In my last blog entry, I concluded with the fascinating observation that the same simple formula can be used as an estimate of what - initially - seem as unconnected areas.
Thus as we have seen the formula 2n(n + 1)/π can be used as an estimate of:
1) the accumulated sum of factors of the composite numbers (up to n);
2) the accumulated sum of the (reduced) value of all roots of 1 (up to n).
So once again in the first case, where n for example = 100, we sum up the factors for each composite number (starting with the sum of factors of 4 which = 2 + 4 = 6), and then accumulate the overall sum for all composite numbers up to 100.
In the second case we obtain the individual t roots of 1 for t = 1, 2, 3, 4,.... 100, and then accumulate the total sum of the reduced values of these roots for each value of t.
Once again in this reduced approach the 3 roots of 3 would be expressed as 1, .5 + .866, and .5 + .866 i.e. .5, 1.366 and 1,336 (correct to 3 decimal places) respectively. so we basically just concentrate on number magnitudes in a positive real manner (ignoring both negative and imaginary signs).
What is surprising here is that the first case of common factors is intimately associated with the Riemann (Zeta 1) zeros. Indeed we showed in earlier blog entries, how a surprisingly accurate estimate of the frequency of these zeros can be obtained through considering the aggregate total of the number of factors involved (up to n).
However the second case of the roots of 1, is intimately associated with the unrecognised (Zeta 2) zeros.
Indeed all the roots of 1 (except the trivial case in all cases where the root = 1) represent the non-trivial zeros (from this Zeta 2 perspective).
What this clearly suggests is that the Zeta 1 and Zeta 2 zeros respectively represent close complementary perspectives of the same underlying reality with respect to the number system.
In other words, in both cases the zeros represent an indirect attempt to provide a numerical measurement of the interdependent nature of the number system.
As we have seen from an isolated analytic perspective, we can indeed attempt to view both the primes and natural numbers in independent terms as number quantities.
However the very nature of the relationship between the primes and natural numbers (which can be viewed from complementary cardinal and ordinal perspectives) entails the qualitative notion of a synchronous form of interdependence that underlies the number system.
Now in direct terms the appreciation of this notion of qualitative interdependence is of a holistic (rather than analytic) nature. However indirectly it can then be represented (from the two related perspectives) in a quantitative manner..
So the zeta zeros (Zeta 1 and Zeta 2) therefore serve as indirect quantitative measurements of the qualitative synchronous nature of the relationship as between the primes and the natural numbers (and the natural numbers and the primes).
However, once again it is strictly futile to attempt to grasp this holistic feature of the number system in the conventional analytic manner.
The true test as to whether one can understand in the appropriate holistic fashion, stems from an enhanced ability to see all fundamental relationships in an inherently dynamic interactive manner (as complementary pairings of opposite poles).
Indeed the truly remarkable conclusion is that the Zeta 1 and Zeta 2 zeros in fact are providing fundamentally the same information regarding the qualitative interdependent nature of the number system. However because this information arises from varying perspectives it has indeed the appearance of being different!
In an earlier blog entry, "Zeta 2 Formulation of the Euler Product" I showed how the famous Euler Product can equally expressed in terms of the Zeta 1 and Zeta 2 Functions.
This indicated therefore that the Zeta 1 and Zeta 2 Functions simply represent two complementary ways of looking at the same reality. So likewise the Zeta 1 and Zeta 2 zeros represent two complementary ways of indirectly measuring the holistic (interdependent) aspect of the number system.
One interesting implication of this finding is, that just as the Zeta 1 zeros can be used to correct the deviations associated with predicting the frequency of primes to a given number, the Zeta 2 zeros in principle can be used to achieve the same result (from an alternative perspective).