We have seen in previous blog entries how the multiplication of primes, with respect to each composite number, is associated with a unique qualitative resonance for each prime (and indeed combination of primes as natural factors of the number involved).
And remarkably we can show a very close relationship as between the qualitative nature of these number combinations (as factors) and the notion of the Riemann (i.e. Zeta 1) zeros.
So put more simply the frequency of the Riemann (Zeta 1) zeros bears a very close relationship with the corresponding cumulative frequency of the natural factors of numbers.
However whereas the (cumulative) frequency of factors of the natural numbers is expressed on a linear scale, the corresponding frequency of Riemann (Zeta 1) zeros is expressed with respect to a circular scale (where the circumference = linear radius * 2π).
So, to convert from the circular notion of the frequency of non-trivial zeros up to a given number (t) to the linear notion of frequency of factors up to given number (n), we set n = t/2π
Therefore, to illustrate with a simple example the cumulative frequency of factors up to n = 10, it should thereby equate with the corresponding frequency of non-trivial zeros to t = 62.83 (approx).
Now using the method I have already suggested, we can manually calculate the cumulative frequency of factors to 10.
So associated with 1, 2 and 3, we have no factors. Remember the rule with primes is to disregard the two factors of 1 and the prime in question as trivial factors!
Then with 4, we have 2 factors i.e. 2 and 4. (Again because 4 is now a composite number representing the combination of primes 2 * 2, we include it also as a factor).
So the important notion to grasp here is that both 2 and 4 (as factors) acquire a unique qualitative resonance through relationship with the no. 4.
Then 5 as a prime again has no factors.
However 6 (as composite) has 3 i.e. 2, 3 and 6. So once again 2, 3 and 6 acquire a unique qualitative resonance through relationship (as factors) to 6.
7 (as prime) has no factors. However 8 (as composite) has 3, i.e. 2, 4 and 8.
Then 9 (as composite) has 2, i.e. 3 and 9.
Finally 10 (as composite) has 3, i.e. 2, 5 and 10.
So if we now accumulate over the 10 natural numbers involved (i.e. n = 10), we get 0 + 0 + 0 + 2 + 0 + 3 + 0 + 3 + 2 + 3 = 13.
Now according to what I have stated, this should then equate well with the corresponding frequency of non-trivial zeros (to t = 62.83).
Now (correct to 2 decimal places) the non-trivial zeros to 62.83 are 14.13, 21.02, 25.01, 30.42, 32.94, 37.59, 40.92, 43.33, 48.01, 49.77, 52.97, 56.45, 59.35 and 60.83.
So this gives a total of 14 zeros (to t = 62.83), which already equates very well with the corresponding frequency of factors (to n = t/2π = 10)) of 13!
In fact, I manually calculated the cumulative frequency of factors to n = 100 which gave 357.
The corresponding frequency of non-trivial zeros to t = 628.3 = 362.
So in fact we can already see a very close relationship as between the two sets of measurements.
Thus we have the extremely interesting phenomenon regarding the non-trivial zeros, which seems to me greatly missing from conventional mathematical appreciation.
Therefore, from a dynamic complementary perspective, though the Riemann (Zeta 1) zeros bear a complementary (opposite) relationship to the primes (without factors), they also bear a direct relationship to the natural numbers (with respect to the cumulative nature of their factors).
One further issue that need to be explained at this point relates to the fact that the non-trivial zeros are measured on an imaginary - as opposed to a real - scale.
Now, the holistic mathematical reason for this is very simple (though also very revealing) in that it points to the fact that these zeros relate to the qualitative - as opposed to quantitative - aspect of number.
Therefore from one perspective, we can attempt to look at the natural numbers as merely quantitative measurements on a linear (1-dimensional) scale.
We can then equally attempt to look at numbers (representing the frequency of factors) equally as quantitative measurements on a linear scale.
However like the directions at a crossroads, the frame of reference has now switched with multiple dimensions (as factors) involved.
Put another way the two measurements are quantitative as to qualitative with respect to each other.
And the basic way of indirectly expressing - what refers to - a qualitative type measurement in quantitative terms, is to use imaginary rather than real units.
So the Riemann zeros - in being directly related to the factor composition of the natural numbers - relate inherently to a qualitative, rather than quantitative notion of number.
In other words, when we view the primes in quantitative terms, the Riemann zeros are thereby of a complementary qualitative nature.
In psychological terms, if we view the primes in a conscious (rational) manner, therefore the Riemann zeros are thereby of an unconscious (intuitive) nature.
In fact this is all deeply relevant, for in very true experiential manner - corresponding well with Jungian type notions - the Riemann zeros represent the (hidden) shadow system of the conventional natural number system (understood in a merely quantitative rational manner).
Put another way, the Riemann zeros - correctly appreciated - represent the (unrecognised) unconscious basis of the natural number system (that is conventionally understood in a merely conscious manner).
However to properly understood the relationship between both the (recognised) conscious and (unrecognised) unconscious aspects, we must view both in a dynamic interactive manner.