Wednesday, September 24, 2014

Using Frequency Formula to Estimate Location of Riemann Zeros

I have remarked before on the stunning accuracy of the formula for calculating the frequency of the Riemann zeros.

Therefore to calculate the frequency of these zeros up to t on the imaginary line (through ½) we use the formula,

t/2π(log t/2π –  1).

I have suggested that in general an even more accurate measurement (in absolute terms) will occur from the addition of 1 to this formula i.e.

t/2π(log t/2π –  1) + 1.

If we use this latter formula to calculate the exact occurrence of just one zero, it occurs at 17.08 (approx) which lies pretty well midway between the first two actual zeros (i.e. 14.13 and 21.02 respectively.

I then went on using the formula to calculate the exact predicted frequency of 2, 3, 4,….10 zeros.

So the table underneath (cols 2 and 3) compares the actual location of the first 10 zeros as against the predicted location. I also show (in cols 4 and 5) the successive deviations of the actual and predicted zeros.


Riemann Zeros
Actual location
Predicted Location
Deviation of Actual Zeros
Deviation of Predicted zeros
       1st
      14.13
     17.08
    
     
       2nd
      21.02
     22.56
     6.89
      5.48
       3rd
      25.01
     27.14
     3.99
      4.58
       4th
      30.43
     31.24
     5.42
      4.10
       5th
      32.94
     35.04
     2.51
      3.80
       6th
      37.59
     38.56
     4.65
      3.52
       7th
      40.92
     41.92
     3.33
      3.36
       8th
      43.33
     45.18
     2.41
      3.26
       9th
      48.01
     48.34
     4.68
      3.16
     10th
      49.77
     51.34
     1.76
      3.00

In one way it is remarkable how this general formula to calculate the frequency of the zeros can be used to estimate the precise value of each zero.

The deviations are also interesting. The deviations with respect to the estimated zeros operate in a smooth consistent fashion with respect to the manner in which they become smaller.

However the corresponding deviations with respect to the actual zeros are much more erratic.

Thus though the average deviation in both cases is roughly similar (3.96 for actual and 3.81 for estimated zeros respectively) considerable variations with respect to the local spread as between any two successive zeros is in evidence.

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