I have been emphasising the truly complementary nature of the two ways of calculating the frequency of prime numbers.
Once again the standard analytic cardinal (Type 1) approach uses the well known formula n/log n – 1, where the frequency of primes is measured up to n on a linear scale.
What is interesting is that this linear measurement is with respect to just one independent set of numbers. So for example in measuring the frequency of primes to 100, the estimate is with respect to one unique set (made up of the cardinal numbers 2, 3, 5, 11,....)
What is equally interesting is that each prime (in this cardinal measurement) is composed of multiple part sub-units.
So 5 for example = 1 + 1 + 1 + 1 + 1.
By contrast in a direct complementary manner, the (unrecognised) holistic ordinal (Type 2) approach uses the formula (2t/π)/(log 2t/π – 1). Here the frequency of primes is measured up to t on a circular scale, with n = 2t/π.
As the very nature of the circular scale is to indicate interdependence with respect to number. Therefore any number point can be taken as the initial starting point with which all other numbers in a number group can then be consistently related with each other.
This means that this circular measurement can be taken with respect to potentially numerous sets of numbers. So for example in measuring the ordinal frequency of primes to 100, the estimate is with respect to - potentially multiple sets of numbers (with any of the 100 points valid as the initial starting point).
However - again in complementary manner - what is interesting about each ordinal prime is that it is composed of just one unit.
So the estimate of prime frequency here is with respect to the 2nd, 3rd, 5th, 11th ... members of the ordinal set of members of 100. So whereas the cardinal notion of 5 (as we have seen) is composed of multiple part units (in a quantitative manner), the ordinal notion of 5th has one unique meaning (in the context of a group of 100)
So once again, both the cardinal and ordinal measurement of the frequency of primes are related to each other in a complementary manner requiring analytic and holistic interpretation respectively.
And as the very notion of complementarity requires a dynamic interactive means of understanding, this implies once again that the fundamental nature of the number system in its two-way relationship as between the primes and natural numbers (in both cardinal and ordinal terms) is necessarily of a dynamic interactive nature.
More than anything else this is the crucial insight that needs to be taken on board by anyone following these blog entries which directly implies that the very nature of Mathematics is fundamentally different from what is customarily envisaged.
Now just as cardinal and ordinal complementarity applies in relation to the estimation of the frequency of primes, equally it also applies to the estimation of the frequency of (natural) factors.
So again n(log n – 1) measures the (combined) frequency of (natural) factors of the composite natural numbers up to n on a real scale.
This represents the standard (linear) analytic approach to such measurement.
However the complementary holistic formula t/2π(log t/2π – 1) as we have seen, provides a stunningly accurate calculation for the estimation of the frequency of the Riemann (Zeta 1) zeros up to t on an imaginary scale (where n = t/2π). The imaginary scale in this context provides the means of expressing what is properly of a qualitative circular nature, indirectly in an analytic manner!
Thus the Riemann zeros provide the alternative holistic manner for estimation of the frequency of prime factors.
So what does this precisely mean?
Well if we take 6 as an example we have 2, 3 and 6 as natural factors.
Now we can treat these factors as representing number quantities, which the formula n(log n – 1) directly measures.
However a unique qualitative aspect also applies, so that where multiplication of factors in involved, a (dimensional) qualitative transformation also takes place (which cannot be captured in a linear manner).
So the Riemann Zeros relate directly to the (natural) factors of the composite numbers with respect to their (dimensional) qualitative aspect.
In one way this finding is remarkable. We are accustomed to linking the Riemann zeros to the primes. However though it is certainly true that a complementary (opposite) relationship connects the primes and Riemann zeros, the direct relationship is between the Riemann zeros and the (natural) factors of the composite numbers.