The Critical Region (1)

As we have seen the Euler Zeta Function

1/(1^s) + 1/(2^s) + 1/(3^s) + 1/(4^s) +..... is defined for all values of s > 1.

However in standard linear terms we cannot give numerical meaning to the function for other values of s.

For example when s = 0, we generate the series 1 + 1 + 1 + 1 +.... which - again in conventional terms - diverges to infinity.

However Riemann showed that in his treatment of the function where s can take on any complex value that it is possible to extend the domain of definition of the Function for all values of x (except 1).

The critical region involves values of s from 0 to 1. It has long been known that all the non-trivial zeros must lie in this region (with the Riemann Hypothesis suggesting that they all lie on the line (for real part of s = .5)


Now this is where qualitative - as opposed to mere quantitative - interpretation of numerical becomes extremely important.

Once again it is not possible to give a finite meaning to the sum of a series such as 1 + 1 + 1 + 1 +.... which clearly gets larger and larger and in conventional terminology diverges to infinity.

However in the Riemann Zeta Function the sum of this series (and a vast range of other divergent series) are indeed given a definite finite value. So this raises the very obvious question as to what such a result can mean. And the fascinating answer is that it points in all cases to an additional holistic qualitative interpretation in accordance with Type 2 Mathematics.

The ultimate implication is that we cannot properly understand the very meaning of the Riemann Hypothesis in the absence of Type 2 mathematical understanding. And once we do establish the true meaning of the Hypothesis it becomes readily apparent that it can neither be proved nor disproved in standard mathematical terms (i.e. in accordance with Type 1 interpretation).


It may be helpful at this stage to raise an area in relation to Fibonacci type number sequences that initially provided for me many of the insights regarding qualitative interpretation of numerical values that are so useful with respect to the Riemann Zeta function.


Yesterday, in my related blog on "The Spectrum of Mathematics" I briefly mentioned these. So I will myself here:


For example the Fibonacci Sequence can be obtained with reference to the simple quadratic equation x^2 - x - 1 = 0.

What we do here is to start with 0 and 1 and then combine the second term (* 1) with with the first term (*1) to get 1. Now these two values are obtained as the negative of the coefficients of the last 2 terms in the quadratic expression. So the last 2 terms in the sequence are now 1 and 1. So again combining the second of these (*1) with the first (*1) we now obtain the next term in the sequence i.e. 2 So the final 2 terms are now 1 and 2 and we continue on in the same manner to obtain further terms.


Now a fascinating aspect of such sequences is that we can then approximate the positive value for x in the original equation (i.e. phi) through the ratio of the last 2 terms in the sequence (taking the larger over the smaller).


The equation x^2 - 1 = 0 gives the correspondent to the pure 2-dimensional case where the values for x = + 1 and - 1.

This corresponds to the general quadratic equation x^2 + bx + c = 0 where b = 0 and c = - 1.

So in starting with 0 and 1 we keep adding zero times the second term to 1 times the first to get 0 as the next term. And it continues in this fashion so that we get 0, 1, 0, 1, 0, 1,....

Now what is interesting here is that we cannot approximate the (positive) value of x directly through getting the ratio of successive terms which will give us either 0/1 or alternatively 1/0.

However we can obtain the value directly through concentrating on the ratios of terms (occuring as each second term in sequence). In this we get either 1/1 or 0/0. The first would give us the conventional rational quantitative interpretation using linear (1-dimensional) logic. However the second actually corresponds to the qualitative holistic interpretation according to circular (2-dimensional) logic. This can be expressed as the complementarity of opposite poles so that 0, which numerically is given here could equally be represented as 1 - 1 where both aspects must be taken as a pairing.

So to sum up:

Thee initial equation x^2 - 1 = 0 i.e. x^2 = 1, is of a pure 2-dimensional nature. Therefore we can only obtain meaningful quantitative solutions by taking the ratio - not of successive terms as in linear terms - but rather the ratio of every second term (as befits 2-dimensional interpretation).


Two results now arise. The first, 1/1 gives us the (positive) quantitative result of the equation (i.e. the square root of 1).
The second 0/0 gives the qualitative basis of this result based on circular 2-dimensional understanding (that entails the complementarity of opposite poles).

When we attempt to obtain the root in linear terms through the ratio of opposite terms we get either 1/0 or 0/1. What both of these indicate is a relationship between two different interpretations, 1/0 (as between 1-dimensional and 2-dimensional and in reverse fashion 0/1 (as between 2-dimensional and 1-dimensional).

Both of these result from the attempt to split up what is inherently of a 2-dimensional nature (in qualitative terms) in a manner amenable to 1-dimensional linear understanding (which is not appropriate in this context).

So we cannot interpret the behaviour of such a sequence without reference to its qualitative dimensional characteristics. Because of the merely reduced quantitative interpretation of symbols employed in Type 1 Mathematics, these qualitative aspects are never properly investigated.


So when s = 0 the Riemann Zeta Function results in the sum of terms,

1 + 1 + 1 + 1 +...... which diverges in linear (1-dimensional) quantitative terms.

However it is possible to provide a finite value for this series. In my piece on "Holistic Values" on "The Spectrum of Mathematics" blog I explain how this is done:


The Zeta Function is defined as

1 + 1/(2^s) + 1/(3^s) + 1/(4^s) +......

If we consider just the even values terms and subtract double of each of these terms from the original series we obtain the well known Eta Function which is defined in terms of alternating terms

1 - 1/(2^s) + 1/(3^s) - 1/(4^s) +......

Now through multiplying each of the even valued terms by 2^s we can derive the original terms in the Zeta Function.

This therefore enables us to establish a simple relationship as between the two Functions so that the Zeta Function = Eta Function divided by {1 -1/[2^(s - 1)]}

When s = 0 the Eta function results in the alternating sequence of terms

1 - 1 + 1 - 1 + 1 - ...

Now the sum of this sequence does not properly converge in conventional terms.

When we add up an even number of terms the value = 0; however when we add an odd number the value = 1. Thus by taking the average of these two results we can come up with a single answer = 1/2.

And then from this Eta value the corresponding Zeta value can be easily calculated = -1/2.


There is in fact another way of doing this: If we attempt to obtain the value of 1/(1 - x) we generate the infinite series:

1 + x + x^2 + x^3 + ....


Clearly this latter series only converges for values of x between - 1 and + 1.

But the former expression can be defined for all values of x (except where x = 1).


Now when x = - 1, 1/(1 -x) = 1/2;

If we attempt to express the equivalent series in terms of x = - 1, we obtain

1 - 1 + 1 - 1 +.... which gives us the result that we have already calculated through another means.

However we have already used this Eta value to calculate the corresponding value for the Zeta series where s = 0

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


Now this series can equally be generated by letting x = 1 in our series

1 + x + x^2 + x^3 + ....


So 1 + x + x^2 + x^3 + .... = - 1/2;


However 1/(1 - x) = 1 + x + x^2 + x^3 + .... = - 1/2 (when x = 1).


However 1/(1 - x) = 1/0 (when x = 1.

What this establishes therefore is that the famed result for the Riemann Zeta Function where s = 0 involves the relationship as between linear and circular type interpretation.

In other words the result, - 1/2 is actually the attempt to express the qualitative nature of circular (2-dimensional) understanding in a linear (1-dimensional) manner. Once again 2-dimensional interpretation involves two poles as an inherent pairing that are positive and negative with respect to each other. So if we take the negative pole and attempt to express it as a fraction of the pairing we get - 1/2.