Once again I am used the slightly modified formula i.e. t/2π(log t/2π – 1) + 1.
And as there are 29 nontrivial zeros up to 100, this means that we have thereby approximated all the nontrivial 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

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 nontrivial 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 nontrivial zeros can predict in absolute terms with a remarkable level of accuracy.
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