EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA IIThe fact that the five element subdivisions algorithm can be applied to each separate 63.6363... segment can immediately be used to great advantage to calculate the zeta zeros for extremely large values of t (1/2 + it). So far, the computations of the Riemann zeta function for very high zeros have progressed to a dataset of 50000 zeros in over 200 small intervals going up to the 10
36-th zero.
The main problem is the calculations of the exponential sums in the Riemann-Siegel formula.
However, the five element subdivisions algorithm suffers from no such restrictions.
The 63.6363... segment can be shifted to any desired height, using arbitrary-precision arithmetic.
Therefore, computations of zeros around the first Skewes number, 1.39822 x 10
316 become possible using the Schönhage–Strassen algorithm for the multiplication/addition of very large numbers.
The Riemann-Siegel requires the addition of all of the terms in the formula, involving the evaluation of cosines, logarithms, square roots, and a complex set of remainders.
With the five element subdivision algorithm, only the following calculations are required: k x 63.6363..., where k can be 1.39822 x 10
316 or 10
10,000 (10 followed by ten thousand zeros). No divisions are required, no evaluation of elementary transcendental or algebraic functions is needed. The five element sequence of proportions are T, 63.6363... x T/250, 3T/10, T/10, T/20: simple multiplications by 1/250, 3/10, 1/10 and 1/20.
The only figure remaining to be calculated very precisely is the actual value of the sacred cubit distance.
14/22 = 0.63636363...
2/π = 0.636619722...
286.1/450 = 0.6357777...
14.134725 x 45 = 636.062625....
π has been calculated to over one million digits, the first zeta zero to over 40,000 digits.
The precise figure can be deduced by using the five element subdivision algorithm to the following heights: 636.63, 6,363.63, 63,636.63, 636,363.63.
Two examples which prove that the 63.6363 segment can be shifted to higher intervals on the critical 1/2 line, with no previous knowledge of the values of the other zeta zeros.
Zeta zero: 79.337375020
14.134725 + 63.63 = 77.7647
L(77.7647) = 2.4975 (average spacing estimate 80.262)
77.7647 + 0.80886 = 78.57356
77.7647 + 3.1815 = 80.9462
141.3947 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 = 80.2836
141.3947 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 = 79.598
141.3947 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 - 0.4661 = 79.1318
3.1815 - 0.80886 = 2.3726
2.3726
0.60332
0.3559
0.23726
0.11863
Adding the bottom four values to 78.57356:
79.177
78.929
78.811
78.6922
79.177 is the first lower bound.
2.3726 - 0.60322 = 1.7694
1.7694
0.45
0.26541
0.17694
0.08847
Adding the bottom four values to 79.177:
79.627
79.442
79.354
79.2655
79.598 is the first upper bound.
0.4661
0.275
0.1833
0.09
Substracting these values from 79.598:
79.1318
79.323
79.4147
79.508
79.323 is the new lower bound.
79.354 is the new upper bound.
0.17694 - 0.08847 = 0.08847
0.08847
0.0225
79.2655 + 0.0225 = 79.288
0.08847 - 0.225 = 0.06597
0.06597
0.016772
79.288 + 0.016772 = 79.30437
0.06597 - 0.016772 = 0.0492
0.0492
0.01251
79.30437 + 0.01251 = 79.3173
0.0492 - 0.01251 = 0.03669
0.03669
0.009328
79.3173 + 0.009328 = 79.32663
79.362663 is the new lower bound.
0.03669 - 0.009328 = 0.027362
0.027362
0.0069565
0.0041043
0.0027362
0.00131681
Adding the bottom four values to 79.3266:
79.3336
79.3307
79.32937
79.328
0.027362 - 0.0069565 = 0.0204055
0.0204055
0.0051879
0.003061
0.00204055
0.00102
Adding the bottom four values to 79.3336:
79.33879
79.336661
79.33564
79.33462
0.0204055 - 0.0051879 = 0.0152176
0.0152176
0.003869
0.00283
0.00152176
0.000761
Adding the bottom four values to 79.33879:
79.34266
79.3416
79.3403
79.3395
The calculations for the second zeta function.
0.275 - 0.1833 = 0.0917
0.0917
0.0233
79.4147 - 0.0233 = 79.3914
0.0917 - 0.0233 = 0.0684
0.0684
0.0174
79.3914 - 0.0174 = 79.374
0.0684 - 0.0174 = 0.051
0.051
0.012966
79.374 - 0.012966 = 79.361
0.051 - 0.012966 = 0.038034
0.038034
0.00967
79.361 - 0.00967 = 79.35133
79.35133 is the new upper bound.
0.038034 - 0.00967 = 0.028364
0.028364
0.00721
79.35133 - 0.00721 = 79.34412
0.028364 - 0.00721 = 0.021154
0.021154
0.0053782
0.003773
0.0021154
0.001058
Substracting the bottom values from 79.34412:
79.338742
79.34095
79.342
79.34306
0.021154 - 0.0053782 = 0.015776
0.015776
0.004011
0.0023664
0.0015776
0.000789
Substracting the bottom four values from 79.338742:
79.33473
79.3364
79.337164
79.33795
Now, the new features/results from the previous message will be used.
0.0917
0.0233
0.013755
0.00917
0.004585
Adding the bottom four values to 79.323:
79.3276
79.3322
79.3367
79.3453
0.17694 - 0.08847 = 0.08847
0.08847
0.022493
0.013271
0.008847
0.0044235
Substracting the bottom four values from 79.354:
79.33151
79.34073
79.3455
79.3496
79.3367 is the new lower bound.
79.33879 is the new upper bound.
Since 79.337164 is a higher figure than 79.3367, 79.337164 is the new lower bound for the entire approximation.
Without any knowledge of the values of the previous zeta zeros, a five digit/three decimal place approximation of the zeta zeros was obtained.
Zeta zero: 143.111845808
14.134725 + 63.63 + 63.63 = 141.3947
L(141.3947) = 2.018 (average spacing estimate 143.4126)
141.3947 + 0.80886 = 142.20356
141.3947 + 0.60322 = 142.8068
141.3947 + 0.45 = 143.2568
141.3947 + 0.335 = 143.592
205.0247 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 = 143.9187
205.0247 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 = 143.2277
205.0247 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.7322 - 2.7834 - 2.0757 - 1.548 - 1.1544 - 0.861 - 0.692 - 0.4661 = 142.7618
142.8068 is the first lower bound.
143.2277 is the first upper bound.
1.7694
0.45
0.26541
0.17694
0.08847
Adding the bottom four values to 142.8068:
143.2568
143.07221
142.984
142.8953
0.4661
0.275
0.1833
0.09
Substracting these values from 143.2277:
142.7618
142.9527
143.0444
143.1377
Now, the new features/results from the previous message will be used.
1.7694
0.45
0.26541
0.17694
0.08847
Substracting the bottom three values from 143.2568:
142.9914
143.0798
143.16833
0.4661
0.275
0.1833
0.09
Adding the bottom three values to 142.7618:
143.0368
142.945
142.8518
143.07221 is the new lower bound.
143.1377 is the new upper bound.
Careful calculations should be performed using the five element subdivisions algorithm to be compared to the known values of the Lehmer pairs:
A treatise which specializes in the calculation of Lehmer pairs (see pages 64-87 for a list):
http://www.slideshare.net/MatthewKehoe1/riemanntex The five element subdivisions algorithm can detect the first Lehmer pair effortlessly:
https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1950822#msg1950822 (very close values right from the first level of the subdivision algorithm, 111.317 and 111.38)
https://arxiv.org/pdf/1612.08627.pdf (pg. 7: 111.02953554 and 111.87465918, the first Lehmer pair)
Then, the five element subdivision algorithm should be applied to the same intervals used in the zeta zeros approximations around the 10
36th zero:
NEW COMPUTATIONS OF THE RIEMANN ZETA FUNCTION ON THE CRITICAL LINE
https://arxiv.org/pdf/1607.00709.pdfAs a byproduct of our search for large values, we also find large values of S(t). It is always the case in our computations that when ζ(1/2 + it) is very large there is a large gap between the zeros around the large value. And it seems that to compensate for this large gap the zeros nearby get “pushed” to the left and right. A typical trend in the large values that we have found is that S(t) is particularly large and positive before the large value and large and negative afterwards.
The calculations involve more than 50000 zeros in over 200 small intervals going up to the 10
36th zero.
S(t) is related to the large gaps between the zeta zeros where high extreme values of peaks occur, where it seems to protect the zeta function from attaining the tightly packed spikes conjectured by mathematicians.