The Zeta 1 (Riemann) can be seen to operate in a complementary fashion from the Zeta 2 zeros, which is just another way of saying that from a dynamic interactive perspective, both sets of zeros are intimately related.
Whereas with respect to the Zeta 2, each zero can be (indirectly) given a relatively independent quantitative identity, the collective sum of zeros (together with the default root of 1) reveals their true qualitative nature.
So for example with respect to "3" as a prime, – 1/2 + .866i , – 1/2 – .866i and 1, represent the 3 roots of 1. These indirectly express in a quantitative manner, the notions of 1st (in the context of 3), 2nd (in the context of 3) and 3rd (in the context of 3) respectively, with the first two representing Zeta 2 zeros.
Thus the 3 roots have a relatively independent identity. However the collective sum of these roots = 0, reveals the true qualitative nature of their relative interdependence with each other, through yielding a result with no quantitative significance!
Now it works somewhat in reverse with respect to the Zeta 2 zeros. Here each individual zero, again indirectly expressed in a quantitative manner, reveals its true qualitative nature.
To be precise, to must see each zero as part of a pairing where in general terms 1/2 + it always implies 1/2 – it likewise as a zeta zero.
So in the simultaneous recognition of both zeros, the imaginary parts cancel out to leave 1.
And just as 0 deeply symbolises (in an indirect additive manner) the qualitative aspect of understanding, likewise 1 deeply symbolises (in an indirect multiplicative manner) the qualitative aspect.
However, associated with the collection of Zeta 1 zeros is - in complementary - terms a relatively independent quantitative aspect. and it is this aspect, which comes into play in correcting the deviations from the general formula to estimate the frequency of primes to a given number, so that ultimately an absolutely correct estimate can be approached.
However it is still important in this context to recognise that such calculations using the wave forms associated with the Zeta 1 zeros, as corrections on a general estimate of frequency, remain strictly of a relative nature.
In other words, regardless of how many zeros are used in the correction process, the answer will never give the exact frequency of primes (which is an absolute number). Rather the answer will eventually approximate so close to the exact frequency, that one can then easily estimate the correct answer with a high degree of confidence.
However, before moving on, let us try to probe more deeply the exact nature of the Zeta 1 zeros.
As is well known, the exact distribution of the primes is represented by a discontinuous step function.
So for example at 1, we have yet encountered no prime. However then at 2 the frequency of primes increases to 1. Then at 3 it increases again by 1 to 2. However at 4 no change takes place, before again at 5 we have another discrete change of 1, as the cumulative frequency of primes increases again by 1 to 3. So the function representing cumulative frequency, remains completely flat and horizontal over the numbers which are not prime and then increases in a discrete manner - always by 1 - when a new prime is encountered.
It is somewhat similar with respect to the corresponding function, representing the cumulative frequency of natural number factors. Here where a number is prime, the cumulative frequency increases in discrete fashion by 1. However where a number is composite it increases - again in a discrete fashion by a number > 1.
Since 1 is a factor of 1 and the common factor of all numbers, which we do not consider in this context, the cumulative frequency of factors = 0. Then at 2 (which is prime) it increases to 1 and at 3 (again prime) it increases, again by 1 to 2. However at 4 (which is composite) it increases by 2 (with factors. 2 and 4); then at 5 (which is prime) it increases by 1, while at 6 (which is composite) it increases by 3 (with factors 2, 3 and 6).
So we have an even more complicated step function for the cumulative frequency of factors, where for each prime, as we have seen, the function steps up in discrete fashion by 1. However for the composite numbers it steps up - again in discrete fashion - by more than 1.
And it is this function, to which the corresponding cumulative frequency of Zeta 1 (Riemann) zeros closely relates.
So once again, the cumulative frequency of factors to n, closely approximates the corresponding cumulative frequency of Zeta 1 zeros to t (where n = t/2π).
Therefore the Zeta 1 zeros can be best seen as an attempt to smooth out in a continuous manner the discontinuous discrete nature of the step function (associated with the cumulative frequency of factors) so that - in a very important sense - the changes in factor frequency due to the primes are no longer distinguishable from the corresponding frequency due to the (composite) natural numbers. And the Zeta 1 zeros therefore represent those points where the factor frequency increases by 1 for both primes and composite natural numbers (which at these points are indistinguishable).
Now, reminding ourselves of the crossroads situation, which can be simultaneously approached from N and S directions so that left and right turns have a circular paradoxical meaning, it is quite similar here.
Therefore, we cannot make dualistic sense, in a linear analytic fashion, of the notion that primes and (composite) natural numbers can be somehow identical; however we can indeed understand this in a nondual holistic manner (that is directly intuitive in nature) .
Therefore, understanding of the true holistic nature of the Zeta 1 zeros comes again from initial appreciation that the relationship as between primes and natural numbers can be defined within two relatively independent reference frames, which are - in my chosen terminology - external and internal with respect to each other. Then when one attempts to - literally - "see" this relationship simultaneously within both reference frames, the true holistic nature of the Zeta 1 zeros (as approaching both pure psychological and physical energy states respectively) is revealed.
And in this direct intuitively inspired revelation, which indirectly can be interpreted intellectually in a circular paradoxical manner, one realises the true ultimate nature of both the primes and natural numbers as perfect mirrors of each other in an inseparable manner.
Once again, from a holistic mathematical perspective, the imaginary aspect simply represents an indirect analytic means of attempting to convey meaning that is directly holistic in nature.
And this is the simple the reason why all the Zeta 1 zeros appear to lie on an imaginary line. We will look at the deeper consequences of this in the next blog entry.