As we have seen the basis of the Zeta 2 Function (in obtaining prime roots of unity with complementary qualitative interpretations) is to provide the unique individual structure of each of the natural number ordinal members of this prime set (in what I refer to as unique circles of interdependence).

Now the interdependence of such prime roots is exemplified by the fact the sum of the roots of 1 = 0. In other words the circular interdependent nature of these roots (when combined) rules out cardinal features, which would imply an independent quantitative aspect!

However a reduced linear measurement of such interdependence can be given basically by ignoring the distinction as between positive and negative values (and also real and imaginary values). And as we have seen this very approach leads to both an alternative prime number theorem and Riemann Hypothesis. So the mean average for both (absolute) cos and sin values (as the measurement of roots) tends to 2/π = i/log i). And the the ratio of deviations of cos to sin average values from 2/π approaches .5!

As we have explained earlier when we express Zeta 2 in terms of Zeta 1 values we have to allow a gap of 2. Therefore s = - 1 for s with respect to Zeta 1 corresponds with s = 3 (with respect to Zeta 2).

Therefore with respect to explaining the denominator (in absolute terms) of ζ(- 1) we refer to the 3 roots of 1 (with respect to Zeta 2).

Now the sum in absolute terms here of these three roots approximates 12/π.

Then converting this to a linear expression of number as a measurement of cardinal independence we decircularise the numerical expression by multiplying by π to obtain 12.

However because this really relates to an initial expression with respect to the ordinal interdependent nature of a group of numbers (as roots) its true nature remains hidden within the context of the standard Zeta 1 Function.

So the best way of looking at the number 12 as the denominator of ζ(- 1) (in absolute terms) is - not as a single number measurement - but rather as a way of numerically measuring interdependence (among several relatively independent numbers).

Not surprisingly, denominators of the Zeta 1 Function (with respect to negative odd integers) tend to be very rich in combinatorial terms.

Now going back to the Zeta 2 Function, the sum of the 3 roots of 1 (representing the natural numbers in ordinal terms i.e. the 1st + 2nd + 3rd roots) = 0.

Then when we add these three natural numbers in cardinal terms they are factors of 12. In other words 1 + 2 + 3 = 6 is a factor of 12.

Now this is a feature that tends (with qualifications) to characterise denominators of the Zeta 1 Function (where the prime numbers are factors).

For example the denominator (absolute) of ζ(- 3) is 120 which is divisible by 5.

And 120 is likewise divisible by 1 + 2 + 3 + 4 + 5 = 15.

The denominator (absolute) of ζ(- 5) is 252 which is divisible by 7. and 252 is divisible by the sum of the first seven natural numbers (= 28).

Then the denominator of ζ(- 9) is 132 which is divisible by 11. And 132 is divisible by the sum of the first 11 natural numbers (= 66).

Finally the denominator of ζ(- 11) is 32760 which is divisible by 13. And this number is divisible by the sum of the first 13 natural numbers (= 91).

This clear pattern which applies only where prime numbers are concerned, breaks down after this (but still holds when suitable small modifications are made).

For example the denominator of ζ(- 15) is 8160 which is divisible by 17. However 8160 is not directly divisible by the sum of the first 17 natural numbers (= 153). However when 8160 is multiplied by one of its prime factors (3) it is indeed divisible!

The sum of the first five roots of 1 (in reduced linear terms) approximates 20/π. Once again we decircularise this expression by multiplying by π to get 20 (and 20 is a factor of 120).

Now 120 is especially rich in combinatorial significance (as the product of the first 5 natural numbers (which - as we have seen is also divisible by the sum of the first 5 natural numbers). Also 120 = (2^3) * 3 * 5. So as a product it includes all the prime factors from 2 to 5 (inclusive).

However there is another quite remarkable feature to this number that directly highlights the cardinal/ordinal interdependence of this number with respect to the primes.

113 (as cardinal number is the 30th prime (as ordinal number); 127 (as cardinal number is the 31st prime (as ordinal).

120 therefore lies exactly half way as between the 30th and 31st prime number.

So so with respect to the 30th it is + 1/2 and with respect to the 31st - 1/2.

Now 30 in turn (now as a cardinal number) lies exactly half way as between the 10th and 11th primes (as ordinal) i.e. 29 (as cardinal) is the 10th prime (as ordinal) and 31 (as cardinal) is the 11th prime (as ordinal).

Now converting once again 10 (now as a cardinal number) lies between - though not exactly half way - between the 4th and 5th prime numbers (as ordinal).

So taking 4 now as cardinal it lies half way between 3 and 5 (i.e. the 2nd and 3rd primes).

So 2 now as cardinal is the 1st prime.

So with respect to the prime numbers involved here there is a there is a perfect cardinal ordinal relationship that winds itself back to the 1st prime.

So 127 (as cardinal) is the 31st prime (as ordinal);

31 (as cardinal) is the 11th prime (as ordinal).

11 (as cardinal) is the 5th prime (as ordinal).

5 (as cardinal) is the 3rd prime (as ordinal)

3 (as cardinal) is 2nd prime (as ordinal)

And finally,

2 (as cardinal) is the 1st prime (as ordinal).

Meanwhile these primes are associated - in the manner demonstrated - with the numbers 120, 30, 10, 4, 2 and 1 (which are all factors of 120).

Now this pattern cannot be shown as clearly with other denominators (which seems related to the fact that not all prime numbers are included in the denominator).

For example the denominator for ζ(- 5) i.e. 252 = (2^2)*(3^2)*7. So 5 is missing as a prime factor here!

So once again the denominators of the Zeta 1 (Riemann) Function for negative odd integers of s do not correspond with normal linear interpretation (where distinct cardinal values of a merely quantitative nature apply).

Rather these numerical values pertain in varying ways to the collective relationship of both ordinal and cardinal type features.

The trivial zeros for negative even integers of s do directly apply to a purely ordinal interpretation of interdependence among numbers of a qualitative nature. However denominator values of the Zeta Function for negative odd integers of s are more complicated in nature as they combine once again the relationship of both ordinal and cardinal type features of number.

## No comments:

## Post a Comment