And likewise I conjectured that both of these in turn have an intimate relationship with the average (absolute) value of the cos and sin parts respectively of the roots of 1.

Now 2/π directly expresses the pure relationship as between a line diameter and the semi-circle with which it is bound.

Therefore in holistic mathematical terms, this symbolises the pure relationship as between linear and circular type understanding of the primes, which are quantitative and qualitative with respect to each other.

Thus we may start by attempting to view each (cardinal) prime in a merely quantitative fashion.

However the quantitative independence of each prime must be balanced with the qualitative nature of all primes (in their collective relationship with each other).

So the fundamental problem which besets conventional mathematical interpretation is that it continually reduces the notion of interdependence, which is inherently qualitative, with that of independence of a quantitative nature.

Therefore to correctly balance the quantitative and qualitative aspects of the primes, we need to employ a dynamic interactive approach (based on complementary Type 1 and Type 2 aspects).

So in properly understanding these results, we must balance quantitative measurements with their holistic qualitative interpretations!

However the connections do not appear to end there!

When I was working on absolute measurements of the various roots of 1, it struck me forcibly that the largest combined result of cos and sin values = √2. This occurs for the angle of 45 degrees

^{}when both cos 45 and sin 45 = 1/√2.

And it seems that these measurements are also relevant to the ratios entailing repeating prime structures.

On my "Spectrum of Mathematics" blog, I recently defined three variations on numbers with repeating prime structures.

Firstly we can count all the prime factors occurring (1)

Secondly we count only (distinct) prime factors (i.e. each prime that recurs is counted only once) (2).

Thirdly we only count those prime factors that occur just once (3).

For example 3150 = 2 * 3 * 3 * 5 * 5 * 7.

So when we count all prime factors the total = 6 (1).

When we count (distinct) prime factors the total = 4 (2).

Finally when we count prime factors that occur just once the total = 2 (3).

Now basically my conjecture is that for numbers with repeating prime structures the ratio of (1) : (2) ~ √2.

Then the ratio of (2) : (3) is similar and ~ √2.

This therefore implies that the ratio of (1) : (3) ~ 2.

Expressed in complementary fashion, this implies that the ratio of (3) : (1) ~ .5.

This could indeed amount to an alternative simple manner of expressing the Riemann Hypothesis!

Indeed in this context, it is fascinating to report that whereas the (absolute) average value of both cos and sin parts of the roots of 1 ~ 2/π, the average cos value always slightly exceeds 2/π, whereas the average sin value is always slightly less than 2/π .

And the ratio of the (absolute) amount by which the average cos value exceeds 2/π to that by which the average sin value falls short of 2/π ~ .5.