Tuesday, June 4, 2013

Reformulating the Riemann Hypothesis

In my last blog I was at pains to explain the significance of 1/2 in the context of the Zeta 2 Function.

So once again,

1/(1 – s) = 1 + s1 + s2 + s3 +……

Now when s = – 1, 1/(1 – s ) = 1 – 1 + 1 – 1 +……  = 1/2.

In this case, the value of s =  – 1, corresponds to the 2nd root of 1, with the values for the first two terms of the series continually repeating in cyclical groups of 2. So as the value of the infinite series keeps alternating as between 0 and 1, the value of 1/2 given by the formula represents the mean average of these two possible values.
Therefore, the series keeps alternating as between an even and odd number of terms.
So when the number of terms is even, the value of the infinite series corresponds to the value of the first two terms.

From a quantitative perspective this implies that that the value of the series = 1 – 1 = 0 (where positive and negative terms cancel out); from  a qualitative perspective it implies interpretation corresponding to the 2nd (of two dimensions) based on the complementarity of opposite poles i.e. circular both/and logic (which implies paradox). 

When however the series is odd, from a quantitative perspective, the value of the infinite series corresponds to the first term = 1; from a qualitative perspective this implies interpretation corresponding to the 1st (of two dimensions) i.e. linear either/or logic (that is unambiguous). 

I have illustrated the nature of these two types of understanding on countless occasions previously with respect to  assigning directions at a crossroads. 

When we apply a single frame of reference to movement e.g. "up" or "down" a road, the designation of  left and right turns is then unambiguous (as either left or right).

So interpretation here corresponds to linear logic based on just one independent  reference frame. 

However, when we consider both frames as interdependent, then interpretation corresponds to circular logic of a paradoxical nature (as both left and right depending on context).     

So insofar as we can consider relationships as independent based on isolated poles of reference, then linear logic (based on the 1st dimension)  is appropriate. However insofar as we consider relationships as interdependent, based on the dynamic interaction of opposite poles, then circular logic (related to the 2nd dimension) is now appropriate.

Thus at a minimum, to successfully deal with both issues of independence and interdependence with respect to the number system, we need 2-dimensional interpretation.

This in turn implies that with respect to numbers, both independence and interdependence are defined in a relative - rather than absolute - fashion.  

Therefore once again the the key problem with Conventional Mathematics is the fact that it is qualitatively defined in a merely 1-dimensional manner.

So from this linear rational perspective, no clear distinction can be made as between independence and interdependence. Clearly when one attempts to view numbers as absolutely independent, then all relationships between numbers (which imply qualitative notions of interdependence) can only be dealt with a reduced manner!         

As the ultimate nature of the number system implies intimate notions of interdependence (with respect to the relationship of the primes to the natural numbers, and the corresponding relationship of the natural numbers to the primes) quite simply it cannot be properly interpreted within the conventional mathematical paradigm (based solely on 1-dimensional  interpretation).

And once again this constitutes the key qualitative explanation as to why the Riemann Zeta Function remains undefined (where s = 1).

Because this interpretation is 1-dimensional and thereby absolute in nature, no meaningful relationship can be established as between analytic notions of number independence (on the one hand) and holistic notions of number interdependence (on the other) both of which are necessarily of a relative nature. 

So not alone can the Riemann Hypothesis neither be proved (nor disproved) from this perspective; more importantly it cannot even be properly interpreted in such limited terms!

So we have looked at interpretation of the value of the infinite Zeta 2 series in the simplest case where s = – 1.

This - as we have seen - corresponds with the 2 roots of 1 (where – 1 represents the single non-trivial root). Then the solution for the infinite equation = 0 (where we take successive groups of two terms).

Now we can extend this thinking to all prime numbered roots.

So for example in the case of the 3 roots of 1, two of these i.e. the two roots other than 1, – .5 + .866i and  – .5 - .866i, will be non-trivial.

So when we now sum the infinite Zeta 2 series in successive groups of 3 terms, once again the sum = 0. And this sum of all terms will be identical with the sum of the first 3 terms i.e. 1 + s1 + s2.

However this leaves two other possible situations where a finite solution to the series arises.

For example if we take 1 term the sum = 1. Now when we take 2 terms (where s1= – .5 + .866i), the sum 

= 1 – .5 + .866i   = .5 + .866i.

However we could equally take  s = – .5 + .866i in which case the sum of 1st two terms =  1 – .5 –.866i = .5 – .866i.

So the average of these two possible values for the second term = (.5 + .866i + .5 –.866i )/2 = 1/2.

Therefore in this way, the series can have 3 possible answers!

When we groups terms from the start with 3 in a group, the sum of the infinite series = 0.

When however we allow for one more term (than a multiple of 3) the sum of the infinite series = 1.

Finally when we allow for 2 more terms (than a multiple of 3) the (average) sum of the infinite series = 1/2.

Therefore if we now again average the value of the infinite series over the 3 possible answers in this case, the sum of the infinite series = 1/2.

So in this way, the value for the infinite series will in all cases = 1/2, where s represents the non-trivial prime numbered roots of 1 .

And these are what I define as the non-trivial solutions for the Zeta 2 Function!

Because both the Zeta 1 and Zeta 2 Functions are complementary, in dynamic interactive terms, this thereby provides the key explanation as to why all the non-trivial solutions for the Zeta 1 expression lie on the line through 1/2.

From a quantitative perspective, the significance of 1/2 is that it represents the midpoint of the additive identity (0) and the multiplicative identity (1).

From a corresponding qualitative perspective, the significance of 1/2 is that it represents the midpoint of both linear interpretation (based on isolated polar reference frames) and circular interpretation (requiring complementary reference frames).

Thus the very means of reconciling addition and multiplication (in quantitative terms) is thereby inseparable from the corresponding means of reconciling linear (analytic) with circular (holistic) type interpretation.

When appreciated in this light, the Riemann Hypothesis can be seen as the central condition for reconciling both addition and multiplication (in quantitative terms) through the corresponding reconcilation of both (linear) analytic and (circular) holistic interpretation (from a qualitative perspective).

In other words the Riemann Hypothesis implies that ultimately both quantitative and qualitative aspects of the number system are inseparable.

No comments:

Post a Comment