In my last blog entry I indicated that the prime number theorem can be viewed in two complementary ways (that dynamically interact with each other).
From the standard (Type 1) perspective it relates to the frequency of the primes with respect to the natural numbers.
However from the corresponding - largely unrecognised (Type 2) perspective - it relates to the average frequency of the natural factors of a number with respect to its prime factors.
Once again what is important to understand here is the truly complementary nature of both aspects (where the relationship as between the primes and natural numbers is directly inverted).
So again from the Type 1 perspective, numbers are treated in an independent quantitative manner. What this implicitly implies is that all numbers are defined with respect to the default 1st dimension.
Thus any number, n, in this approach is defined more fully as n1. Therefore 2 is 21, 3 is 31, 4 is 41 and so on.
From this perspective log n measures the (estimated) average spread (or gap) as between each prime (up to the number n).
This means that the intervening numbers between the primes are all composite. So log n (strictly log n – 1) therefore represents an unbroken sequence of composite natural numbers!
Then n/log n, measures the (estimated) frequency of the primes among the natural numbers.
It is also important to recognise the additive nature of the relationship in that the frequency of primes (up to n) combined with the remaining frequency among composite natural numbers is connected through addition (with the sum = n).
However from the Type 2 perspective, numbers are treated - relatively - in an interdependent qualitative manner. What this implies is that all numbers relate directly to a dimension (power or exponent) that is defined with respect to the default base quantity of 1.
Thus any number in this approach is defined as are defined as 1n. Therefore 2 is now 12, 3 is 13, 4 is 14 and so on.
The distinct prime factors can be seen here as representing the corresponding dimensional power of the number (with which the Type 2 perspective is directly concerned).
So using the same example as in yesterday's entry the number 12 has two distinct prime factors i.e. 2 and 3.
This therefore constitutes 2 possible dimensions which could be geometrically illustrated as the sides of a rectangle (resulting from multiplying 2 by 3 (i.e. 2 * 3).
However 12, as we have seen (according to my manner of definition) has 5 natural factors i.e. 2, 3, 4, 6 and 12.
Thus through multiplication of these factors 5 possible dimensions result (geometrically represented by the hypercube with sides 2, 3, 4, 6 and 12 respectively).
Thus both the prime and natural factors give rise directly here from the Type 2 perspective to the notion of number as representing dimensional powers; however, as we have seen, with the (standard) Type 1 aspect both prime and natural numbers are treated directly as number quantities (defined with respect to the default dimension of 1).
And what is vital to recognise is that, in dynamic interactive terms, The Type 1 and type 2 aspects are quantitative and qualitative (with respect to each other).
Therefore whereas the Type 1 aspect relates directly to notions of number independence (suited to addition), the Type 2 aspect relates directly to the notion of number interdependence (in the dimensional changes brought about through the multiplication of differing numbers.
Whereas from the Type 1 perspective, log n relates to an unbroken sequence of (composite) natural numbers, in Type 2 terms it relates in complementary fashion directly to the number of prime factors of a number.
And whereas from the Type 1 perspective, n/log n relates to the frequency of primes among the natural numbers (which is connected to the naturals through an additive relationship), again in a complementary Type 2 manner, n/log n relates to the frequency of the natural factors (with respect to the prime factors) with the relationship between both in this case due to a multiplicative relationship. So again n/log n in this case measures the number we multiply the (average) frequency of the prime factors to obtain the corresponding (average) frequency of the natural factors of a number.
Finally whereas in Type 1 terms we move from the individual numbers to consideration of relationships with respect to the overall collective set of numbers, in Type 2 terms we move in reverse fashion from the overall set of numbers to consideration of the results for the (average) individual number.
So this illustrates well how with respect to the dynamic interactive nature of the number system we have the the two-way relationship of the part with the whole and the whole with the part respectively.
So we have in fact two complementary perspectives on the prime number theorem which in dynamic interactive terms are the direct opposite of each other.
Again in the (standard) Type 1 approach, it is customary to view the primes as the basic (quantitative) building blocks of the natural number system.
However from the Type 2 perspective, this is directly reversed with the natural numbers in a (qualitative) ordinal manner seen as the building blocks of each prime number.
So once again if we look at the number 6 for example in Type 1 terms this natural externally represents the quantitative product (in cardinal terms) of its two prime factors (i.e. 2 and 3).
Thus we treat each cardinal number here as an indivisible whole unit!
However, when we look internally, as it were at the composition of both prime numbers, we find that they necessarily already comprise a sequence of natural numbers in a qualitative ordinal manner.
Thus 2 necessarily consists of a 1st and 2nd member, whereas the number 3 necessarily consists of a 1st, 2nd and 3rd member!
Therefore we cannot explicitly even refer to a prime externally in cardinal terms, without implicit recognition of its natural number composition in an ordinal manner.
Of course from the opposite perspective, we cannot explicitly refer to the ordinal natural number composition of a prime, without implicit recognition of its prime identity in cardinal terms!
Thus in dynamic interactive therms both the cardinal and ordinal aspects of number are ultimately fully interdependent with each other (in an ineffable manner).
It is important to remember that the quantitative also has a qualitative aspect (when viewed from an opposite perspective) and vice versa the qualitative a quantitative aspect.
Thus the Type 2 perspective I offer here of the prime number theorem is presented now in a quantitative type manner.
However it is vital to keep remembering that in dynamic interactive terms, complementary (opposite) poles are always quantitative as to qualitative (and qualitative as to quantitative) with each other.
Thus the importance of the relationship between the primes and natural numbers when seen from this perspective, is that they serve as the crucial means through which both the quantitative (and qualitative aspects qualitative aspects) of the number system - and indeed ultimately all created phenomena - are communicated. with each other.