Where s = – 1

Just to clarify a small point!

In the past two blogs I have been addressing the nature of the Riemann zeta function for s = 0.

However strictly, as in dynamic interactive terms two complementary functions i.e. Zeta 1 and Zeta 2 always interact, I have been addressing the nature of the Zeta 1 function i.e. ζ(s₁), where s₁ = 0.

And just as an important functional equation connects ζ(s₁) and ζ(1 – s₁), likewise in corresponding manner, an important, though very simple, functional equation connects  ζ(s₂) and - in this case -
ζ₂(1/s₂), where ζ(s₂)  = 1 – ζ(1/s₂).

Now, again as we have seen, the one value for which ζ(s₂) is undefined is where s₂ = 1. However this implies also that 1/s₂ = 1 with both sides of the functional equation remaining undefined in this limiting case.

And we have also seen in yesterday's entry, how both the Zeta 1 and Zeta 2 functions are connected through the corresponding Eta 1 (related to the Zeta 1 function).

So, ζ(s₁) = η(s₁)/(1 – s₂).   And as ζ(s₂) =  1/(1 – s₂), this implies that,

ζ(s₁) = η(s₁) * ζ(s₂).

Alternatively we can say that,

ζ(s₂) = ζ(s₁)/η(s₁).


We now move on to the intriguing case of the Riemann zeta function where s (i.e. s₁) =  – 1.

So ζ(s₁)  = 1/1^{– 1} + 1/2^{– 1} + 1/3^{– 1} + 1/4^{– 1} + ...

= 1 + 2 + 3 + 4 + ...

So this represents the sum of the natural numbers, which in the standard linear manner of interpretation, diverges to ∞.

However through  analytic continuation, the Riemann zeta function can be extended from "intuitively meaningful" results for s > 1 to "non-intuitive" results for negative values < 0 (including the limiting value of s = 0). 

As I have stated on so many occasions, this apparent problem with respect to these "non-intuitive" results, reflects a key limitation of conventional mathematical interpretation, where numbers are explicitly understood in absolute terms with respect - merely - to their quantitative aspect.

However in truth all numbers possess both a quantitative aspect, relating to their independent and a qualitative aspect, relating to their corresponding interdependent identity. 

Numbers therefore must be appropriately interpreted in a dynamic interactive manner, where both quantitative and qualitative aspects are understood in a relative complementary manner.

So results of the Riemann zeta function appear as "intuitively meaningful" for values of s > 1, because these concur with the standard quantitative (or analytic) aspect of number interpretation.

However by the same token, results of the Riemann zeta function appear as non-intuitive (for s ≤ 0) precisely because these do not now concur with the standard quantitative interpretation. 

In others words, the meaning of these results is of a direct qualitative (or holistic) nature, which is related to the unconscious - rather than the conscious - aspect of understanding.


However, we can indirectly attempt to convey what is implied by the value of ζ(s₁) =   – 1/12.

Once again we start by attempting to obtain the corresponding Eta 1 expression for,
ζ( – 1)  = 1 + 2 + 3 + 4 + ...

So η( – 1) = 1 – 2 + 3 – 4 + ... 
  
One simple approach, which does not require even simple differentiation, is to start by grouping the terms in pairs. 

Therefore

η(– 1) = (1 – 2) + (3 – 4) + (5 – 6) + …

=  – 1 – 1 – 1 – …

Now we have already seen that ζ(0) = 1 + 1 + 1 + …  =  – 1/2.   

This might suggest therefore that η(– 1)  = – (– 1/2) = 1/2.

However because of the pairing of terms with respect to η(– 1), this means that it contains only half the terms contained in ζ0).   

Therefore, η(– 1) = 1/2 * 1/2  = 1/4.

One can just about intuitively accept - in terms of conventional linear interpretation - that η(0) = 1 – 1 + 1 – … = 1/2.

However in then “converting” this value to obtain the corresponding value for ζ(0) =

– 1/2, the intuitive nature of the new result is lost.

This is due to the fact - as I explained in my last entry - that a proper holistic interpretation is now required.

So what is interesting here is that in obtaining ζ(– 1) we have to now “convert” a value i.e. η(– 1) that is already non-intuitive.

So in a very real sense, the ultimate value obtained for ζ(– 1) = – 1/12 is thereby “doubly non-intuitive” from the conventional linear quantitative perspective.

Therefore the very important point that needs to be stressed is that that an increasingly more refined holistic ability is required to properly interpret the numerical results for ζ(1 – s), where the value of s (representing a positive even integer) continually increases.

So in the case of ζ(– 1), s = 2 which is still relatively simple to interpret in a refined holistic manner. However for ζ(– 3), ζ(– 5), ζ(– 7), … where s = 4, 6, 8, … true holistic interpretation becomes progressively more demanding.

This is something that is not at all yet appreciated in present Mathematics. There are really two forms of interpretation - analytic and holistic - respectively, that properly apply to all relationships. Because for millennia now Mathematics has undergone development largely in just one direction, considerable specialisation has now taken place with respect to its analytic aspect! However its holistic aspect - which in truth is equally important - is not yet formally recognised by the profession!

So quite understandably, true holistic development remains remarkably undeveloped, which explains why the results of the Riemann zeta function - for the simplest of negative (odd) integers -  appear so “non-intuitive”. Again appropriate intuition based on true holistic - rather than conventional analytic - understanding is required.

As we have seen, ζ(s₁) = η(s₁) * ζ(s₂).

Thus we “convert” the result for η(– 1) to obtain ζ(– 1) for the Zeta 1 function by multiplying by ζ(s₂) = 1/(1 – 1/2^{s – 1}).   

When s = – 1, ζ(s₂) = 1/(1 – 4) = – 1/3.

Therefore ζ(– 1) = 1/4 * (– 1/3) = – 1/12.

s₂ = 1/2^{s – 1} = 1/(1/4) = 4 (when s = – 1).

So ζ(s₂) = 1 + 4 + 16 + …  = – 1/3

Again let us imagine a cake divided into 4 portions, where the first portion chosen (= 1 slice) represents 1/4 of total cake.

Then with respect to another of the four starting portions take a quarter portion

(= 1/16 of total cake) and divide it into 4 slices.

Then with respect to one of the remaining (latest) portions again divide into 4 new portions (with each = 1/64 of total cake) and divide one of these into 16 slices.

So when one sums up the (part) fractions of the cake, we get,

1/4 + 1/16 + 1/64 + … = 1/3.

However when we now alternatively add up the whole slices, we get,

1 + 4 + 16 + … =  – 1/3.

Now the first answer literally represents the (conscious) quantitative manner of understanding number relationships, where the whole is viewed in terms of its related parts.

The second answer represents the (unconscious) qualitative manner of understanding number relationships where the part is now viewed in terms of its related wholes. So the very nature of addition is now distinct!

And because unconscious recognition represents the (dynamic) negation of consciously posited reality, the result carries a negative sign.  

Thus again, it takes a true holistic way of appreciating number relationships to make this latter result intuitively accessible.

However in the next entry we will look in more detail at what the important result for

ζ(– 1) properly entails.