In earlier blog entries, Simple Estimate of Frequency of Riemann Zeros 1 and Simple Estimate of Frequency of Riemann Zeros 2, I demonstrated the very close link that connects the (natural) factors of the composite numbers with the Riemann (Zeta 1) zeros.
So again if we measure the frequency of the (natural) factors (on the real scale) up to n and then compare this with the corresponding frequency of the Riemann zeros (on the imaginary scale) up to t, where n = t/2π, the two totals will approximate very closely with each other.
Indeed we can then simply convert the Riemann zeros to measurements on the corresponding real scale by dividing each value (on the imaginary scale) by 2π.
So we now can draw direct comparison with the frequency of the (natural) factors of the composite numbers up to n, with the corresponding frequency of these "converted" Riemann zeros also up to n (both now measured on a real scale).
Once again, to illustrate. whereas the (distinct) prime factors of a composite number such as 12 are 2 and 3, the corresponding natural factors of 12 are 2, 3, 4, 6 and 12. Thus in compiling the natural factors of a composite number, we include all factors (including the number itself) except 1.
Now when it comes to prime numbers we do not consider any factors. So for example though a prime number such as 7 is - by definition - divisible by 7, we do not in this case include 7 as a factor precisely because it is not a distinct factor (i.e. is not distinct from the number in question) and has no factors.
By contrast in the case of 12 for example, though 12 is not a distinct factor in this sense, it is indeed comprised of sub-factors.
So in the light of these definitions, the prime numbers all share the same characteristic of possessing no natural factors, whereas - by definition - the composite numbers are comprised of at at least two factors.
This highlights the discontinuous nature of the number system, which alternates as between randomly occurring primes (with 0 factors) and non-random occurring composite numbers (with at least 2 factors).
Thus on the natural number scale we start with 1 (which serves as the unit measurement for cardinal numbers, both prime and composite). Then we encounter the 1st prime i.e. 2 (0 factors), then the next prime 3 (0 factors) then the 1st composite number 4 (2 factors), another prime 5 (0 factors) then the next composite number 6 (3 factors) another prime 7 (0 factors), the next composite number 8 (3 factors) then another two composite numbers i.e. 9 (2 factors) and 10 (3 factors).
Therefore the frequency of the total number of natural factors encountered in the 1st 10 natural numbers = 2 + 3 + 3 + 2 + 3 = 13.
This compares very closely with the corresponding frequency of the "converted" Riemann zeros (up to 10) = 14.
And, as I demonstrated in the earlier entries, the two measurements approximate ever closer to each other (in relative terms) as we increase the value of n.
So once again the formula for calculating the original (non-converted) Riemann zeros, is
t/2π(log t/2π – 1), which is remarkably accurate - not only in relative - but also in absolute terms, though I have suggested that t/2π(log t/2π – 1) + 1 provides an even more accurate measurement in the majority of cases.
Now the corresponding formula for calculation of the "converted" Riemann zeros is,
n(log n – 1), which again is remarkably accurate (in both absolute and relative terms).
This also provides a very good estimate (though not of the same absolute accuracy) of the combined frequency of natural factors up to n.
In one way it is quite remarkable that we are here showing a very close direct relationship as between the Riemann zeros and the (natural) factors of composite numbers, when generally the connection is drawn as between the Riemann zeros and the primes!
Therefore though the relationship of the zeros with the (natural) factors is of a direct nature, the corresponding relationship of the zeros with the primes is of a complementary (i.e. dynamically opposite) nature.
This is why indeed I have frequently used the Jungian notion of the "shadow" to describe the relationship of the Riemann zeros to the primes in that the zeros in fact comprise the collective "shadow" number system to the natural number system (based on the product of primes).
Alternatively, we can express the Riemann zeros as the holistic number system counterpart to the analytic system of the (cardinal) natural numbers.
In other words, customary analytic interpretation of the (cardinal) natural numbers is based on quantitative notions of number independence. The deeper significance of this is that analytic notions imply linear unambiguous interpretations (based on using just one polar reference frame)
However holistic interpretation by contrast is based on the qualitative notion of number interdependence, where at least two complementary polar reference frames are simultaneously adopted in a circular paradoxical type manner.
So in analytic terms, the natural composite numbers are clearly contrasted with the primes. Thus, as we have seen, we can interpret the primes as comprising 0 (natural) factors whereas the composite numbers have least 2 or more factors.
We could also see this as the contrast between notions of randomness and order, with the primes comprising the random and the composites the ordered aspects of the cardinal number system respectively.
Therefore the holistic appreciation of the Riemann zeros relates to the paradoxical notions of both prime and composite aspects (or equally randomness and order) being simultaneously combined in a dynamic interactive manner.
In my recent blog entries, I dealt at length with the paradoxical notion of how the Riemann zeros can be seen as the points, where the notions of randomness and order are simultaneously combined with respect to the cardinal number system.
Indeed, a fascinating alternative way can now be used to suggest the same dynamic notion.
As we have seen in standard analytic terms, the primes and the composite natural numbers are clearly divided with once again the primes having 0 factors and the composites 2 or more.
However whereas the factors of the composite numbers occur in discontinuous blocks (i.e. of 2 or more) the Riemann zeros occurs always as single figures.
Now, remarkably all the Riemann zeros can thereby be defined as numbers with 1 factor. Therefore they fall neither into the category of the primes or composite numbers as separate but rather as combining what is common to both primes and composites. Thus though in analytic terms, the primes and composites are necessarily separate (with respect to factor composition) in a dynamic holistic manner they can be seen as approximating total identity with each other. And this recognition comes from the ability to simultaneously combine both cardinal and ordinal appreciation - which are the reverse of each other - in a common intuitive recognition.
This holistic recognition is what happens in a more obvious way, when we we able to see left and right turns at a crossroads as interdependent (so that what is left from one perspective is right from the other and vice versa).
So in this regard it is exactly similar with respect to the primes and the (composite) natural numbers when the two reference frames of cardinal and ordinal recognition are simultaneously combined.
However, quite remarkably there is no formal recognition whatsoever within Conventional Mathematics of this vitally important holistic manner of dynamic interpretation.
Put more bluntly, we have now been training ourselves for millennia to look at mathematical relationships in a fundamentally distorted manner.