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.

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