Another Interesting Connection

As we have seen we can use the analytic formula n/log n – 1) to estimate the frequency of primes up to n.

Then we can use the complementary analytic formula n(log n – 1) to calculate the corresponding frequency of natural factors up to n.

Therefore if we obtain the ratio of the (estimated) frequency of the natural factors and primes respectively (both up to n) it will be given as (log n – 1)².

For example the actual frequency of natural factors (to n) = 357 and the actual frequency of primes 25.

Therefore the ratio = 357/25 = 14.28.

This compares fairly well with the estimated ratio = (3.60517)² = 13.00 (correct to 2 decimal places).

Alternatively we could express the ratio of the (estimated) frequency of primes to natural factors (to n) as 1/(log n – 1)².

Then when n is very large the two ratios would approximate closely to (log n)² and
1/(log n)² respectively.


Again as we have seen, we can use the holistic formula t/2π(log t/2π – 1) to calculate the frequency of Riemann (Zeta 1) zeros to t on an imaginary scale (where n = t/2π).

We can then "convert" these zeros in an analytic manner on the real scale (by setting n = t/2π).

Thus the formula for the "converted" Riemann zeros i.e. n(log n – 1), then serves in a direct manner as a means for estimating the frequency of natural factors.

Therefore, we can equally express the ratio of the formula for calculating the frequency of "converted" Riemann zeros to that for calculating the corresponding frequency of primes as (log n – 1)².

Alternatively, we can express the ratio of the formula for calculating the frequency of primes to that for calculating the frequency of "converted" Riemann zeros as 1/(log n – 1)². 

In conclusion, I would like to highlight the truly complementary nature of both the Zeta 1 and Zeta 2 zeros.

The zeta function for calculating the Zeta 2 zeros is of a finite nature and given as:

ζ₂(s)  =  1 + s¹ + s² + s³ + … + s^{t – 1} = 0 (where t is prime).

The extended infinite version of the formula is then given as:

1/2 = 1 + s¹ + s² + s³ + .......

For example, we can perhaps easily see why in the simplest case where t = 2,

 ζ₂(– 1) = 1 – 1 = 0.

However with the infinite extended version of this formula,

1/2 = 1 – 1 + 1 – 1 +.......

Remarkably, for all other prime roots (except 1) representing the Zeta 2 non-trivial zeros, the expected value of the infinite series = 1/2.

This then readily provides an important connection with the corresponding Zeta 1 function i.e

ζ₁(s)  =  1 ^{– s}  + 2 ^{– s} + 3 ^{– s} + 4 ^{– s} + ...... = 0, where all solutions for s are of the form 1/2 + it and 1/2 – it respectively.

Thus the requirement that all the Zeta 1 non-trivial zeros lie on the line through 1/2 (which is true if the Riemann Hypothesis holds) is directly linked with the Zeta 2 zeros.

The Zeta 2 zeros are then in turn directly linked with the Zeta 1 zeros is enabling all natural numbered solutions for

ζ₂(s)  =  1 + s¹ + s² + s³ + … + s^{t – 1} = 0 (i.e. where t can be a natural number > 1).

So in dynamic interactive terms, the Zeta 1 and Zeta 2 are truly interdependent in a holistic manner.

Therefore the Riemann Hypothesis cannot be proved in the standard analytic manner of Conventional Mathematics (which assumes independence of polar reference frames).

Put another way, the Riemann Hypothesis directly relates to a key requirement for the consistency of both the cardinal and ordinal aspects of the number system (in quantitative and qualitative terms). This cannot be proved using mere quantitative notions based on the cardinal interpretation of number. 

Conventional proof in Mathematics thereby is already built on the implicit assumption that the Riemann Hypothesis is indeed true. However this represents essentially an act of faith in the ultimate consistency of the mathematical system (which cannot therefore be proved or disproved within this system).