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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