Return of Shadow Deviations

I have mentioned in  previous entries - see "Mysterious Role of 6 and 12" and "Calculating the Shadow Deviation"- how a very simple formula could be used to calculate to a high degree of accuracy the deviation from 2/π (= i/ln i) of both the (reduced) average cos and sin values of the n roots of 1.

So the formula again for the cos part deviation = π /12(n ^ 2) and the sin part = π /6(n ^ 2). Now the cos part is always positive (> than 2/π) and the sin part is always negative (< than 2/π). Therefore when we combine both cos and sin parts the combined deviation will be negative = π /12(n ^ 2).

Strictly speaking these formulas only hold for odd numbered values of n.


To calculate the corresponding deviations for the even numbered values a simple adjustment is required.

If a number is even it will then be divisible by 2 (or a power of 2).

So the general formula for calculating the deviations for the cos part of even number,

= π * 2^k /12(n ^ 2)  (where n is divisible by 2^k ).  The general formula  for the sin part is also
π * 2^k /12(n ^ 2). However in contrast to odd numbered values, both of these deviations will now be negative. 

Therefore the deviation for combined cos and sin parts is π * 2^k/6(n ^ 2)   which again is negative.
For example where n = 8 the number is thereby divisible by 2³. So the deviation for the cos part = π * 2³ /12(8 ^ 2)  = – .0327249...

This already compares well with the actual deviation = – .03306338... with the relative accuracy improving for larger n.

Likewise the calculated deviation for the sin part = – .0327249... (as opposed  again to the actual deviation of – .03306338... )

The combined deviation of both cos and sin parts = – .0327249... * 2 = – .0654498...

The actual deviation in this case (this time from 4/π) = – .03306338 * 2 = – .06612676...

These deviations, which ultimately entail a modified (reduced) estimate for Zeta 2 zeros, should then in principle be capable of being used in a much simpler fashion that the corresponding Zeta 1 zeros to precisely predict primes (up to any given number).

However whereas the Zeta 1 zeros are used to directly correct a general estimate of prime number frequency (to a given number) these latter (modified) estimates can be used to directly adjust the weighting of each individual number (as slightly different from 1).

Then in principle with each individual number appropriately weighted (through use of these deviations) we should be able to exactly predict the number of primes with respect to the weighted - as opposed to unweighted - individual units.