In yesterday's blog entry, I mentioned how a complementary version of the Riemann Function exists. And we will see in a later entry how this alternative Function is - in a sense - already contained in the Riemann waiting to be unpacked.
And in reverse the Riemann Function is contained within this new Function likewise waiting to be unpacked. Indeed this is vitally necessary from both perspectives to demonstrate the true nature of the interaction as between quantitative and qualitative aspects.
Yesterday, I mentioned an equivalent Riemann Hypothesis with respect to this complementary Zeta Function. In fact to avoid confusing it with the recognised Function we will refer to it as Zeta 2 (with the existing Function Zeta 1). And the when finally both are brought together in full interactive glory we will refer to this combined Function as Zeta 3.
In fact this "new" Function is not new at all. Rather its newness related to its (unrecognised) intimate interdependence with the Zeta 1 Function (and the Zeta 1 with it in turn). So with the help of the Zeta 2 we will be able to resolve that old problem of giving a coherent explanation for values of the Zeta 1 Function (where s < 0).
Now once again the Zeta 1 Function is concerned with the precise prediction of the (cardinal) primes among the natural numbers.
Zeta 2 in reverse is concerned with precise prediction of the ordinal natural numbers among the primes.
In other words here we are looking at the distribution of the roots of 1 - which are listed in an natural number ordinal fashion - for each of the prime numbers.
So we have switched from a linear to a circular type perspective here.
As in the Zeta 1 case, the accuracy of the distribution quickly improves as the prime number (with respect to the extraction of all its natural numbered roots) increases.
Its value zones in on 2/π (= i/log i) just as the distribution of (cardinal) primes among the natural numbers approaches in complementary manner can be given as n/log n.
We also saw that the deviation from this number of the (absolute) cos value as a ratio of the corresponding (absolute) sin value quickly approaches .5, which in this context of the Zeta 2 Function is the complementary condition to the Riemann Hypothesis in Zeta 1!
I also spent some time working out how these deviations themselves are distributed and discovered an interesting pattern.
So concentrating here on the average cos deviation for some prime p1, if we wish to approximate the deviation for a larger prime p2 we multiply the existing deviation by the square of p1/p2.
So the new deviation d2 ~ d1 * (p1/p2)^2.
This simple procedure is surprisingly accurate though it does not fully predict the new deviation.
I will briefly illustrate!
We will start with the very simple case of the prime number 3 which has 3 corresponding roots i.e. the first second and third that are listed as 1, 2 and 3 in ordinal terms.
Now with respect to cos part, the 3 roots (expressed in absolute terms) are 1, 1/2 and 1/2. The sum of these roots is 2 so that the average is .66666666
The absolute deviation from 2/π (i/log i) = .0030046893...
Now I manually calculated all these deviation ratios for all prime root values (up to 127).
So if we wished to estimate the smaller deviation of the average cos value from 2/π where p2 = 127, then,
d2 ~ d1 * (p1/p2)^2
and d1 * (p1/p2)^2 = (.030046893...) * (3/127)^2
Now given that we have based our calculation on an very early prime value 3, this compares extremely well with the true deviation = .00001623...
Of course if we estimated the deviation (in the case of 127) from a later deviation we would get a much better approximation.
Now we will later discover that a whole new world opens up from this absolute treatment of root values both with respect to the addition and multiplication of roots which can play a great role in understanding the true nature of the Riemann Zeta Function (i.e. Zeta 1).
However I mentioned that the Riemann Hypothesis lends itself primarily to qualitative interpretation so that one for example - from understanding of its nature - can establish in fact that in has no proof (in Type 1 terms) using quantitative Mathematics. However a full explanation does indeed require the complementary blending of both quantitative and qualitative aspects.
As I have repeatedly stated the Riemann Hypothesis points to the central aspect of the primes which is the ultimate identity of quantitative and qualitative aspects in an ineffable manner.
In fact when looked on appropriately, as simultaneous from both perspectives, the natural numbers and the primes are thereby understood as perfect mirrors of each other.
However this cannot be achieved through understanding based on just one logical system (as in Conventional Mathematics).
So we properly require two means of interpretation that are linear and circular (and circular and linear) with respect to each other. As we have seen linear logic is based on isolated (uni-polar) frames of reference that is ill-suited for appreciation of interdependence; circular logic is always based on the pairing of complementary opposites (in a bi-polar fashion).
So the appreciation of the simultaneous identity of both the natural and prime requires complex interpretation in a qualitative sense, where one can keep shifting as between real (1-dimensional) and imaginary (as the indirect rational expression of 2-dimensional) understanding respectively.
Now the sin part of a complex root is imaginary and the cos real.
So the condition of .5 simply relates to this fundamental relationship as between 1-dimensional and 2-dimensional appreciation. In other words, the mystery of the Riemann Hypothesis does not reside in appreciation in accordance with linear or circular understanding (as separate) but in the relationship between both, where ultimately they become identical as perfect mediators of an ineffable reality.
The point at the centre of a circle is equally the centre of its line diameter at the midpoint (1/2) of this diameter.
With Zeta 2 we are directly encountering the qualitative counterpart of this central point.