**EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA II**The 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

^{36}th 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

^{36}th 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.