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. Therefore in the attempt to eliminate this bias I decided to use a new adjustment factor (based on the deviations of the 1st set of approximations).

Therefore 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 behaviour 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.