I have been discussing the "linear" and "circular" versions of a formula that can calculate both the combined natural factors up to a given number n (on a real scale) and a corresponding frequency of Riemann zeros up to a corresponding number t (on an imaginary scale) where n = t/2π.
Thus the frequency of Riemann zeros here provides the equivalent holistic interpretation to the frequency of factors in analytic terms.
So the analytic formula for frequency of factors = n(log n – 1) whereas the corresponding holistic formula for frequency of trivial zeros = t/2π(log t/2π – 1).
Now of course there is an even better known simple analytic formula that complements that for frequency of factors.
This is n/log n – 1), a more accurate version of the well known formula (n/log n) for estimation of primes up to a given number.
Because of the complementarity as between analytic and holistic explanations, this suggests that a parallel formula exists for measuring the frequency of the Zeta 2 zeros.
In fact, the dynamic use of complementarity can suggest the precise nature of this formula.
In the corresponding holistic formula for calculation of Riemann (Zeta 1), n (in the analytic formula) is replaced by t/2π.
However in the two analytic formulae, complementarity applies in this manner.
If a = n and b = log n – 1, then ab is replaced by a/b.
So n(log n – 1) for calculation of frequency of natural factors becomes n/log n – 1) for calculation of frequency of prime numbers.
Notice how natural factors complement prime numbers in this case!
However complementarity also applies to 2π with respect to the holistic formula for calculation of Riemann (Zeta 1) zeros, which now becomes 2/π with respect to the corresponding holistic formula for calculation of the Zeta 2 zeros.
Thus our equivalent holistic formula that complements n/log n – 1 is (2t/π)/(log 2t/π – 1).
So what does this formula precisely measure?
Well, I have mentioned before in previous blog entries how the Zeta 2 zeros correspond directly with the prime roots of 1 (with the exception of 1 which is always a root).
The significance of these prime roots of 1 (i.e. Zeta 2 zeros) is that - by definition - they comprise a unique set for all prime numbers. In other words these roots can never repeat themselves (where prime numbered roots are concerned).
Now clearly when we add up all the roots of 1 (including the trivial root 1) we always get zero.
However there is a fascinating way for converting these roots in a (reduced) real manner where we simply ignore both negative and imaginary signs treating both sin and cosine parts of all roots in a positive real manner.
So for example the 3 roots of 1 are 1, .– 5 + .866i... and – .5 – .866i respectively. Strictly the latter two roots here comprise the Zeta 2 zeros as solutions to the equation,
1 + s1
+ s2 = 0. (Again, 1 is always a common root) .
Now the sum of the 3 roots (as always with the sum of the n roots of 1) = 0.
However, indirectly we can convert to a meaningful quantitative measurement by ignoring both real and imaginary signs.
So the "converted" real values of the 3 roots are 1, .5 + .866, and .5 + .866
Now the sum of the "converted" cos parts = 1 + .5 + .5 = 2 and the sum of the "converted" sin parts =
.866 + .866 = 1 .732...
Now what is striking is that we when we then obtain the average of both "converted" cos and sin parts that they approximate 2/π = .6366... (with the approximation quickly improving as the value of t increases).
Already with just 3 values the average of the cos values = .666... and the average of the sin values (also dividing by 3) = .5773...
Now the actual average of "converted" cos values will always exceed 2/π while the corresponding average of sin values will always be less than 2/π.
Remarkably, the ratio of the (absolute) difference of cos and sin values quickly approximates .5, bearing comparison with the corresponding situation with respect to the Riemann zeros (as lying on the imaginary line through .5).
Indeed in this case (with just 3 values), the absolute ratio of differences = .03.../.0593... = .506...
So the approximation to .5 is already very close!
Because the average of both cos and sin parts approximates to 2/π, this means that if we sum up all t roots for both cos and sin parts they will approximate 2t/π in both cases.
If we now were to express the sum of the prime valued cos and sin parts (of t) in relation to the sum of all roots of t, we would use our required formula,
(2t/π)/(log 2t/π – 1) .
Inbuilt in this is the important assumption that the value of "converted" prime roots (both cos and sin parts) would be random and thereby unbiased with respect to all "converted" roots!
So we have now arrived at the fascinating conclusion that there are in fact two ways of expressing the relationship of prime number frequency.
We can, as in the conventional manner, express this in Type 1 cardinal terms as the frequency of the primes (in relation to the natural numbers) up to n.
However we can equally in a Type 2 ordinal manner express this relationship as the combined value of the "converted" prime roots (cos and sin parts) of 1 in relation to the combined value of all t roots.
Thus for example if we take the 100 roots of 1 to illustrate we would get "converted" real values for both the cos and sin parts of all these 100 roots.
We would then estimate the combined value of the prime roots (2nd 3rd 5th, 7th,..., 97th roots ) in relation to the combined value of all 100 roots for both "converted" cos and sin parts.
This would then give an alternative "ordinal" measurement of the frequency of the prime numbers in relation to the natural numbers.