## Wednesday, June 8, 2016

### Approximating the Non-Trivial Zeros (2)

Having approximated the first 10 of the non-trivial zeros, I decided to continue on an calculate the first 30.

Once again I am used the slightly modified formula i.e. t/2π(log t/2π –  1) + 1.

And as there are 29 non-trivial zeros up to 100, this means that we have thereby approximated all the non-trivial zeros for t to 100!

However in the original approximation of values, where I adjusted the first calculation for each zero downward (by half the deviation from the next value), a bias still remained in that the overall sum of the actual zeros tended to be consistently overshoot that of the corresponding approximations. Therfore in the attempt to eliminate this bias I decided to use a new adjustment factor (based on the devations of the 1st set of approximations).

Therfore to more accurately approximate the nth zero, I decided to multiply the deviation as between the nth and (n + 1)st value by (1 – 2/π) and then subtract this from the original 1st approximation.

So below, I have provided a table showing the three different approximations, together with the acual values for the trivial zeros.

I have then highlighted the most recent approximations and actual values in bold type for easier comparison.

 Riemann Zeros Predicted Location (1) Deviation of Zeros Predicted Location (2) Predicted Location (3) Actual Location 1st 17.08 14.34 15.09 14.13 2nd 22.56 5.48 20.27 20.90 21.02 3rd 27.14 4.58 25.09 25.65 25.01 4th 31.24 4.10 29.34 29.86 30.43 5th 35.04 3.80 33.28 33.76 32.94 6th 38.56 3.52 36.88 37.34 37.59 7th 41.92 3.36 40.28 40.73 40.92 8th 45.20 3.28 43.64 44.06 43.33 9th 48.33 3.13 46.82 47.23 48.01 10th 51.36 3.03 49.89 50.29 49.77 11th 54.31 2.95 52.87 53.26 52.97 12th 57.19 2.88 55.78 56.17 56.45 13th 60.00 2.81 58.62 59.18 59.35 14th 62.76 2.76 61.40 61.78 60.83 15th 65.47 2.71 64.14 64.51 65.11 16th 68.12 2.65 66.81 67.17 67.08 17th 70.74 2.62 69.45 69.80 69.55 18th 73.32 2.58 72.05 72.40 72.07 19th 75.86 2.54 74.61 74.95 75.70 20th 78.36 2.50 77.12 77.46 77.14 21st 80.83 2.47 79.60 79.94 79.34 22nd 83.28 2.45 82.07 82.40 82.91 23rd 85.70 2.42 84.50 84.83 84.74 24th 88.09 2.40 86.91 87.23 87.43 25th 90.46 2.37 89.29 89.61 88.81 26th 92.80 2.34 91.64 91.95 92.49 27th 95.12 2.32 93.96 94.28 94.65 28th 97.43 2.31 96.28 96.60 95.87 29th 99.72 2.29 98.59 98.90 98.83 30th 101.98 2.26 100.86 101.17 101.32 31st 104.22 2.24

Once again, I consider it striking how the simple general formula provides such a convenient means for calculating, with stunning accuracy, not only the frequency of zeros up to any given t, but likewise a ready means for approximating the value for each one of the trivial zeros.

The difference as between the actual values for the zeros and their corresponding approximations is due to the local random nature of the zeros.

However this randomness is at the other extreme from the primes. In fact both the primes and trivial zeros complement each other in a dynamic interactive manner.

So the behviour of individual primes is as independent as possible consistent with maintaining an overall collective interdependence with each other (through the natural numbers).

However the collective behaviour of the non-trivial zeros is as interdependent (i.e. ordered) as possible, consistent with each zero maintaining an individual local independence.

Therefore whereas the simple general formula for frequency of  primes can only hope to predict with a strictly relative degree of accuracy, the corresponding formula for frequency of non-trivial zeros can predict in absolute terms with a remarkable level of accuracy.