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.