The Strange Case of η( - 1)

The Eta Series for s = - 1 is,

η( - 1) = 1/1^(- 1) - 1/2^(- 1) + 1/3^(- 1) - 1/4^(- 1) + ....

Therefore

η( - 1) = 1 - 2 + 3 - 4 + 5 - 6 + ....

Withe reference to the Riemann Zeta Function the value for this Eta series = 1/4.


The question then arises as what meaning can we give this result!

It is perhaps better in illustrating to start with η(0) = 1 - 1 + 1 - 1 + 1 - 1 + ...

The value of this alternating series = 1/2

It is easy enough in this case to see how this value might arise!

If we take an even number of terms, the sum of the series = 0.

However if we take an odd number of terms the sum = 1.

Therefore it seems reasonable - where the number of terms is unspecified - to average the two values as 1/2. However in illustrating this we need to consider a finite number of terms.

In more general terms the answer here is n/2 (where the nth term when it is odd = 1).

However what is interesting is that when we now consider the series (in Type 1 mathematical terms) as infinite

i.e. 1/(1 + x) = 1 + x + x^2 + x^3 + x^4 + ...,

by setting x = - 1 we obtain 1/2 unambiguously as the correct answer.

i.e. 1/2 = 1 - 1 + 1 - 1 + 1 - ..


So the value here from the infinite perspective as 1/2 can be simply considered as n/2 (where the quantitative value of n is set at a default value of 1).


What I am getting at here is very significant indeed in terms of properly interpreting the nature of the Riemann Hypothesis!

Conventional (Type 1) Mathematics is inherently defined with respect to a default dimensional (qualitative) value of 1. In other words the nature of Type 1 Mathematics is qualitatively linear (1-dimensional) where all quantitative values are ultimately reduced in 1-dimensional terms.
So for example 2^2 = 4^1 (in Type 1 terms).

However Holistic (Type 2) Mathematics - in inverse fashion - is inherently defined with respect to a default base quantitative value of 1. So in concentrating on the nature of qualitative transformation (as with the switch from finite to infinite series) Type 2 interpretation is not directly concerned with the quantitative nature of number but rather as its representation of a holistic transformation (where dimensional powers other than + 1 are entailed)!

Now this will perhaps become clearer when we look at both the finite and infinite interpretation of terms corresponding to η(- 1).

η(- 1) = 1 - 2 + 3 - 4 + .....

We consider this series initially as finite and attempt to sum its value in linear (Type 1) terms.

For example we will initially derive η(0) with 10 terms.

i.e. y = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9

Then when we differentiate y with respect to x we obtain 9 remaining terms on the RHS

i.e. 1 - 2x + 3(x^2) - 4(x^3) + 5(x^4) - 6(x^5) + 7(x^6) - 8(x^7) + 9(x^8)

Setting x = - 1 we obtain

1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9

Then in summing this finite series by grouping terms in pairs
the sum of the first 8 terms is - 1 - 1 - 1 - 1

So, if the number of terms is even the sum of the series = - (n - 2)/2

Thus as we originally started with n = 10 the sum of the first 8 terms = - (8/2) = - 4.
However if we sum the first n - 1 terms (which is now odd) we obtain - (n - 2)/2 + (n - 1).

Thus the sum of the first 9 terms = - 4 + 9 = 5.

As we did before for η(0), since there is a 50:50 chance of obtaining the positive value for the sum (associated with an odd number of terms). So the average = {(n - 1) - (n - 2)/2}/2 = (2n - 2 - n + 2)/4 = n/4

So again with originally n = 10, the sum of the first 9 terms = 5. As there is a 50:50 chance of getting this value, the expected value is therefore 2.5. And this is the value corresponding to n/4 (where n = 10).

What is fascinating is that when we then consider the series in infinite terms through differentiating both sides of original expression we get

1/(1 + x)^2 = 1 + 2^x + 3(x^2) + 4(x^3) + .....

By setting x = - 1,

1/4 = 1 - 2 + 3 - 4 + 5 - .....


So once again the sum for the infinite series is the same as derived for the finite, with the important difference that n is here given a default value of 1.

This strongly suggests that in transforming from finite to infinite expressions the very representation of number itself switches from a specific (quantitative) to a holistic (qualitative) meaning. In other words the expected value of the nth term (when n is odd) is n/4. So when referring to the general ratio (rather than its specific quantitative value) we get 1/4!