Incredible Nature of the Zeta Zeros (25)

We can perhaps throw further light on the nature of the Type 1 non-trivial zeros with respect to calculation through the general formula for their frequency.

So for example using this formula we can calculate how many zeros would be obtained on the imaginary line up to 100!


Riemann himself suggested the formula in his famous 1859 paper on the primes as

t/2π*log(t/2π) - t/ 2π.

Now in a more complete manner it is given as:

t/2π*log(t/2π) - t/2π + O(log t).

However ignoring the last term it still gives – unlike its counterpart for the frequency of primes (t/log t) - surprisingly accurate results (even over a very limited range).

So the formula predicts 28.17 non-trivial zeros up to t = 100 (as against the correct result of 29)!


What is perhaps surprising is that this formula can be derived with reference to my holistic (Type 2) approach, with the added benefit of throwing much greater light on the true nature of these non-trivial zeros.

We start here by considering the general distribution of primes. As we have seen the spread between these primes steadily increases as we ascend the natural number scale and is approximated by log t.

Put another way the frequency of primes up to t is approximated by 1/log t.

Now in an inverse manner as the frequency of prime numbers decreases, the frequency of remaining composite numbers increases. So for example as the general frequency of primes falls to ½ of its previous level the corresponding average unbroken frequency of composite numbers roughly doubles.

However there is not an exact inverse relationship here.

For example if the average frequency of primes is 1 in 4, the average unbroken sequence of composite numbers would thereby be 4 – 1 = 3.

So therefore is we measure the average frequency of primes as 1/log t, the average unbroken sequence of composite numbers is log t – 1.


Now again the key distinction as between the primes and the composite natural numbers is that whereas the primes represent a measure of number independence (with no factors other than the prime in question and 1), the composite natural numbers represent a measure of number interdependence (where each natural number represents a unique combination of prime factors).

So the average number of primes up to t is obtained by multiplying t by the average frequency of primes i.e. t * (1/log t) = t/log t.

In inverse fashion the average frequency of composite (interdependent) numbers
= t * (log t – 1). It must be remembered in this context of interdependence that each (individual) member is counted as whole group. Thus each individual member of a group of 3 (for example) - through such interdependence - counts as 3!

However whereas independent notions properly relate to a linear, interdependent notions properly relate to a circular scale (where again each number through relationship occupies all delineated points on the circle).

So the interdependence of an unbroken group of composite numbers would thereby be represented on the circumference of the circle of unit radius.

So the circular unit of measurement here = 2π.

An unbroken group of 8 composite numbers would thereby be geometrically represented as 8 equidistant points on this unit circle.

Thus expressing interdependence between them by the total circular length 2π there would be 8 different starting points (as - relatively - independent) for connecting with all other points as representing interdependence. So our numerical measurements here necessarily reflect a reduced measurement of this dynamic relationships as between (quantitative) independence and (qualitative) interdependence respectively!


Then to convert to linear units we would divide by 2π.

So therefore the formula for average frequency for such interdependent numbers

= t/2π{log (t/2π) – 1} = t/2π*log (t/2π) – t/2π.

So we can see that the formula for general frequency of the trivial zeros represents an inverted version of the general frequency of primes so that as the frequency of primes for example halves the corresponding unbroken sequence of composite numbers roughly doubles.

Then, whereas primes relates to linear (independent), the composite numbers relate to circular (interdependent) notions.


Therefore we can clearly see here that whereas the primes represent the independent aspect of the number system, the non-trivial zeros represent the corresponding interdependent aspect.

This indeed is why the non-trivial zeros are dual to the primes (and vice versa).
In other words the primes and the non-trivial zeros in this sense represent the two extreme poles with respect to the number system.

So at one extreme we can consider each number as independent; however at the other extreme we can consider all numbers as necessarily interdependent with each other.
In between these two extremes number possesses both independent aspects (as quantitative) and interdependent aspects (as qualitative) in dynamic relationship with each other.


So conventional understanding of number represents but an extreme case where we attempt to understand its nature in an absolute independent manner (as quantitative).

The non-trivial zeros however directly point to the opposite extreme, where understanding of number is so dynamic that we can no longer separate its quantitative aspect (as independent) and its qualitative aspect (as interdependent) respectively. So we are attempting here to understand number simultaneously as both quantitative and qualitative in both analytic and holistic terms respectively!


Now we can only approximate this extreme in phenomenal terms. However this is what the non-trivial zeros represent i.e. the closest approximation in the phenomenal number realm to pure ineffable reality (i.e. where quantitative and qualitative aspects in the combined analytic and holistic appreciation of number, can no longer be distinguished from each other).

So from a physical perspective, the non-trivial zeros represent approximations to pure energy states; in complementary psycho spiritual terms they represent approximations to pure (intuitive) energy states (where reason operates seamlessly in such a refined manner with intuition as to become fully transparent).

And if we are to understand the nature of the non-trivial zeros, without gross reductionism, such highly refined understanding (seamlessly blending both rational and intuitive aspects) will be ultimately required!


And of course once again this explains why the non-trivial zeros line up neatly on an imaginary line.

Again in qualitative terms, the mathematical notion of the imaginary relates to the indirect representation of holistic circular notions in a linear manner!

So even though the non-trivial zeros are given a numerical measurement, these properly represent a (linear) representation of relative type approximations with respect to the (circular) integration of both the (quantitative) analytic and (qualitative) holistic aspects of number (where both are understood in complementary terms as ultimately identical).