Yesterday I was dealing in my blog with the shadow approach to the relationship of the primes and natural numbers (using the Type 2 aspect of the number system). More correctly, it should be referred to (in qualitative terms) as the imaginary counterpart of the customary real Type 1 approach to the Riemann Zeta Function.
However we will gradually see that it is also subtly embedded - as it were - in the customary Zeta Function and that indeed it is vitally necessary to give meaning to the values of the Zeta Function for s < 0.
This alternative system though it has indeed a quantitative aspect - which I will presently deal with - is primarily geared to deal with the qualitative nature of the primes.
And as I have long maintained that the Riemann Hypothesis lends itself more readily to qualitative - rather than quantitative - appreciation it does indeed warrant further exploration. And here again - quite remarkably - we will quickly unravel both an alternative Prime Number Theorem and Riemann Hypothesis requiring nothing more, from a quantitative standpoint, than elementary Mathematics!
From the Type 1 perspective we start with the individual primes (as cardinal) and then attempt to express the natural numbers as entailing the product of a unique combination of these primes. However, when looked at more closely, the holistic capacity of primes with respect to the natural numbers is of a qualitative rather than quantitative nature. And appreciation of this requires looking at this relationship from the "imaginary" Type 2 perspective.
We then showed yesterday - in what initially is experienced as a quite remarkable discovery - that the relationship of the primes to the natural numbers is completely inverted from this alternative standpoint.
Here - viewing number now from an ordinal perspective - we start with the natural numbers (as representing the various roots of a given prime number). We then show how each prime is viewed as the unique product of all the natural numbers roots (from 1 up to and including the prime in question).
So to obtain the 17 roots of 1, we must raise the 17th root of 1 to each of the natural numbers from 1 to 17. So each natural number root here is qualitatively unique. However the prime number 17 (as number of roots) is of a cardinal nature.
So just, as properly speaking, the collective nature of the primes when using the Type 1 perspective, is of a qualitative, here in reverse, the individual nature of each prime (as comprising its 17 roots) is properly of a quantitative nature. This suggests that in the more comprehensive Type 3 approach, the two-way interdependence of both systems is required. So here the Type 2 would also be needed to explain fully the Type 1 and likewise the Type 1 to fully explain the Type 2, with eventually both seen as entirely interdependent.
So at this final stage, the primes and natural numbers are seen as dynamically identical, with the apparent difference between them due to the fact that their distinctive nature arises from attempting to look at them in relative separation, whereas their mutual identity requires using both lenses simultaneously (Type 3).
I will now mention briefly a result I came up up with some years ago. (I have dealt with it before in other blog entries but not specifically in this context!) I felt at the time that it was important, but had to wait some time before realising that it actually belongs to this Type 2 formulation of the Riemann Zeta Function.
I knew that the circular study of primes (as the prime roots of 1) should offer a complementary approach to the standard linear approach.
Now, the problem with these complex roots is that when one takes them as a collection i.e. all the roots of a prime number, they add up to 0 and when multiplied give a product of 1.
So the first insight was to decide to look at their numerical properties with respect to absolute values of the roots.
I then decided to sum up both sin and cos values separately to see if a different pattern applied to both.
So the idea was - say for the simple case of the 3 roots of 1 - to first get the sum, in absolute terms, of the three cos parts of the roots, followed by the sum of two sin parts (Note for prime valued roots there will always be one less sin value)!
Having obtained a total, I then divided by the prime number in question to get the mean of the 3 roots (with each part as separate).
I then continued on methodically in this fashion for each prime number, summing up all the roots (both cos and sin absolute values) correct to 8 and 9 places, before obtaining the average, going all the way to the 127 roots of 1.
It struck me quite quickly that the average for both cos and sin values was zoning in on a fixed value which I worked out was 2/pi.
I then made the interesting discovery that 2/pi can alternatively be expressed as i/log i.
Deriving this is quite simple:
e^(2iπ) = 1^1
Therefore, raising both sides to i,
e^(- 2π) = 1^i
Next raising each side to power of 1/4,
e^(- π/2) = i^i
Taking logs of each side,
- π/2 = i log i
π/2 = - i log i
2/π = 1/(- i log i)
= i/log i
Now this bears direct comparison with the simple expression for the distribution of primes among the natural number i.e. n/log n.
In fact, it is the imaginary counterpart of the alternative real expression.
The idea of increasing to infinity is the Type 1 way of dealing with the relationship as between finite and infinite (in quantitative terms).
The notion of i, as a finite means of embodying the infinite in qualitative terms, is the corresponding Type 2 manner of incorporating infinite with finite notions.
So we have in fact a parallel prime number theorem here arising from the circular study of prime number roots.
So what we are saying is that when we obtain the average of the prime roots of 1 (measuring cos and sin values in absolute terms), the average of these prime roots sums ever closer to i/log i (i.e. 2/π) in both cases. 2/π = .636619772.. .
When I then studied the deviations of the average cos and sin values from 2/π, I made an interesting discovery, for it became quickly apparent that the cos average were always in excess of 2/π while the sin average was always less, with the ratio between cos and sin quickly approaching the value .5. Does this sound familiar?
Needless to say this ratio steadily improves as we get the average cos and sin values for higher numbered primes.
Though I thought it might be coincidental initially, I now have come to the opinion that in fact this does indeed represent an alternative way (from a Type 2 perspective) of stating Riemann's Hypothesis. As always there is a reversal in Type 2 terms. From the Type 1 perspective, calculations of the general distribution of primes according to formula n/log n, yield an approximation that steadily improves with larger n, while the .5 condition for the Riemann Hypothesis is given as a fixed value.
Here in reverse the value for general distribution of roots of prime number roots is given as a fixed value 2/π, while the deviations from 2/π, approximate ever closer to .5.
What is again remarkable however about this is that it represents a truly elementary approach (in contrast to the high duty Mathematics required in the Type 1 case).
Furthermore, its qualitative significance is much easier to appreciate (as befits a circular Type 2 approach).
So we will return again to this topic!