deeply engaged in the what I refer to as Holistic Mathematics i.e. the Type 2 aspect.

The final 8 types fall into the "complex" mode where orientation is primarily geared to the balanced integration of both analytic and holistic type experience. I often refer to this understanding in mathematical terms as Radial Mathematics i.e. the Type 3 aspect.

My strong contention in all these blog entries is that proper understanding of the Riemann zeta function - and indeed ultimately all mathematical relationships - requires Type 3 understanding.

The Myers Briggs classification deals well with the first two groups. So the conscious "realists" are denoted as S (sense) with the more unconscious orientated "imaginary" denoted as N (intuitive) types. 

Then the "complex" types attempt to achieve the balanced reconciliation of both conscious and unconscious, which in mathematical terms attempts entails the dynamic interaction of both analytic and holistic aspects.

So for example, whereas someone with respect to the other groups is classified as either an extrovert or introvert, in relation to the last group a person is primarily a centrovert.

However in secondary terms such a person could still manifest - predominantly - extrovert or introvert tendencies.

Therefore, in a qualified sense, 4 of this latter group still conform in a secondary manner to "real" S types, while 4 others conform to "imaginary" N types.

Therefore with respect to the 24 Personality Types, 12 are predominantly of an unconscious orientation entailing the negation of the conscious (posited) direction of phenomena. 

In this sense there are 12 ways of negating the positive 1-dimensional direction of experience associated with the conventional rational understanding of the infinite sum of natural numbers = 1 + 2 + 3 + 4 + ... 
And as we have seen, this is the means by which qualitative holistic - as opposed to quantitative analytic - interpretation, takes place.

In this way each of these 12 (negative) "dimensions" - representing a distinct intuitive vibration of the personality (in psycho spiritual energy terms) -  can be represented as – 1/12.

Once again in conventional mathematical understanding, it is assumed that there is just 1 common dimension through which all relationships are understood (in a linear rational manner).

However because there are 24 distinctive personalities, this means that each type represents a unique dimension through which understanding takes place. And whereas 12 of these dimensions are posited in a linear conscious manner, the other 12 represent the corresponding unconscious negation, in a holistic intuitive fashion, of the former understanding. 

There are then close complementary parallels here with respect to the behaviour of a string, where the ground energy of the infinite sum of natural number vibrations is likewise given as – 1/12. For more information on this latter aspect, see this interesting YouTube video "John Baez on the number 24"