We saw yesterday how there is a direct relationship as between the accumulated frequency of the natural factors of each number (i.e divisors) and the corresponding frequency of the the non-trivial (Zeta 1) zeros.

Therefore to estimate the frequency of natural factors to n, we calculate the corresponding frequency of non-trivial zeros to t, on an imaginary scale (where n = t/2π)

The well-known formula for calculation of non-trivial zeros is given as:

(i) t/2π(log t/2π – 1)

Now, I have suggested a simple amendment through the addition of 1 i.e. (ii) t/2π(log t/2π – 1) + 1.

This gives an amazingly accurate estimate of the frequency of non-trivial zeros, which when rounded, even at the highest numbers known (for existence of the zeros) gives estimates that are either exactly correct (in absolute terms) or in error by no more than 1!

Therefore, with respect to formula (i) where n = t/2π, the corresponding formula for calculation of accumulated frequency of (Type 2) natural number factors is given as

n(log n – 1).

This of course then bears a complementary type relationship with the corresponding formula for calculation of the frequency of (Type 1) primes i.e. n/(log n – 1).

In even simpler terms we can see the dual importance of log n in both Type 1 and Type 2 terms.

From the well known Type 1 perspective, log n approximates the average spread or gap as between primes among the natural numbers.

However from the corresponding Type 2 perspective, log n approximates the average frequency of natural number factors (within each number).

However, though as we have seen the frequency of non-trivial zeros (to t) bears a direct relationship to the corresponding frequency of natural number factors (to n), it is not an exact relationship.

When we look at the frequency of natural number factors, they certainly do not occur in a random fashion.

So each prime (with no non-trivial factors) is followed by a natural number (or numbers) where an accumulation of factors occurs.

Thus again illustrating up to n = 10, we have 0, 0, 0, 2, 0, 3, 0, 3, 2, 3 factors for 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 respectively.

So the non-trivial zeros can best be seen as the attempt to balance the independent behaviour of the primes (with 0 factors) and the corresponding interdependent behaviour of the (composite) natural numbers (with 2 or more factors).

Now, if one thinks back to the example of a crossroads, the recognition that a turn can be both left and right (depending on context) is paradoxical in terms of normal conventional logic, where a turn must be unambiguously either left or right (separately).

In other words, the paradoxical recognition of the two-way interdependence of left and right - which is directly of an intuitive rather than rational nature - comes from the ability to simultaneously "see" the situation from two opposing polar reference frames.

So once again, when one approaches the crossroads in terms of just one such reference frame (i.e. from a S or N direction) unambiguous identifications of left and right turns can be made. However when one simultaneously then tries to view these turns from both N and S directions, left and right turns have a merely paradoxical interpretation.

In principle the appreciation of the nature of the non-trivial zeros is exactly similar.

As we have seen, there are two distinct reference frames (Type 1 and Type 2 respectively) through which we can view the relationship of the primes to the natural numbers.

From the Type 1 perspective, we concentrate on the primes as the building blocks of the natural numbers in their distinct separate identities.

However from the Type 2 perspective, we concentrate on the natural factors within each number with respect to their interdependent identity.

So again from the Type 1 perspective, we are viewing the number system in a quantitative manner composed of prime building blocks.

Then - relatively - from the Type 2 perspective we are viewing each number in a qualitative manner in its relationships expressed through a variety of natural number factors!

And again in particular we have the dual significance of log n, which in Type 1 terms approximates the average gap as between individual primes and then in corresponding Type 2 terms approximates the average frequency of the natural factors of each number.

Therefore the key to appreciation of the very nature of the non-trivial (Zeta 1) zeros is that they represent the simultaneous recognition of the two-way relationship as between the primes and natural numbers (and natural numbers and primes) in both Type 1 and Type 2 terms.

Thus the essential "seeing" of their nature is directly of an intuitive - rather than rational - nature.

In other words, when we attempt to translate the nature of the non-trivial zeros in a rational manner, they appear paradoxical.

Thus we could validly maintain that the non-trivial zeros represent points on an imaginary line where both the primes and natural numbers are dynamically identical with each other (which of course makes no sense in Type 1 or Type 2 terms, as considered separately).

We could equally maintain that these zeros represent points where both the quantitative and qualitative aspects of number are identical (again in a dynamic approximate manner).

Put in an equivalent fashion, we could say that they represent the dynamic identity of both the cardinal and ordinal aspects of number.

From another important perspective, we could say that the zeros represent the dynamic identity of both notions of randomness and order with respect to the number system.

Also from yet another important perspective, we could say that they represent the points, where both addition and multiplication with respect to the number system are mutually reconciled.

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