Product Formulation of Alternative Zeta 2 Function

In a previous entry I drew attention to a complementary (alternative) version of the Zeta 2 function which I now refer to as Alt Zeta 2.

So the Alt Zeta 2 function is associated with the (infinite) sum of reciprocals of the unique number sequences associated with the general polynomial equation (x – 1)ⁿ = 0.

So for example the unique digit sequence associated with (x – 1)³ = 0 is,

1, 3, 6, 10, 15, …

And the corresponding (infinite) sum of reciprocals is

1 + 1/3 + 1/6 + 1/10 + 1/15 + …    = 2.

So I refer to this sequence as Alt ζ₂(3).

Now in general terms, I represent the Zeta 2 function as  ζ₂(s₂) = 1 + s₂¹ + s₂² + s₂³ + …

And where n is an integer > 2,  s₂ = 1/(n – 1) and ζ₂(s₂) = Alt ζ₂(n).

Therefore, when n = 3, s₂ = 1/2;

So ζ₂(1/2) = 1 + (1/2)¹ + (1/2)² + (1/2)³ + …  = 2; 

Thus ζ₂(1/2) = Alt ζ₂(3), and in general terms ζ₂{1/(n – 1)} = Alt ζ₂(n).

Now I have commented before on the fact that Alt ζ₂(n) represents numbers (as denominators) that result from an ordered summation of natural numbers.

Thus for example with respect to Alt ζ₂(3) = 1 + 1/3 + 1/6 + 1/10 + 1/15 + …

1                               = 1

3   = 1 + 2                = 1 + (1 + 1)

6   = 1 + 2 + 3          = 1 + (1 + 1) + (1 + 1 + 1)

10 = 1 + 2 + 3 + 4    = 1 + (1 + 1) + (1 + 1 + 1) + (1 + 1 + 1+ 1) .

So we can see a definite order to these numbers where the denominator of the k^th term represents the sum of the first k natural numbers.

And when n > 3 the denominator of the k^th term still represents a compound ordered sum involving the first k natural numbers.

For example when n = 4, the (infinite) sum of unique associated reciprocals is,

1 + 1/4 + 1/10  + 1/20 + 1/35 + …

Now if to illustrate, we take the denominator of the 3rd term it can be shown to represent a compounded ordered sum entailing the first 3 natural numbers.

So 10 = 1 + 3 + 6 (the sum of the first 3 denominators terms of the previous sequence for n = 3.

Thus 10 = 1 + (1 + 2) + (1 + 2 + 3).

However there is another remarkable feature associated with these denominators entailing the ordered product of natural numbers.

So for example 3 = 3/1;

6   =  (4 * 3)/(1 * 2);

10 = (5 * 4 * 3)/(1 * 2 * 3);

15 = (6 * 5 * 4 * 3)/(1 * 2 * 3 * 4).

So starting with the initial value of denominator (= n/1), subsequent values are given as {(n + 1) * n}/(1 * 2), {(n + 2) * (n + 1) * n}/(1 *2 * 3), {(n + 3) * (n + 2) * (n + 1) * n}/(1 * 2 * 3 * 4), ...

In fact we can express these in an alternative fashion which shows that each number (as denominator) represents a unique combination.

So in general terms nC_r  = the number of r possible combinations taken from a group of n items.  

Now each number as denominator likewise represents a unique combination with respect to a certain number of items (defined by the value of n₁) = n₁!/{r!(n₁ – r)!}

So for example 6 = (4 * 3 * 2 * 1)/(2 * 1)(1 * 2)  = 4!/{2!(4 – 2)!}   

Thus in general terms, n₁ = (n + k – 2) where k = cardinal number of k^th term and n represents the denominator of the 2^nd term in the infinite series associated with (x – 1)ⁿ = 0, r = k – 1 and n₁ – r = n – 1 respectively.

Thus in terms of our definitions, n₁!/{r!(n₁ – r)!} = (n + k – 2)!/{(k – 1)! * (n – 1)!}

Thus when n = 3, the denominator of the 4^th term (i.e. where k = 4)

= (3 + 4 – 2)!/(3! * 2!)

= 5!/(3!* 2!) = (5 * 4)/(1 * 2) = 20/2 = 10.

So the denominator, i.e. 10  represents the total number of possible combinations of 3 taken from 5 items.

And by definition the denominator of every term, in the infinite series of reciprocals of the unique numbers associated with (x – 1)ⁿ = 0, represents a unique combination of numbers.

This then provides a ready means for calculating any term in the infinite sequence for a given value of n.

Thus for example when n = 6, the denominator of the 5^th term,

= (6 + 5 – 2)!/(5 – 1)!(6 – 1)!   =  9!/(4! * 5!) 

= (9 * 8 * 7 * 6)/(1 * 2 * 3 * 4) = 3024/24  = 126.

 Thus the 5^th term of the (infinite) sum of reciprocals based on the unique numbers associated with 

(x – 1)⁶ = 0 = 1/126.

Thus to sum up, the Alt ζ₂(n) series can be defined both in terms of an ordered sum of the 1^st k natural numbers (to a given term k) relating to the denominators of the terms of its associated series and an ordered product of the natural numbers representing the factorials of numbers involved in the respective unique combinations associated with (n + k – 2)!/(k – 1)!(n – 1)!

And the importance of these formulations in turn relates to the fact that every individual term of the Zeta 1 (Riemann) function i.e. ζ₂(s₁) with s₁ > 1 - both in its sum over natural numbers and product over primes formats - can be expressed in terms of the Alt ζ₂(n) series. 

Analytic and Holistic Interpretation of Mathematical Dimensions

Just as there are two complementary ways of expressing the Zeta 1 (Riemann) function i.e. as a sum over (all) the natural numbers and sum over (all) the primes, equally there are two complementary ways of expressing the Zeta 2 function.

And in this entry, I wish to probe deeply the precise nature of the latter two complementary expressions.

The starting point here is remarkably simple, though quickly becomes much more intricate.

Let us start with the simplest of all expressions viz. x = 1.

Then by the standard laws of conventional algebra, x – 1 = 0.

However if we now square both sides of each expression, something strange happens.

For in the first case, x² = 1, so that x² – 1 = 0; however in the second case, (x – 1)² = 0.

And the latter equation, when expanded is x² – 2x + 1 = 0.

So though starting with two similar expressions i.e. x = 1 and x – 1 = 0, we quickly find through squaring both, that two new distinct expressions emerge.

And, remarkably what has really happened is that the two final expressions i.e. x² – 1 = 0 and x² – 2x + 1 = 0, relate to two distinctive notions of number that are circular and linear with respect to each other. Using a fruitful analogy from quantum physics, they represent thereby the wave and particle aspects, respectively, of number.

However the deeper implications here is that the understanding of number itself can only be properly understand in a truly interactive manner entailing both analytic (quantitative) and holistic (qualitative) appreciation that keep switching in the dynamics of experience.

And Conventional Mathematics is completely unsuited to this new form of understanding as it reduces (in every context) the holistic (qualitative) aspect of understanding in a merely analytic (quantitative) manner.

Therefore to understand properly what happens when we square our original expressions i.e. x = 1 and x – 1 = 0 we must allow for two distinct aspect to the number system, I refer to as Type 1 and Type 2 respectively.

Thus expressed in Type 1 (analytic) terms the natural numbers are defined as,

1¹, 2¹, 3¹, 4¹, …

In other words, they are defined in linear (1-dimensional) cardinal terms as fixed independent quantitative entities i.e. as points on the real number line.

So the natural numbers are defined with respect to a base number that varies against a fixed (default) dimensional number = 1. 

However in Type 2 (holistic) terms the natural numbers are defined in circular (multi-dimensional) ordinal terms as relatively interdependent qualitative relationships entailing the unique sub-units of each number, 

1¹, 1², 1³, 1⁴, …

So for example from the Type 1 perspective 2, i.e. 2¹ = 1 + 1, where the units are considered in quantitative terms as independent and homogeneous, thereby lacking any qualitative distinction.

However from the Type 2 perspective 2, i.e. 1² = 1^st + 2^nd where the units are considered in qualitative terms as interdependent (i.e. interchangeable) and unique, thereby lacking any quantitative distinction. So what is 1^st in one context can be 2^nd in another related context (and vice versa).

Now the clue to what truly happens when we square the expression x = 1 (i.e. x¹ = 1¹) is that we now switch directly from the Type 1 to the Type 2 system.

So again in conventional terms, when we square both sides x² = 1 (i.e. as a number still interpreted in Type 1 quantitative terms). So as this conventional mathematical interpretation ignores the qualitative aspect, 1² is thereby reduced in a quantitative (1-dimensional) manner as 1¹.

However, properly understood x² = 1² (i.e. as a number now interpreted in a Type 2 qualitative manner).

Then when we square x – 1 = 0, we now interpret in complementary fashion this relation in a Type 1 quantitative manner.

So (x – 1)² = 0, i.e. x² – 2x + 1, has two linear roots i.e. + 1 and + 1 respectively (as the same two points on the real number line).

However x² = 1², has two circular roots i.e. + 1 and – 1 respectively (as two points on the unit circle).

In fact what we have here are two distinct mathematical notions of dimension that are analytic and holistic with respect to each other.

The 1st linear notion of dimension is the one that is conventionally recognised in mathematics.

So 1², can be geometrically represented in 2-dimensional terms as a square (with side 1 unit).

So if one side represents the length, the other side represents the width.

So the two roots of the equation (x – 1)² = 0, i.e. + 1 and + 1 represent thereby both the length and width respectively of the square of 1 unit.

And by extension the three roots of the equation of (x – 1)³ = 0, i.e. + 1, + 1 and + 1 represent the length, width and height respectively of a cube of 1 unit.

And though we cannot envisage this in pictorial terms the n roots of the equation (x – 1)ⁿ = 0  represent the n sides respectively of a hypercube of I unit.

However the true nature of the 2^nd circular notion remarkably, is not properly understood in conventional mathematical terms as it is in fact directly associated with an entirely distinctive holistic form of dimension.

Of course, circular notions e.g. with respect to the various roots of 1 are indeed recognised, but invariably in a merely analytic manner (where they are considered as separate from each other).

However the true holistic notion of dimension requires that the various roots of 1 - as indirect quantitative representations of qualitative notions - be interpreted in an interdependent manner (where they are understood as interchangeable with each other).

I will illustrate this again briefly with an oft-quoted example regarding the interpretation of a crossroads.

Now when one approaches a crossroads along a straight road - say heading N - then a left turn for example has an absolute unambiguous meaning.

This is because the frame of reference i.e. the direction of movement, is one-dimensional. So there is only one direction considered here in terms of approaching the crossroads i.e. N.

So we can unambiguously identify the left turn in this context as + 1 with the other right turn (which by definition is not a left turn) thereby as – 1.

Thus + 1 (a left turn) and – 1, as the two conventional roots of 1, carry here a strictly analytic meaning.

Now, if alternatively we were to approach the crossroads from the other direction (heading S) then again left and right turns can be given an unambiguous meaning represented as + 1 and – 1 respectively (as the interpretation is again 1-dimensional with only one direction of approach to the crossroads considered).

However if now consider the approach to the crossroads simultaneously from both N and S directions, then circular paradox is clearly involved for what is left from one direction is right from the other; and what is right from one direction is left from the other.

So in numerical turns what is + 1 from one direction (e.g. a left turn), continually switches to – 1 from the other (i.e. right) turn and vice versa.

Thus what we have here is a holistic 2-dimensional interpretation of left and right (i.e. + 1 and – 1) which are fully relative and thereby interchangeable with each other.

Now whereas 1-dimensional interpretation from one fixed reference frame is absolute and analytic, 2-dimensional interpretation (from two polar reference frames simultaneously) is by contrast relative and holistic in nature.

We could validly equate then 1-dimensional with (linear) rational and  2-dimensional with (circular) intuitive interpretation respectively.

However indirectly we can give intuitive appreciation an indirect rational interpretation in a paradoxical logical fashion.

So whereas with linear logic opposite polarities such as + and – are clearly separated, with circular logic, + and – are understood as fully interdependent (and thereby interchangeable) with each other.

Now the importance of this is that all experience is conditioned by polarities such as external and internal and whole and part that continually interact in dynamic fashion with each other.

This intimately applies also to mathematical understanding.

Therefore through conventional mathematical interpretation is based on the assumption of the abstract independent existence of “objects” (such as number), strictly these have no meaning apart from subjective mental constructs that are used in their interpretation. And both objective and subjective aspects are thereby external and internal with respect to each other.

However once again conventional mathematical interpretation is based on the misguided belief that “objects” such as numbers can be properly understood in an external (1-dimensional) manner possessing an absolute quantitative identity.

And in general, n-dimensional interpretation from the holistic perspective entails highly refined interdependent relationships entailing n interchanging reference frames.

However, though not strictly valid, the corresponding analytic approach attempt to give independently viewed objects a succession of higher dimensions in space.

Though in experiential terms, it is not possible to go beyond 3 space dimensions in this manner, the extension of the linear notion of dimensions can then be abstractly extended to n dimensions.

So the key point again regarding the two complementary expressions of the Zeta 2 function is that they relate to the analytic and holistic notion of dimension respectively.

Thus the infinite sum of reciprocals of the unique numbers associated with (x – 1) ⁿ corresponds to the analytic (linear) notion of dimension (envisaged as n independent linear directions in space).

The complementary approach relates to xⁿ – 1 or 1 – xⁿ =  0. Then to get rid of the one linear dimension, we divide by 1 – x to obtain 1 + x¹ + x²  + x³ + … + x^{n – 1}  = 0.

This then represent the finite expression of the Zeta 2 function.

And then for the geometric series expressions used to define each term of the Zeta 1 (Riemann) function, we use the infinite version of this function, i.e.

1 + x¹ + x²  + x³  + …

Missing Piece of the Jigsaw

In recent posts on my related blog-site “Spectrum of Mathematics” I have drawn attention to an important “missing piece of the jig-saw” with respect to a full explanation of the nature of the Riemann zeta function.

From a conventional perspective the Riemann Zeta function is identified solely with - what I refer to as - the Zeta 1 function. 

And as is well known this function can be expressed in two ways, both as a sum over natural numbers and a product over primes.

So in general terms,

ζ₁(s) = ∑ 1/n^s   = ∏ 1/(1 – p^–s)   
            ^{n = 1            p}   

So for example, when s = 2,

ζ₁(2)  = 1/1² + 1/2² + 1/3² + …    = 1/(1 – 2^{– 2}) * 1/(1 – 3^{– 2}) * 1/(1 – 5^{– 2}) * …

= 1 + 1/4 + 1/9 + …  = 4/3 * 9/8 * 25/24 * …   = π²/6 .

However my strong contention throughout is that the Zeta function can only be properly understood in a dynamic relative manner, entailing the dynamic interaction of two related aspects, which I refer to as Zeta 1 and Zeta 2 functions.

Properly understood, this also requires two distinctive types of mathematical understanding that are analytic (quantitative) and holistic (qualitative) with respect to each other.

And the analytic aspect (in this newly defined context) relates to the notion of number as relatively independent (of other numbers) whereas the holistic aspect relates to the complementary appreciation of number as relatively interdependent (with other numbers).

So in the dynamics of understanding, the very nature of mathematical symbols keeps switching as between analytic and holistic appreciation, i.e. their particle and wave aspects - which are complementary opposite in nature.   

Therefore in conventional mathematical interpretation, a crucial distortion is at work, whereby the holistic (qualitative) aspect - in every formal context - is reduced in a merely analytic (quantitative) manner. 

When one fully grasps the significance of this observation, then it becomes apparent that the conventional understanding of number, despite all the admitted great advances that have been made, fundamentally cannot be fit for purpose.  

Whereas I refer to the Zeta 1 as ζ₁(s) - strictly ζ₁(s₁) - I refer to the Zeta 2 function as ζ₂(s), or again more accurately as ζ₂(s₂).

In general terms ζ₂(s₂) in its infinite expression  = 1 + s¹₂ + s₂² + s₂³ + …   = 1/(1 –  s₂ ).

So in fact it represents an infinite geometric series with common ratio = s₂ .

However the significance here is that each of the individual terms in both the sum over natural numbers and product over primes expressions of the Zeta 1 function, can be expressed in terms of the corresponding Zeta 2 aspect.  

In this way the Zeta 1 function, seen from one important perspective can be viewed as representing a collective sum (over all the natural numbers) or alternatively a collective product (over all the primes) of individual Zeta 2 functions.

So again to briefly illustrate, let us take the 3^rd term of the Zeta 1 function above (for s₁ = 2)!

Now, in the sum over natural numbers expression, this is given as 1/9, which can be stated in terms of the Zeta 2 function as,

{1 + 1/10 + (1/10)² + (1/10)³ + …} – 1 = {1/(1 – 1/10)} – 1  = 10/9 – 1 = 1/9.

The link here with the 3^rd natural number 3 can be shown through rearranging the denominator of each term of the expression in the following manner,

{1 + 1/(3² + 1) + 1/(3² + 1)² + 1/(3² + 1)³ + …} – 1.

And this approach can be fully generalised.

Therefore the 4^th term of the Zeta 1 (sum over natural numbers expression) = 1/16.

And {1 + 1/17 + (1/17)² + (1/17)³ + …} – 1 = {1/(1 – 1/17)} – 1  = 17/16 – 1 = 1/16.              

Again {1 + 1/17 + (1/17)² + (1/17)³ + …} – 1 

=  {1 + 1/(4² + 1) + 1/(4² + 1)² + 1/(4² + 1)³ + …} – 1.

In this way the 4^th natural number (i.e. 4) is directly associated with the Zeta 2 expression for this 4^th term (of the Zeta 1 function).

And the Zeta 2 function can equally be used in place of each individual term of the product over primes expression of the Zeta 1.

So again when s = 2, the 3^rd term of the product over primes expression (for the Zeta 1) = 25/24.

Then in terms of the Zeta 2,

1 + 1/25 + 1/25² + 1/25³ + …  = 1/(1 – 1/25) = 25/24.

And we can then show directly the link here in the Zeta 2 (with respect to the 3^rd prime = 5) by rewriting the expression in the following manner i.e.

1 + 1/5² + 1/5⁴ + 1/5⁶   + …  = 1/(1 – 1/5²) = 25/24.

Thus once again, the Zeta 1 function - both with respect to its sum over natural numbers and product over primes expressions - can be completely written as the collective sum and product respectively of individual Zeta 2 functions.

Thus in general terms, 

∑ 1/n^s   = ∏ 1/(1 – p^–s)  = ζ₁(s)  =   ∑{ζ₂(1/n) – 1}^s =   ∏{ζ₂(1/p)^s}
^{n = 1            p                                     n = 2                                p}

However, though in its own way remarkable, this formulation of the Zeta 1 function (as the collective sum and product respectively of individual Zeta 2 functions) is not yet complete.

Though we have been able to express the Zeta 1 function in two related manners (again as both the sum over natural numbers and product over primes respectively), so far internally, we have expressed the Zeta 2 function in just one way as the sum of repeated multiplied terms.

However to complete the picture, we need to show an alternative formulation for the Zeta 2 function where the value of the function, in complementary fashion, results from the sum of repeated added terms.

And this is where the recent entries on the Spectrum of Mathematics web-site have borne fruit, as they have finally made this latter piece of the jig-saw readily apparent.

In those entries, I consider - rather in the manner of the Fibonacci sequence - the unique infinite number sequences associated with the general polynomial equation,

(x – 1)ⁿ = 0.

Now in the case of the Fibonacci, the unique number sequence,

0, 1, 1, 2, 3, 5, 8, 12, 21, … is associated with the equation x² – x – 1 = 0.

Corresponding unique infinite number sequences are likewise associated with (x – 1)ⁿ = 0.

For example when n = 2, we obtain (x – 1)² = 0, i.e. x² – 2 x + 1 = 0.

And the unique number sequence associated with this equation is the set of natural numbers, i.e. 0, 1, 2, 3, 4, 5, ….

Then when n = 3, we obtain (x – 1)³ = 0, i.e. x³ – 3x²  + 3x – 1 = 0.

And the unique number sequence associated with this equation is the set of triangular numbers,

0, 0, 1, 3, 6, 10, 15, ….

There are strong links here with the binomial theorem, and indeed the diagonal rows (and columns) of Pascal’s triangle can be used as an alternative way of determining the unique number sequence associated with (x – 1)ⁿ for each natural number value of n.

So for n = 4, we obtain the so-called tetrahedral numbers,

0, 0, 0, 1, 4, 10, 20, 35, ….

Now the significance of all these number sequences in the present context, is with respect to the their (infinite) sum of their reciprocals.

So for example, in the first case when we sum the reciprocals of the natural numbers, we obtain

1 + 1/2 + 1/3 + 1/4 + …

And of course, this represents the well-known harmonic series, which is the value of the Zeta 1 function i.e. ζ₁(s₁), where s₁  = 1.

Though the sum of this series diverges to infinity, in all other cases for (x – 1)ⁿ where n > 2, the sum of reciprocals of the unique number sequences involved, converge to a finite rational number.

Furthermore a simple general pattern relates to these sums, with the value depending solely on n and given by the simple expression (n – 1)/(n – 2).

Therefore the sum of reciprocals of the triangular numbers associated with (x – 1)³, i.e.

1 + 1/3 + 1/6 + 1/10 + 1/15 + …  =  (3 – 1)/(3 – 2) = 2/1.

Now to show that this sum of reciprocals involves all the natural numbers, we can rewrite it as follows

1 + 1/(1 + 2) + 1/(1 + 2 + 3) + 1/(1 + 2 + 3 + 4) + 1/( 1 + 2 + 3 + 4 + 5) + …

Therefore the n^th term in this manner contains the sum of the 1st n natural numbers!

Then the denominators in reciprocals of number sequences for (x – 1)ⁿ, where n > 3, contain compound combinations of all the natural numbers (to n) for the nth term.

The importance (in this context) of these sums of reciprocals is that they can then be used as the alternative Zeta 2 expressions, where each individual term of the Zeta 1 - both in its sum over natural numbers and product over primes expressions - now represents the sum of additive terms with respect to the Zeta 2 infinite series.     

So again with respect to the Zeta 1 function, where s = 2, the 3^rd term of the sum over natural numbers expression = 1/9.

This can now be expressed through the alternative formulation of the Zeta 2 (representing the sum of compound natural number terms).

So we use here the unique digit sequence associated with (x – 1)¹¹ = 0,

i.e. 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 11, 66, 286, 1001, 3003, 8008, …

There is a relatively quick way of working out the terms in all these sequences based on the universal ratio of the (t + 1)^th to the t^th term = (t + n – 1)/t.

Therefore as 8008 (the last number given) is the 7^th  term, the next term = 8003 * (7 + 11 – 1)/7 = 19448.

These sequences can all be found, listed to a large number of terms at

The On-line Encylopeadia of Integer Sequences.

So the sum of reciprocals of the sequence associated with (x – 1)¹¹ = 0, is

1 + 1/11 + 1/66 + 1/286 + 1/1001 + 1/3003 + 1/8008 + 1/19448 + …

= (11 – 1)/11– 2) = 10/9.

So the 3^rd term in the Zeta 1 sum over natural numbers expression (where s = 2) =

(1 + 1/11 + 1/66 + 1/286 + …) – 1

And the denominators 11, 66, 286 represent in turn, ordered compound combinations of the first 2, first 3 and first 4 natural numbers respectively.

Then the corresponding 3^rd term in the Zeta 1 product over primes expression (where s = 2) = 25/24

This turn is associated with the alternative Zeta 2 functions based relating to the sum of reciprocals of the unique number sequence associated with (x – 1)²⁶ = 0,

i.e. 1 + 1/26 + 1/351 + 1/3276 + …

= (26 – 1)/26– 2) = 25/24.

And alternative Zeta 2 functions are available for the individual terms in the corresponding Zeta 1 functions (both in the sum over natural numbers and product over primes expressions) for all integer values of s ≥ 2.

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"