Advanced Flat Earth Theory

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #540 on: June 15, 2018, 08:52:25 AM »
SAGNAC EFFECT VS CORIOLIS EFFECT MYSTERY SOLVED III



The most ingenious experiment performed by Professor Yeh: light from a laser is split into two separate fibers, F1 and F2 which are coiled such that light travels clockwise in F1 and counterclockwise in F2.

https://www.researchgate.net/publication/26797550_Self-pumped_phase-conjugate_fiber-optic_gyro

Self-pumped phase-conjugate fiber-optic gyro, I. McMichael, P. Yeh, Optics Letters 11(10):686-8 · November 1986 

http://www.dtic.mil/dtic/tr/fulltext/u2/a170203.pdf (appendix 5.1)

The first phase-conjugate Sagnac experiment on a segment light path with a self-pumped configuration.

The Sagnac phase shift for the first fiber F1:

+2πR1L1Ω/λc

The Sagnac phase shift for the second fiber F2:

-2πR2L2Ω/λc

These are two separate Sagnac effects, each valid for the two fibers, F1 and F2.

The use of the phase conjugate mirror permits the revealing of the final formula, the total phase difference:

φ = -2(φ2 - φ1) = 4π(R1L1 + R2L2)Ω/λc

To obtain the correct Sagnac effect for two separate segments (which feature different lengths and different speeds) of an interferometer which is located away from the center of rotation, one has to add (not substract) the two distinct components.

Let us go back now to the derivation provided by A. Michelson for the 1925 MGX experiment:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2024700#msg2024700

dt = l1/(c - v1) - l1/(c + v1) - (l2/(c - v2) - l2/(c + v2))

Of course, by proceeding as in the usual manner for a Sagnac phase shift formula for an interferometer whose center of rotation coincides with its geometrical center, we obtain:

2v1l1/(c2 - v21) - 2v2l2/(c2 - v22)

l = l1 = l2

2l[(v1 - v2)]/c2

2lΩ[(R1 - R2)]/c2

R1 - R2 = h

2lhΩ/c2

By having substracted two different Sagnac phase shifts, valid for the two different segments, we obtain the CORIOLIS EFFECT formula.

The Coriolis effect means that the phase shift will be caused by the physical modification of the light paths (inflection and deflection due to the Coriolis force effect on the light beams).

l1/(c - v1) - l2/(c - v2) (we combine the terms which feature c - v1,2)

(l1c - l1v2 - l2c + l2v1)/(c2 - cv2 - cv1 +v1v2)

Factoring out c and observing that the terms l1v2/c, l2v1/c and v1v2/c can be neglected, we obtain:

(l1 - l2)/(c - v2 - v1)

Since l1 ~= l2, we can see that the velocity addition equations for the true Sagnac effect, c + v and c - v (in this case c + v1 + v2 and c - v1 - v2) are not applicable to the Coriolis effect situation.

The Coriolis effect is caused by the physical deflection/inflection of the light beams, not by the modification of the velocities.


Let us proceed now exactly as Professor Yeh did in the phase conjugate mirror experiment described above:

dt = l1/(c - v1) - l1/(c + v1) + (l2/(c - v2) - l2/(c + v2))

2[(l1v1 + l2v2)]/c2

Now, we have the correct, true Sagnac effect formula valid for an interferometer which is located away from the center of rotation.

Averaging (v1 + v 2)/2 = v, and (l1 + l2)/2 = l, v1 ~= v2 = v, l1 ~= l2 = l, we obtain:

4lv/c2

Moreover, we can see that now the velocity addition equations are valid:

(l1 + l2)/(c - v2 - v1)

To obtain the Coriolis effect phase shift, we substract the phase differences for each separate segment.

This formula is proportional to the area and the angular velocity.

To get the Sagnac effect phase shift, we have to add the phase differences for each separate segment

This formula is proportional to the linear velocity (and the radius of rotation), and will feature the addition of the two separate speeds and segment lengths. We can average the lengths and the velocities, to get a final formula which features one length and one velocity.

This is the great omission in the calculation done by A. Michelson.

Instead of adding the phase differences to get the true Sagnac effect, he substracted the phase differences and obtained the formula for the Coriolis effect.

The two separate segments are distinct entities, with different velocities and lengths, thus they have two non-identical Sagnac phase differences. They have to be added to get a final phase formula, which, by averaging the lengths and the velocities, can feature a single length and a single velocity.

« Last Edit: June 15, 2018, 11:51:30 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #541 on: June 18, 2018, 02:27:13 AM »
SAGNAC EFFECT VS CORIOLIS EFFECT MYSTERY SOLVED IV



The turning of the MGX area at the hypothetical rotational speed of the Earth takes place a distance of some 4,250 km from the center of the Earth (latitude 41°46').

And yet, this crucial fact was totally omitted by Michelson, even though the calculations offered a very important clue.

dt = l1/(c - v1) - l1/(c + v1) - (l2/(c - v2) - l2/(c + v2))

For each separate segment/arm of the interferometer, each with a slightly different length and a slightly distinct velocity, the calculations proceed as follows:

l1/(c - v1) - l1/(c + v1) = 2l1v1/c2

l2/(c - v2) - l2/(c + v2) = 2l2v2/c2

The phase differences have already been obtained.

By substracting these phase differences, one is actually going to derive the Coriolis effect formula:

https://www.ias.ac.in/article/fulltext/pram/087/05/0071

Spinning Earth and its Coriolis effect on the circuital light beams

Since the phase differences have already been calculated, one has to ADD them in order to get the final, total Sagnac effect:

2[(l1v1 + l2v2)]/c2

This fact has never been observed to the present day.

For a Sagnac interferometer, located away from the center of rotation, one has to ADD the separate phase differences in order to obtain the full Sagnac effect:

dt = l1/(c - v1) - l1/(c + v1) + (l2/(c - v2) - l2/(c + v2))


FULL CORIOLIS EFFECT FOR THE MGX:

4AΩsinΦ/c2

FULL SAGNAC EFFECT FOR THE MGX:

4Lv(cos2Φ1 + cos2Φ2)/c2


Sagnac effect/Coriolis effect ratio:

R((cos2Φ1 + cos2Φ2)/hsinΦ

R = 4,250 km

h = 0.33924 km

(actually, here is how to exactly calculate the radius of the hypothetical spherical Earth at a certain latitude:
https://web.archive.org/web/20150919165338/http://www.usenet-replayer.com/faq/comp.infosystems.gis.html )

The rotational Sagnac effect is much greater than the Coriolis effect for the MGX.

Michelson proceeded with his calculations AS IF he had placed a rectangular interferometer with the dimensions of 2010 ft (612.65 m) by 1113 ft (339.24 m) with its center of rotation coinciding with the center of rotation of the Earth (radius of 6,376.164 km).

Once the interferometer is moved away from the center of rotation, the Coriolis effect will be measured first: the formula involves the substraction of the separate phase differences.

By contrast, the much greater Sagnac effect formula will be derived by adding the separate phase differences.

« Last Edit: November 06, 2019, 01:19:50 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #542 on: June 19, 2018, 02:47:26 AM »
GENERALIZED SAGNAC FORMULA

‘We dance in a circle and suppose, while the secret sits in the centre and knows.’

Robert Frost

Sagnac formula for an interferometer whose center of rotation coincides with its geometrical center:

Δt = l/(c - v) - l/(c + v)

Sagnac formula for an interferometer located away from the center of rotation (different radii, different velocities):

Δt = (l1 + l2)/(c - v1 - v2) - (l1 + l2)/(c + v1 + v2)

Proof:

Δt = l1/(c - v1) - l1/(c + v1) + (l2/(c - v2) - l2/(c + v2))

The use of the phase conjugate mirror has permitted, for the very first time in 1986, the derivation of the Sagnac formula for an interferometer which features two different lengths and linear velocities.

https://www.researchgate.net/publication/26797550_Self-pumped_phase-conjugate_fiber-optic_gyro

Self-pumped phase-conjugate fiber-optic gyro, I. McMichael, P. Yeh, Optics Letters 11(10):686-8 · November 1986 

http://www.dtic.mil/dtic/tr/fulltext/u2/a170203.pdf (appendix 5.1)

φ = -2(φ2 - φ1) = 4π(R1L1 + R2L2)Ω/λc

Since Δφ = 2πc/λ x Δt, Δt = 2(R1L1 + R2L2)Ω/c2.


l1/(c - v1) + l2/(c - v2) = (l1c - l1v2 + l2c - l2v1)/(c2 - cv1 - cv2 + v1v2)

l1/(c + v1) + l2/(c + v2) = (l1c + l1v2 + l2c + l2v1)/(c2 + cv1 + cv2 + v1v2)

Since we have already added the correct Sagnac differences, corresponding to the (l1 + l2)/(c - v1 - v2) and (l1 + l2)/(c + v1 + v2) terms, now the final phase difference can be correctly derived:

(l1c - l1v2 + l2c - l2v1)/(c2 - cv1 - cv2 + v1v2) - (l1c + l1v2 + l2c + l2v1)/(c2 + cv1 + cv2 + v1v2) = 2[(l1v1 + l2v2)]/c2

The Coriolis effect formula by contrast is just the physical effect of the Coriolis force upon the light beams, a modification of the paths of the light beams leading to a final formula where the effect is directly proportional to the area and to the angular velocity.

The Sagnac effect is an electromagnetic effect, the modification of the velocities of the light beams, c + v1 + v2 and c - v1 - v2, leading to the final formula where the Sagnac effect is directly proportional to the linear velocity (radius of rotation x angular velocity) and the length of the segments of the interferometer.

Michelson only measured the Coriolis effect and not the rotational Sagnac effect, since he substracted twice within the same derivation, he obtained a Coriolis effect formula:

http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1925ApJ....61..137M&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf


Sagnac formula for an interferometer whose center of rotation coincides with its geometrical center:

Δt = l/(c - v) - l/(c + v) = 2lv/c2

Sagnac formula for an interferometer located away from the center of rotation (different radii, different velocities):

Δt = (l1 + l2)/(c - v1 - v2) - (l1 + l2)/(c + v1 + v2) = 2[(l1v1 + l2v2)]/c2

« Last Edit: June 19, 2018, 03:14:52 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #543 on: June 21, 2018, 03:23:01 AM »
GENERALIZED SAGNAC FORMULA II

Velocity addition equations for the rotational Sagnac effect: c + v1 + v2 and c - v1 - v2.

The Heaviside-Lorentz equations are not invariant under Galilean transformations.

However, the original J.C. Maxwell dynamical equations are invariant under Galilean transformations:





https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2058884#msg2058884 (Dynamical Maxwell equations)

Equation (2.24) represents two waves: one wave propagating forward at a speed of (c+u) in the direction of the positive x axis and another wave propagating backward at a speed of (c-u) in the direction of the negative x axis.




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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #544 on: June 22, 2018, 12:42:40 AM »
GENERALIZED SAGNAC FORMULA III



Sagnac effect formula (center of rotation coincides with the geometrical center of the interferometer):

Δt = l/(c - v) - l/(c + v) = 2lv/c2

l = 2πr




Sagnac effect formula (interferometer located away from the center of rotation):

Δt = (l1 + l2)/(c - v1 - v2) - (l1 + l2)/(c + v1 + v2) = 2[(l1v1 + l2v2)]/c2

Coriolis effect formula:

4AΩ/c2




Sagnac effect formula (interferometer located on the surface of the Earth, at a certain latitude):

Δt = (l1 + l2)/(c - v1 - v2) - (l1 + l2)/(c + v1 + v2) = 2[(l1v1 + l2v2)]/c2

Coriolis effect formula:

4AΩsinΦ/c2


It is interesting to note that Michelson, Silberstein, Lorentz, Miller, Post could not derive the correct Sagnac effect formula (interferometer located away from the center of rotation).

« Last Edit: June 22, 2018, 01:09:17 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #545 on: June 22, 2018, 01:46:44 AM »
TESLA'S NONLINEAR CIRCUIT THEORY

http://www.cheniere.org/references/TeslaOSC.pdf

Barrett, T.W., "Tesla's nonlinear oscillator-shuttle-circuit (OSC) theory."   Annales de la Fondation Louis de Broglie, 16, pp. 23-41, 1991

Dr. Terence Barrett (Stanford Univ., Princeton Univ., U.S. Naval Research Laboratory, Univ. of Edinburgh) analyzes Tesla's electrical circuit design theory, conclusion: Tesla did not use either Kirchhoff's laws or Ohm's law.


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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #546 on: June 23, 2018, 12:11:34 AM »
ETHER BLACKBODY RADIATION

“Now the apparent temperature of the Sun is obviously nothing but the temperature of the solar rays, depending entirely on the nature of the rays, and hence a property of the rays and not a property of the Sun itself. Therefore it would be not only more convenient, but also more correct, to apply this notation directly, instead of speaking of a fictitious temperature of the Sun, which can be made to have meaning only by the introduction of an assumption that does not hold in reality”

M. Planck

On the Temperature of the Photosphere: Energy Partition in the Sun

http://vixra.org/pdf/1310.0140v1.pdf

If the  local thermal equilibrium and its extension of Kirchhoff’s formulation fails to guarantee that a blackbody spectrum is produced at the center of the Sun, then the gaseous models have no mechanism to generate its continuous emission. In part, this forms the basis of the solar opacity problem.

Stellar Opacity: The Achilles’ Heel of the Gaseous Sun

http://vixra.org/pdf/1310.0139v1.pdf

Given the problems which surround solar opacity, it remains difficult to understand how the gaseous models of the Sun have survived over much of the twentieth century. Local
thermal equilibrium does not exist at the center of the Sun. Both Kirchhoff and Planck require rigid enclosure which is not found in the Sun. Planck has also warned that the Sun fails to meet the requirements for being treated as a blackbody.

On the validity of Kirchhoff's law of thermal emission

https://ieeexplore.ieee.org/document/1265348/

https://www.libertariannews.org/2014/04/04/kirchhoffs-law-proven-invalid-the-implications-are-enormous/

Further, all blackbodies are limited to solids, since only they can be perfect absorbers, and unlike liquids, they cannot sustain convection.  Prof. Robitaille also explains why gases do not follow these laws because they do not emit radiation in a continuous manner, further discrediting the standard model of stars.  The emissivity of a real gas drops with temperature. Planck’s equation remains the only fundamental equation that has yet to be linked to physical reality, which is a direct result of Kirchhoff’s error.

Prof. Robitaille notes that the standard gaseous Sun model uses equations of radiative transfer, and those equations all have, at their source, KLTE.  The invalidity of KLTE means there cannot be blackbody radiation at the center of the Sun, which means the entire standard model of the gaseous Sun is invalid.

https://principia-scientific.org/new-study-invalidates-kirchhoff-s-law-of-thermal-emission/

https://web.archive.org/web/20160211150839/http://www.ptep-online.com/index_files/2015/PP-41-04.PDF

“The Theory of Heat Radiation” Revisited:
A Commentary on the Validity of Kirchhoff’s Law of Thermal Emission
and Max Planck’s Claim of Universality


The ether's blackbody radiation = CMBR:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1828839#msg1828839


Potential gravitational waves blackbody radiation:

https://arxiv.org/pdf/1612.04199.pdf


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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #547 on: June 25, 2018, 03:02:58 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION

In the currently accepted theory, the average gap/spacing formula between consecutive zeros is just a side note; in my opinion, it plays a far greater role in determining the values of the zeros of the zeta function.

2π/log(z/2π) = 4/{sc x lnz  +  [sc x ln(sc/4)]}

π = 2/sc (sc = sacred cubit)

2π/log(t/2π) = 4/{ln[(t ⋅ sc)/4]}sc

It is actually a sacred cubit formula.

In the conventional theory of Riemann's zeta function, this formula represents the average spacing between the zeros of the zeta function on the 1/2 line.

In the theory which uses the fact that the height of the Gizeh pyramid is 141.34725 meters (14.134725 is the value of the first zero of the zeta function), this formula becomes an exact sacred cubit spacing equation, if the ln term can be expressed directly in terms of integers and sacred cubit constants.

ln 20sc = 4 sc

4/{ln[(t ⋅ sc)/4]}sc, t = 80

4/{ln[(80 ⋅ sc)/4]}sc = 1/sc2

Zeta zeros

79.337
82.91

80 + 1/sc2 = 82.47

82.47 - 79.337 = 3.13288
82.91 - 82.47 = 0.44
80 - 79.337 = 0.663

0.44 = 0.6632

Sacred cubit constants:

1/sc2 = 2.47, 2.4753, 0.24753...

1/(1-sc) = 2.744, 0.2744...

1.309 = 2.618/2 (2.618 x 12 = 2/sc x 10)

0.309 = 0.618/2

0.84 = 0.535/sc

3.39 = 0.535sc , also 3.4025 = 2.5 x 1.361

7.2738 = missing apex height (286.1 si, sacred inches)

1.4305 (2.861/2 = 1.4305)

2.5442 or 2.5424 (100 sacred inches)

3.1815 = 5sc

3.8178 = 6sc

1.1444 = 0.2861 x 4


ln 4sc = 2sc/(2 - sc)

ln2 = 8sc2- 4sc

ln 80sc = 16sc2 - 4sc

ln 1000 = 5 + 3sc

ln 2.5 = 1/3(1-sc)

ln 3 = 2/2.861sc

ln 5 = (2/3sc + 7)/5

ln 7 = 1/2sc3

ln sc = 2sc/(2 - sc) - 2ln2


4/{ln[(t ⋅ sc)/4]}sc

t = 2n10

n= 2

4/{ln[(40 ⋅ sc)/4]}sc = 3.3945

Zeta zeros

37.586
40.9187
43.327

40 + 3.3945 = 43.3945

43.3945 - 37.586 = 5.8085 = 8 x 0.726

43.3945 - 40.9187 = 2.4758 = 1/sc2

In this case, the sacred cubit formula gives the exact distance to the zeta function zero.

43.3945 - 43.327 = 0.0675

2.4758 + 0.0675 = 2.5433 = 100si

These precise mathematical relationships can be observed only if the average spacing formula is written in terms of sacred cubits.

4/{ln[(t ⋅ sc)/4]}sc

n = 3 (case examined earlier)

4/{ln[(t ⋅ sc)/4]}sc

n = 4

4/{ln[(160 ⋅ sc)/4]}sc = 1.9408

Zeta zeros

158.85
161.189

160 + 1.9408 = 161.9408

161.9408 - 158.85 = 3.0908 = 6.1816/2

161.189 - 161.9408 = 0.7518

0.7518 x sc = 0.4784
3.0908 x sc = 1.9666

1.9666 x 0.4784 = 0.9408

4/{ln[(t ⋅ sc)/4]}sc

n = 5

4/{ln[(320 ⋅ sc)/4]}sc = 1.5986

Zeta zeros

318.853
321.16

320 + 1.5986 = 321.5986

321.5986 - 318.853 = 2.7456 = 1/(1-sc)

321.5986 - 321.16 = 0.4386

4/{ln[(t ⋅ sc)/4]}sc

n = 6

4/{ln[(640 ⋅ sc)/4]}sc = 1.359

Zeta zeros

639.928
640.6948
641.945

641.359 - 639.928 = 1.431 = 2.862/2
641.359 - 640.6948 = 0.6642
641.945 - 641.359 = 0.586

1.431 + 0.6642 = 1/(1/sc2 - 2)

4/{ln[(t ⋅ sc)/4]}sc

n = 7

4/{ln[(1280 ⋅ sc)/4]}sc = 1.18177

Zeta zeros

1279.3328
1280.1559
1281.828

1281.18177 - 1279.3328 = 1.849
1281.18177 - 1280.1559 = 1.02587
1281.828 - 1281.18177 = 0.64723

1.849 + 1.02587 = 2.87487
1.849 + 0.64723 = 1/0.28602 - 1

4/{ln[(t ⋅ sc)/4]}sc

n = 8

4/{ln[(2560 ⋅ sc)/4]}sc = 1.045475

Zeta zeros

2559.234
2560.2633
2561.4056

2561.045475 - 2559.234 = 1.8115
2561.045475 - 2560.2633 = 0.782175
2561.4056 - 2661.045475 = 0.360125

0.782175 + 0.360125 = 1.1423 =~ 1.1444 = 0.2861 x 4

4/{ln[(t ⋅ sc)/4]}sc

n = 9

4/{ln[(5120 ⋅ sc)/4]}sc = 0.93736

Zeta zeros

5119.366
5120.24
5121.0835

5120.93736 - 5119.366 = 1.57136 = 1/sc

4/{ln[(t ⋅ sc)/4]}sc

n = 10

4/{ln[(10240 ⋅ sc)/4]}sc = 0.84952

Zeta zeros

10239.5405
10240.3278
10241.2142
10241.492

10240.84952 - 10239.5405 = 1.309017 = 2.618034/2, again an exact value
10241.2142 - 10240.84952 = 0.364683 = 1 - 1sc

4/{ln[(t ⋅ sc)/4]}sc

n = 11

4/{ln[(20480 ⋅ sc)/4]}sc = 0.776725

Zeta zeros

20479.1428
20479.8897
20481.07
20481.91135

20480.776725 - 20479.1428 = 1.634
20480.776725 - 20479.8897 = 0.887
20481.07 - 20480.776725 = 0.2933
20481.91135 - 20480.776725 = 1.1346

1.1346 - 0.887 = 0.2476 = 1/10sc2

4/{ln[(t ⋅ sc)/4]}sc

n = 12

4/{ln[(40960 ⋅ sc)/4]}sc = 0.7154

Zeta zeros

40959.6213
40960.28154
40961.49
40961.796

40960.7154 - 40959.6213 = 1.094
40960.7154 - 40960.28154 = 0.43388
40961.49 - 40960.7154 = 0.7746

0.7746 - 0.43388 = 0.34072 = 0.5354sc = 2.5 x 1.362

Now, a more difficult test.

http://www.dtc.umn.edu/~odlyzko/zeta_tables/index.html

4/{ln[(t ⋅ sc)/4]}sc

n = 13

4/{ln[(81920 ⋅ sc)/4]}sc = 0.6631

Zeta zeros

81919.8862
81920.0323
81921.321

81920.6631 - 81919.8862 = 0.77689
81920.6631 - 81920.0323 = 0.6308
81921.321 - 81920.6631 = 0.6579

0.77689 + 0.6579 = 1.43479 = 2.86595/2

4/{ln[(t ⋅ sc)/4]}sc

n = 14

4/{ln[(163840 ⋅ sc)/4]}sc = 0.6178905 = 1/sc x 2.5424

Zeta zeros

163839.8017
163840.154
163840.632
163841.17

163840.6178905 - 163839.8017 = 0.81619
163840.6178905 - 163840.154 = 0.46389
163840.632 - 163840.6178905 = 0.01411
163841.17 - 163840.6178905 = 0.55211

0.81619 + 0.55211 = 1.3683
0.46389 - 0.01411 = 0.45 = 0.2861sc

4/{ln[(t ⋅ sc)/4]}sc

n = 15

4/{ln[(327680 ⋅ sc)/4]}sc = 0.57846

Zeta zeros

327679.5804
327680.803
327681.19467

327680.57846 - 327679.5804 = 0.99806
327680.803 - 327680.57846 = 0.22454
32768119467 - 327680.803 = 0.61614

0.99806 - 0.61614 = 0.38192 = 6sc/10

4/{ln[(t ⋅ sc)/4]}sc

n = 16

4/{ln[(655360 ⋅ sc)/4]}sc = 0.54376

Zeta zeros

655359.3515
655359.697
655359.9748
655360.9358
655361.3667
655361.8966

655360.54376 - 655359.3515 = 1.19226
655361.8966 - 655360.54376 = 1.35284

1.35284 + 1.19226 = 2.5451 = 100si


4/{ln[(t ⋅ sc)/4]}sc

Let t = 100sc

4/{ln[(100sc ⋅ sc)/4]}sc = 2.71376

Zeta zeros

60.8317
65.1125
67.079

66.34376 - 60.8317 = 5.51836
66.34376 - 65.1125 = 1.23756 = 2.47512 = 1/sc2
67.079 - 66.34376 = 0.72894, where 7.2738 = 286.1si


Lehmer pairs

4/{ln[(t ⋅ sc)/4]}sc

175sc = 111.353

4/{ln[(175sc ⋅ sc)/4]}sc = 2.1856

Zeta zeros

111.03
111.87
114.32

113.5386 - 111.03 = 2.508
113.5386 - 111.87 = 1.6681

2.508 - 1.6681 = 0.84 = 0.535/sc

4/{ln[(t ⋅ sc)/4]}sc

7005.1 = 11008sc (sc = 0.63636446)

11008 = 256 x 43

ln43 = 1.4134725 x 2.66
ln43sc = 2.618/2 + 2

4/{ln[(7005.1 ⋅ sc)/4]}sc = 0.895485

Zeta zeros

7005.062866
7005.100564
7006.74

7005.995485 - 7005.062866 = 0.93262
7005.995485 -  7005.100564 = 0.894921
7006.74 - 7005.995485 = 0.7445

0.93262 - 0.894921 = 1/26.53 26.66 = 53.33/2
0.93262 + 0.894921 = 2.872sc
0.93262 - 0.7445 = 0.18812 = 1/(2 x 2.658)

The hidden structure of the zeros of the zeta function:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1855591#msg1855591 (18 parts)

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006274#msg2006274 (5 parts)

« Last Edit: June 26, 2018, 08:49:55 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #548 on: June 30, 2018, 05:42:54 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION II

The average number of zeta zeros on the entire critical strip:

N(T) = (T/2π)(lnT/2π) - T/2π + S(T)

http://inspirehep.net/record/1245512/files/arXiv%3A1307.8395.pdf (average number of zeros on the critical line)

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2007826#msg2007826

http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros.pdf

http://web.yonsei.ac.kr/haseo/Rikkyo.pdf

http://www.mat.uniroma3.it/users/pappa/sintesi/24_Menici.pdf

The main term is a sacred cubit formula:

N(T) = (T ⋅ sc)[ln(t ⋅ sc)/4 - 1)\]/4

If L(T) = 4/{ln[(t ⋅ sc)/4]}sc (average spacing formula), then

N(T) = T(1/L(T) - sc/4)

If N(T) can be expressed directly in terms of sacred cubit constants, then it becomes an exact sacred cubit spacing formula.

T = 40

N(T) = 5.41765

L(T) = 3.3945

Zeta zeros

40.9187
43.327

8N - 43.327 = 0.1 - 3x0.02861

8N - 40.9187 = 1.361 x 5.33/3

43.3945 - 8N = 0.0533


T = 80

N(T) = 20sc(4sc - 1) = 19.661

L(T) = 1/sc2 = 2.47

Zeta zeros

79.337
82.91

82.47 + 79.337 = 161.807

4N = 78.644

80 - 4N ~= 1.361

79.337 - 4N = 0.44/sc

82.91 - 82.47 = 0.44

82.91 - 4N = 2 x 5.343 x 0.4


T = 160

N(T) = 56.9723

Zeta zeros

161.189
158.85

3N - 161.189 = 3.4 x 2.861

3N - 158.85 = 12.06688

63.6363 - 16.1773 = 47.459 (five elements subdivision)

47.459
12.066
7.11885
4.7459
2.373


T = 320

N(T) = 149.246

Zeta zeros

318.853
321.16

318.853 - 2N = 32sc = 8 x 2.545 = 2 x 10.18

10.18 = 4 x 2.545

321.16 - 2N = 136.01/6 = 20 + 2.668


T = 640

N(T) = 369.096

Zeta zeros

639.928
640.6948
641.945

2N - 639.928 = 154.43sc

617.72/4 = 154.43

617.72 x sc = 1/0.0025442

2N - 641.945 ~= 136.1 x 1.41347/2

2N - 640.6948 = 27.45 x 1/0.25424


T = 1280

N(T) = 879.3987

Zeta zeros

1279.3328
1280.1558
1281.828

1279.3328 - N = 400

1280.1558 - N = 7.5 x 53.434

1281.828 - N ~= 2.4 x π x 53.43 = 4.8/sc x 53.43


T = 100sc

N(T) = 13.32147

Zeta zeros

60.8317
65.1125
67.079

60.8317 - 4N = 7.54582 = 2.4 x π

5N - 65.1125 = 1/(1-sc) - 2

67.079 - 5N = 1/sc2 - 2

« Last Edit: June 30, 2018, 06:56:00 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #549 on: July 07, 2018, 02:06:32 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION III

N(T) = (T ⋅ sc)[ln(t ⋅ sc)/4 - 1)\]/4

L(T) = 4/{ln[(t ⋅ sc)/4]}sc (average spacing formula)

N(T) = T(1/L(T) - sc/4)


14.1347 + 6.3636 = 20.4977

Zeta zeros:

21.022
25.0108

N = 0.5951616

40N = 23.80646

25.0108 - 40N = 100si (1/sc2 - 2) = 4.5 x 0.2676

40N - 21.022 = 4.5 x 0.61877


14.134725 + 6.363 + 0.477225 = 20.975

3.1815
0.80886
0.477225
0.31815
0.159075

N = 0.68586

40N - 21.022 = 12 x 0.5344

40N - 25.0108 = 6 x sc2


14.134725 + 9.545 = 23.68

N = 1.231466

20N - 21.022 = 4 - 1/100si

25.0108 - 20N = 6sc/10


14.134725 + 9.545 + 0.995 = 24.675

16.1773 - 9.545 = 6.6328

6.6328
1.68632
0.99492
0.66328
0.33164

N = 1.444851

20N - 25.0108 = 1/sc3

30.424 - 20N = 2.4 x sc

20N - 21.022 = 2.861 x 1/(1-sc) = 5sc x 1/sc2


T = k10n x sc, 2n10, or other suitable values involving the sacred cubit

Let T1 = T + L(T)

N(T1) = (T + L(T))/2π {ln[(T + L(T)) ⋅ sc/4] - 1} = N1


T = 22 x 10 = 40

L(T) = 3.3945 = 5.335sc

T1 = 43.3945

N = 5.41765

N1 = 6.44

Zeta zeros

37.586 = z1
40.9187 = z2
43.327

8(N + N1) - 2z1 = 19.6892

8(N + N1) - 2z2 = 13.0238

8(N + N1) - z1 - z2 = 16.3565

16N - z1 - z2 = 8.1777

16N1 - z1 - z2 = 24.5353


8(N + N1) - z1 - z2 = 2 x (16N - z1 - z2) = 32N - 2z1 - 2z2

z1 + z2 = 32N - 8(N + N1)

This is the first exact formula ever for the sum of two consecutive zeta zeros obtained in terms of N, N1 and integers.

This means, in my opinion, that there exists a recurrence formula, precise to the nth decimal, which features Nk, Lk(T), and which has to involve the second zeta function.

For T = 20 and T = 80 there are symmetrical equations involving N and N1, but not the precise formula obtained for T = 40; an exact formula might be more complex for these values and for T = k10n (as an example).


T = 14.134725

N = -0.425722

L = 7.74977

N1 = 0.86345

z1 = 14.134725
z2 = 21.022

20N1 = 17.269
40N1 = 34.538

N1 - N = 1.2892

N1 + N = 0.43773

-20(N1 - N) + z1 + z2 = 9.362175

z1 + z2 - 40N1 = 0.618725 = 2.4749/4

40N1 - z2 = 13.516


40N1 - z2 - z1 - z2 + 40N1 ~= (N1 - N)10

80N1 - 10(N1 - N) - z1 ~= 2z2

Since we know the value of z1 (height of the Gizeh pyramid is 141.34725), we can immediately find the value of z2.


Again, a confirmation of the fact that there must exist a hidden recurrence formula, starting with 14.134725, which then will provide each subsequent zeta zero, one at a time. Obviously, such a recurrence formula, involving perhaps both zeta functions, would be very complex, since it must detect the Lehmer pairs if needed.

Also, there are shortcut formulas, for values such as 2n10 or k10nsc, suitable to obtain the adjacent zeta zeros values.


« Last Edit: July 07, 2018, 04:28:14 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #550 on: July 08, 2018, 12:56:20 AM »
RADIUS OF THE SUN PARADOX

Within the context of modern solar theory, the Sun cannot have a distinct surface. Gases are incapable of supporting such structures. Modern theory maintains the absence of this vital structural element. Conversely, experimental evidence firmly supports that the Sun does indeed possess a surface. For nearly 150 years, astronomy has chosen to disregard direct observational evidence in favor of theoretical models.

Dr. P.M. Robitaille

http://www.ptep-online.com/2011/PP-26-08.PDF

On the Presence of a Distinct Solar Surface: A Reply to Herve Faye



Spectacular images of the solar surface have been acquired in recent years, all of which manifest phenomenal structural elements on or near the solar surface. High resolution images acquired by the Swedish Solar Telescope reveal a solar surface in three dimensions filled with structural elements.

Beyond the evidence provided by the Swedish Solar Telescope and countless other observations, there was clear Doppler confirmation that the photosphere of the Sun was behaving as a distinct surface. In 1998, Kosovichev and Zharkova published their Nature paper X-ray flare sparks quake inside the Sun. Doppler imaging revealed transverse waves on the surface of the Sun, as reproduced in Figure 2: “We have also detected flare ripples, circular wave packets propagating from the flare and resembling ripples from a pebble, thrown into a pond”. In these images, the “optical illusion” was now acting as a real surface. The ripples were clearly transverse in nature, a phenomenon difficult to explain using a gaseous solar model. Ripples on a pond are characteristic of the liquid or solid state.


http://vixra.org/pdf/1310.0159v1.pdf

Commentary on the Radius of the Sun:
Optical Illusion or Manifestation of a Real Surface?

Observational astronomy continues to report increasingly precise measures of solar radius and diameter. Even the smallest temporal variations in these parameters would have profound implications relative to modeling the Sun and understanding climate fluctuations on Earth. A review of the literature convincingly demonstrates that the solar body does indeed possess a measurable radius which provides, along with previous discussions (Robitaille P.M. On the Presence of a Distinct Solar Surface: A Reply to Herve Faye. Progr. Phys., 2011, v. 3, 75–78.), the twenty-first line of evidence that the Sun is comprised of condensed-matter.

For theoretical solar physicists, any variation in the dimensions of the Sun would have severe consequences with respect to the gaseous models. The latter would be hard-pressed to account for fluctuations in radius. This helps to account for the reassurance experienced when the solar radius is perceived as constant.


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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #551 on: July 25, 2018, 03:57:10 AM »
STANDARD ATMOSPHERE VALUE FOR A COLUMN OF WATER PARADOX



https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3768090/

According to the official chronology of history, the effects of the "atmospheric pressure" upon a column of water of considerable height were measured as early as 1640 AD.



Illustration of Gasparo Berti's experiment using a very long lead tube containing water.

A column of water 32.381' (π2 m) (cross-section 1 square inch) generates a pressure at the bottom of one atmosphere.

The magnitude of the value is exactly equal to that of the g acceleration.

Even if we take the value of 10.336 meters to be true, then 10.336/π2 = 1.0472 which is exactly π/3!

Under the spherical Earth hypothesis, this is supposed to be just a random coincidence.

However, this fact cannot be true.

Modern science assumes that the proportions of the ingredients of the air (distribution of gases) have varied since the formation of the Earth, yet we are to believe that now the atmospheric pressure on a column of water will generate a height of the liquid in the glass tube of exactly π2 m (the value of the g acceleration).

In addition, there is the atmospheric escape which takes place every year.

How could these random fluctuations in the chemical composition/mass of the Earth's atmosphere have lead to the precise value of π2 m for the height of the water in the glass tube experiment?

In the flat earth theory, this fact is easily explained: the effect of the laevorotatory waves upon the column of water (π2 meters) will equal exactly the magnitude of the value of the g acceleration (π2).

The height of the liquid column does not rise because of atmospheric pressure. It is an extraordinary antigravitational effect and a proof of the existence of laevorotatory subquark strings.

g acceleration = π2 m/s2

http://www.aetherometry.com/Aetherometry_Intro/pratt_aether_grav.php#g3 (section 3, Gravitational Pendulums, g related to π2)

Again, even if we take the value of 10.336 meters to be true, then 10.336/π2 = 1.0472 which is exactly π/3!

« Last Edit: July 25, 2018, 05:02:07 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #552 on: July 25, 2018, 08:10:24 AM »
TECHNICAL ATMOSPHERE VALUE FOR A COLUMN OF WATER PARADOX II

The standard atmosphere, defined as being exactly equal to 101,325 pascals, is the reference value for the average atmospheric pressure at sea level.  A torr is fixed by definition as being precisely equal to 1/760th of a standard atmosphere. The value of a millimeter of mercury is determined by: 1) the definition of gravity, 2) to the density of mercury (13.595 078(5) g/ml @ 0 °C, NIST value), and 3) to the temperature at which mercury's density is taken. However, the barometric pressure can vary by thousands of Pa in a single week.

The older concept of a technical atmosphere was phased out even though it worked very well in practice.

https://sizes.com/units/atmosphere-technical.htm

Using the technical atmosphere, we get 28.96 inHg and exactly 10 meters of water.

100,000/101,325 = π2/10

http://www.aetherometry.com/Aetherometry_Intro/pratt_aether_grav.php#g3

The true gravitational acceleration at the Earth's surface corresponds to the gravitational field intensity E, and not to the net resultant acceleration, which varies with latitude.

"Traditionally, this field intensity is considered to be counteracted by the centrifugal force created by the Earth's rotation; the centrifugal acceleration is zero at the poles and reaches a maximum of 0.03392 m/s2 at the equator. One of the problems in the current understanding of gravity is that the difference between the gravitational acceleration at the poles and at the equator is greater than any centrifugal reaction can account for. This discrepancy is conventionally explained by the Earth being not a perfect sphere but an oblate spheroid, or rather a triaxial spheroid.

Assuming that g = π2m/s2, and taking account of the centrifugal reaction, the value of g at the equator should be 9.83568 m/s2, whereas the measured value is far lower: 9.780524 m/s2. Modern technology permits more exact determinations of the measured values of net g at the poles and the equator, along with better determinations of the polar and equatorial radii. This makes it possible to accurately determine the angular velocity function (Ω) that is a constituent of the gravitational field intensity. It is pointed out that if we employ the values for net g at the poles (where no centrifugal reaction exists) along with the polar radii to determine the value of Ω, and then use this value together with the known equatorial radius to determine the gravitational field intensity at the equator, this will be found to be exactly π2m/s2, to the fourth digit!

This rules out geometric explanations for the actual value of net g at the equator, as the differences in terrestrial geometry are already taken into account. So something besides the centrifugal force or geometry must account for the counteraction of gravity at the equator by Δ = (π2 - 0.03392) - 9.780524 = 0.05516 m/s2."

Therefore, this extra factor (which could be accompanied by other antigravitational factors to be accounted for) has to be substracted as well from the height of the column of water.

In the end, the magnitude of the true value of g, π2, will equal the calculated height of the column of water, or be extremely close to it.

The mysterious antigravitational factor discovered by Dr. P. Correa is directly related to the effects exerted by the laevorotatory subquark strings.


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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #553 on: July 26, 2018, 01:51:30 AM »
TECHNICAL ATMOSPHERE VALUE FOR A COLUMN OF WATER PARADOX III



The sacred cubit is designated in the form of a horseshoe projection, known as the "Boss" on the face of the Granite Leaf in the Ante-Chamber of the Pyramid. By application of this unit of measurement it was discovered to be subdivided into 25 equal parts known now as: Pyramid inches.

The value chosen in 1954 by the 10th Conférence Générale des Poids et Mesures (CGPM) for the standard atmosphere is directly related to the sacred cubit.

https://www.bipm.org/jsp/en/ViewCGPMResolution.jsp?CGPM=10&RES=4

1013250 dynes per square centimetre (101325 Pa).

4 x 101,325 = 405,300

405,3001/2 = 636.63176

2/π = one sacred cubit = 0.636619772

A four digit perfect match.

100,000/101,325 = 0.9869233

π2/10 = 0.98696044

A four digit perfect match.

Dr. C. Goldblatt, one of the foremost experts on atmospheric physics in the world (Space Science and Astrobiology Division, NASA Ames Research Center) explains the total and complete random nature of the Earth's atmosphere evolution.

https://arxiv.org/pdf/1710.10557.pdf

Then, he explains these facts in the context of the faint young sun paradox:

https://www.clim-past.net/7/203/2011/cp-7-203-2011.pdf

"Geology has been viewed as a collection of events derived from insignificant causes, a string of accidents."

Yet, out of this string of accidents, we obtain a four digit perfect match between the value of the standard atmosphere and the magnitude of the g acceleration, and between the sacred cubit and the value of the standard atmosphere.

The main reason why the technical atmosphere (one kilogram-force per square centimeter) was phased out is connected in a direct way to the fact that by using this value, the figure for the column of water will be exactly 10 meters, a fact impossible to explain in the context of the random fluctuations of the atmosphere's chemical composition/mass.

980.665 mbar = 98.0665 kPa technical atm = 28.959136 inHg = 32.8093 ft of water = 10.00027464 m

The ratio 100,000/101,325 equals exactly π2/10 (g acceleration divided by 10, the height of the column of water using the technical atmosphere).

In 1982, the International Union of Pure and Applied Chemistry (IUPAC) recommended that for the purposes of specifying the physical properties of substances, “standard pressure” should be precisely 100 kPa (1 bar) = 100,000 Pa.

http://goldbook.iupac.org/html/S/S05921.html


The missing apex of the Gizeh pyramid measures 286.1 sacred inches (7.2738 meters), where 286.1 = 450 sacred cubits, and 100 sc = 45 x 1.4134725, 141.34725 = the height of the Gizeh pyramid frustum.

727 Torr = 28.622156 inHg = 9.8839 m of water, a value very close to the g acceleration figure π2 = 9.8696

How many inhg in 1 torr? The answer is 0.039370072825186.

One sacred inch = 0.025424 meters.

1/3.9370073 = 0.254


Now, here is another reason why the technical atmosphere was phased out.

https://www.sensorsone.com/kpa-to-mh2o-conversion-table/

The conversion factor from pascals to meters of water involves this value: 980,665 Pa (one technical atmosphere).

1/9.80665 = 0.1019716213

2/π = 0.636619772

32/100π = 0.10185916

0.1019716213 - 010185916 = 0.00011246129 = 2.861 x 0.0000393083852

1/3.93083852 = 0.2543986

0.2543986/4 = 0.063599661

Then, the conversion factor can be evidenced directly in terms of sacred cubits.

1013250 dynes per square centimetre (101325 Pa).

4 x 101,325 = 405,300

405,3001/2 = 636.63176

2/π = one sacred cubit = 0.636619772

101,325 = (2000/π)2/4 + 6sc

Then, the value of the height of column of water, corresponding to 101,325 Pascals can be expressed directly in terms of sacred cubits.

« Last Edit: July 28, 2018, 05:17:15 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #554 on: July 29, 2018, 04:07:07 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION IV

z1 = 14.134725

L(z1) = 7.74977

14.134725 + 7.74977 = 21.884497

Now, all of the previous results will be applied to obtain the value of the second zero of the zeta function, to four decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

9.5445 - 6.36363 = 6.36363 - 3.1815 = 3.1815

3.1815
0.80886
0.477225
0.31815
0.159075


14.134725 + 6.36363 = 20.4977 (lower bound)

First estimate, using the zeta function directed to the left, the lower bound

20.4977 + 0.80886 = 21.30656 (upper bound)

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (basic subdivision of the first 63.63636 sacred cubit interval into five elements ratios)

2.7834

22.29945

Upper bound of first estimate using the zeta function directed to the right.

2.0757

20.22945

Lower bound

Both lower and upper bounds of the estimates both zeta functions  will be used to refine the approximation.


Since the lower bound of the second zeta function has a smaller value than the corresponding figure of the lower bound of the first zeta function, the next value in the five element subdivision is substracted to get an UPPER BOUND.

10.9478
2.7834
1.6422
1.90478
0.5474

8.1644
2.0757
1.22466

0.81644
040822


1.22466

21.069425

This is the new UPPER BOUND for the approximation.

3.1815
0.80886
0.477225
0.31815
0.159075

20.4977 + 0.477225 = 20.975

2.0757 - 1.22466 = 0.85104

0.85104
0.21637
0.127656
0.085104
0.042552

Substracting these bottom four values successively from 21.069425:

20.853055
20.94177
20.984321
21.026873

Since now 20.984321 exceeds the estimate from the other zeta function (20.975), this will be new LOWER BOUND of the approximation.

So far:

Lower bound: 20.984321
Upper bound: 21.069425


0.80886 - 0.477225 = 0.331635

0.331635
0.084315
0.049745
0.0331635
0.016582

Adding 0.084315 to 20.975 will equal 21.0593, a figure which already exceeds the upper bound.

Adding 0.049745 to 20.975 will equal 21.024745.

Adding 0.0331635 to 20.975 will equal 21.0081635.

21.024745 will be the new upper bound of the approximation.

0.085104 - 0.042552 = 0.042552

0.042552
0.01081842
0.0063828
0.0042552
0.0021276

Substracting the bottom four figures from 21.026873 we obtain:

21.016055
21.0205
21.02262
21.024745

21.024745 is the SAME VALUE obtained from the five element subdivision for the first zeta function, this is how we know it is the upper bound of the entire approximation.

The lower bound is 21.016055.

To get the lower bound for the first zeta function, we have to subdivide the interval further.

The last estimate was 21.0081635.

0.084315 - 0.049745 = 0.03457

Using only the first two subdivision values:

0.03457
0.0087891

21.024545 + 0.0087891 = 21.03353, a figure which is too large.

0.049745 - 0.0331635 = 0.016582

Again, using only the first two subdivision values:

0.016582
0.0042157

21.0081635 + 0.0042157 = 21.01238

Continuing in this way we obtain:

21.01786

This will be the new lower bound of the entire approximation.

Continuing even further:

21.0226217 (this corresponds to the subdivision 0.00285253 and 0.00072523; the previous subdivision is 0.003825 and 0.00097247).

This is the same value as the one obtained from the other subdivision.

This will be new UPPER BOUND of the entire approximation.

0.0063828 - 0.0042552 = 0.0021273

0.0021273
0.000540845
0.0003181
0.00021273
0.000106365

21.02262 - 0.000540845 = 21.02207916

21.02262 - 0.00021273 = 21.02240727

To get the new lower bound, a figure higher than 21.016055 has to be obtained from the first zeta function subdivision.

0.003825
0.0009725
0.00057375
0.0003825
0.00019125

The value corresponding to 0.0009725 is 21.0218965.

This now is the new lower bound.


So far:

21.0218965 = lower bound

21.0226217 = upper bound

0.00285253 - 0.00072523 = 0.0021273

This is the same value as that obtained earlier from the second zeta function.

Since 21.02207916 exceeds 21.0218965, it will become the new UPPER BOUND of the entire approximation.

0.0009725 - 0.00057375 = 0.00039875

0.00039875
0.000101378

21.0218965 + 0.000101378 = 21.02199788

This figure will be the new lower bound.

The true value for the second zeta zero is:

21.022039639

Already we have obtained a five digit/three decimal place approximation:

21.02207916

« Last Edit: July 30, 2018, 03:26:00 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #555 on: July 30, 2018, 03:22:36 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION V

z2 = 21.022

L(z2) = 5.2026

21.022 + 5.2026 = 26.2246

The third zeta zero, to four decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

14.134725 + 9.545 = 23.6747

16.1773 - 9.5445 = 6.6328

6.6328
1.68632
0.99492
0.66328
0.33164

23.6747 + 1.68632 = 25.36602

23.6747 + 0.99492 = 24.67462

23.6747 is the first lower bound.

25.36602 is the first upper bound.

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (basic subdivision of the first 63.63636 sacred cubit interval into five elements ratios)

3.7322

25.099425

1.64

23.459425

1.09478

24.001945

0.5474

24.552025

The values are taken from the subdivision:

14.68
3.7322
2.202
1.468
0.734

14.68 - 3.7322  = 10.9478

10.9478
2.7834
1.6422
1.09478
0.5474

Upper bound of first estimate using the zeta function directed to the right:

25.099425

Lower bound:

24.552025

Just like before, we search for a higher lower bound in both subdivisions, and for a lower upper bound in both subdivisions (a comparison, in order to locate the precise and correct subdivision interval for the zeta zero).

So far:

Upper bound

25.099425

Lower bound

24.67462


1.68632 - 099492 = 0.6914

0.6914
0.17578
0.10371
0.06914
0.03457

Adding these bottom four values successively to 24.67462:

24.8504
24.7783
24.77376
24.7092

0.6914 - 0.17578 = 0.51562

0.51562
0.1311
0.077343
0.051562
0.025781

Adding these bottom four values successively to 24.8504:

24.9815
24.9277
24.902
24.8762

From the other zeta function:

0.5474
0.139171
0.08211
0.05474
0.02737

Substracting these bottom four values successively from 25.099425:

24.96025
25.0173
25.044685
25.072055

24.96025 is the new LOWER BOUND.

Since 24.9815 (from the other zeta function) is a higher lower bound, this value will become the new lower bound for the entire approximation.

In order to obtain the new upper bound:

0.51562 - 0.1311 = 0.38452

0.38452
0.09776
0.057678
0.038452
0.019226

Adding these bottom three values successively to 24.9815:

25.039178
25.019952
25.000726

Then, 25.000726 becomes the new lower bound, while 25.019952 is the new upper bound for the first zeta function.

Since 25.0173 (second zeta function) is a lower value than 25.019952, this then is the new UPPER BOUND for the entire approximation.

So far:

25.000726 is the lower bound

25.0173 is the upper bound

0.038452 - 0.019226 = 0.019226

0.019226
0.004888
0.002884
0.0019226
0.0009613

Adding these bottom four values successively to 25.000726:

25.005614
25.00361
25.002649
25.00169

Using the second zeta function:

0.139171 - 0.08211 = 0.057061

0.057061
0.0145072
0.00856
0.0057061
0.00285305

Substracting these bottom four values successively from 25.0173:

25.0028
25.00874
25.0116
25.014447

The new lower bound is 25.00874 (a higher lower bound than 25.005614).

The new upper bound is 25.0116.

0.019226 - 0.004888 = 0.014338

0.014338
0.0036453
0.0021507
0.0014338
0.0007169

Adding these bottom four values successively to 25.005614:

25.00926
25.00776
25.00705
25.006331

0.014338 - 0.0036453 = 0.0106927

0.0106927
0.0027185
0.001604
0.00106927
0.000534635

Adding these bottom four values to 25.00926:

25.012
25.010864
25.01033
25.0098

25.010864 is the new upper bound (a lower upper bound than 25.0116).

25.01033 is the new lower bound.

The true value for the third zeta zero is:

25.01085758

Already we have obtained a six digit/four decimal place approximation:

25.010864

« Last Edit: July 30, 2018, 03:26:34 AM by sandokhan »

*

sandokhan

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Re: Advanced Flat Earth Theory
« Reply #556 on: July 31, 2018, 01:34:57 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION VI

z3 = 25.0108

L(z2) = 4.54832

29.55912

The fourth zeta zero, to three decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (basic subdivision of the first 63.63636 sacred cubit interval into five elements ratios)

14.134725 + 16.1773 = 30.312

16.1773 + 2.373 = 32.685

5.0045

28.81

2.95

30.8694

30.312 is the first lower bound.

30.8694 is the first upper bound.

2.373
0.60331
0.35595
0.2373
0.11865

Adding the bottom three values to 30.312:

30.668
30.55
30.43065

5.0045 - 2.95266 = 2.05184

2.05184
0.509798
0.307776
0.205184
0.102592

Substracting the bottom four values from 30.8694:

30.3596
30.5616
30.664216
30.76681

30.3596 is the new lower bound.

30.5616 is the new upper bound for the second zeta function.

30.43065 is the new upper bound for the first zeta function; since this figure is smaller than 30.5616, it is the upper bound of the entire approximation.

0.11865
0.0301656
0.0177975
0.011865
0.0059325

Adding the bottom four values to 30.312:

30.34216
30.3298
30.324
30.318

0.11865 - 0.0301656 = 0.0884844

Using only the first two subdivisions (corresponding to 534 and 136.1):

0.0884844
0.022496

30.342 + 0.022496 = 30.36465


The subdivisions for the second zeta function.

0.509796 - 0.307776 = 0.202022

0.202022
0.051362

30.5616 - 0.051362 = 30.510238

In order to make a new comparison between the two zeta functions, we have to subdivide further in order to determine the correct upper and lower bounds using subdivisions which have a very close value.

0.202022 - 0.051362 = 0.15066

0.15066
0.038304

30.510238 - 0.038304 = 30.47193

0.15066 - 0.038304 = 0.112356

0.112356
0.0285654

30.47193 - 0.0285654 = 30.44336

0.112356 - 0.0285654 = 0.083791

0.083791
0.021303
0.012569
0.0083791
0.0041895

Substracting the bottom four values from 30.44336:

30.422057
30.43079
30.435
30.43917


The subdivisions for the first zeta function.

0.0884844
0.022496

30.342 + 0.022496 = 30.36465

0.0884844 - 0.022496 = 0.065988

0.065988
0.016777

30.36465 + 0.016777 = 30.38143

0.065988 - 0.016777 = 0.049211

0.049211
0.0125114

30.38143 + 0.0125114 = 30.39394

0.049211 - 0.0125114 = 0.0366996

0.0366996
0.00933051

30.39394 + 0.00933051 = 30.40327

0.0366996 - 0.00933051 = 0.0273691

0.0273691
0.0069583

30.40327 + 0.0069583 = 30.410228

0.0273691 - 0.0069583 = 0.0204108

0.0204108
0.005189242

30.410228 + 0.005189242 = 30.41542

0.0204108 - 0.005189242 = 0.01522156

0.01522156
0.00387

30.41542 + 0.00387 = 30.41928

0.01522156 - 0.00387 = 0.01135156

0.01135156
0.002886

30.41928 + 0.002886 = 30.422176

By comparison with the subdivisions obtained from the second zeta function, we can see that 30.422176 is the new lower bound.

0.01135156 - 0.002886 = 0.0084656

0.0084656
0.0021523

30.422176 + 0.0021523 = 30.424328


Returning to the subdivisions for the second zeta function.

0.112356 - 0.0285654 = 0.083791

0.083791
0.021303
0.012569
0.0083791
0.0041895

Substracting the bottom four values from 30.44336:

30.422057
30.43079
30.435
30.43917

0.021303 - 0.012569 = 0.008734 (this is the interval of the subidivision where the upper and lower bounds of the second zeta function are located)

0.008734
0.0022205
0.0013101
0.0008734
0.0004367

Substracting the bottom values from 30.43079:

30.42857
30.42948
30.429917
30.43035


Returning to the subdivisions for the first zeta function.

0.0084656
0.0021523

30.422176 + 0.0021523 = 30.424328

0.0084656 - 0.0021523 = 0.0063133

0.0063133
0.0016051
0.000947
0.00063133
0.000315665

Adding the bottom three values to 30.424328:

30.425275
30.424959
30.424684


Returning to the subdivisions for the second zeta function.

0.008734 - 0.0022205 = 0.0065135

0.0065135
0.001656

30.42857 - 0.001656 = 30.4269

0.0065135 - 0.001656 = 0.0048575

0.0048575
0.001235

30.4269 - 0.001235 = 30.42566

0.0048575 - 0.001235 = 0.0036255

0.0036255
0.000921
0.00054383
0.00036255
0.0001813

Substracting the four bottom values from 30.42566:

30.42474
30.42512
30.4253
30.42548


30.424684 is the new lower bound.

30.424959 is the new upper bound.

The true value for the fourth zeta zero is:

30.424876126

Already we have obtained a five digit/three decimal place approximation:

30.424684

*

sandokhan

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Re: Advanced Flat Earth Theory
« Reply #557 on: July 31, 2018, 03:54:21 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION VII

z4 = 30.4247

L(z4) = 3.98331

34.408

The fifth zeta zero, to three decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (basic subdivision of the first 63.63636 sacred cubit interval into five elements ratios)

16.1773 + 2.373 = 32.685

4.7459 - 2.373 = 2.373

2.373
0.6033
0.356
0.2373
0.118645

Adding to the bottom four values to 32.685:

33.2883
33.041
32.9223
32.8036


1.968

31.8494

0.984

32.8294

6.7106

33.8

32.8294 is the first lower bound.

Since 32.9223 is a higher lower bound, this value is the lower bound of the entire approximation.

To find the first upper bound, we need to subdivide the intervals for the second zeta function further, in order to find a lower upper bound than 33.041.

0.98422
0.25023

33.8 - 0.25023 = 33.55

0.98422 - 0.25023 = 0.734

0.734
0.18661

33.55 - 0.18661 = 33.364

0.734 - 0.18661 = 0.5474

0.5474
0.139171

33.364 - 0.139171 = 33.225

0.5474 - 0.139171 = 0.40823

0.40823
0.103788

33.225 - 0.103788 = 33.1212

0.40823 - 0.103788 = 0.304442

0.304442
0.0774

33.1212 - 0.0774 = 33.0438

0.304442 - 0.0774 = 0.227042

0.227042
0.05772

33.0438 - 0.05772 = 32.9861

32.9861 is the new upper bound of the entire approximation.


0.356 - 0.23729 = 0.11871

0.11871
0.0302
0.01781
0.011871
0.0059355

Adding the bottom four values to 32.9223:

32.9525
32.9401
32.9342
32.928

32.9401 is the new upper bound.


Returning to the subdivisions for the second zeta function.

0.227042 - 0.05772 = 0.16932

0.16932
0.04305

32.9861 - 0.04305 = 32.94305

0.16932 - 0.04305 = 0.12627

0.12627
0.0321
0.01894
0.012627
0.0063135

Substracting the bottom four values from 32.94305:

32.911
32.9241
32.9304
32.93673

32.93672 is the new upper bound.

0.012627 - 0.0063135 = 0.0063135

0.0063135
0.0016052
0.000947
0.00063135
0.000315675

Substracting the bottom four values from 32.93673:

32.935125
32.935783
32.9361
32.936414


Returning to the subdivisions for the first zeta function.

0.01781 - 0.011871 = 0.0059355

0.0059355
0.001509
0.000891
0.00059355
0.000297

Adding the bottom four values to 32.9342:

32.93571
32.935091
32.9348
32.9345

Since 32.935091 is a lower value than 32.935125, this figure is the new upper bound of the entire approximation.

0.0063135 - 0.0016052 = 0.0047083

0.0047083
0.00119704
0.000706245
0.00047083
0.000235415

Substracting the last figure from 32.935125 we obtain 32.93489.

Since this is greater value than 32.9348, it becomes the new lower bound of the entire approximation.

This is further proof that 32.935125 was an upper bound, and that 32.935091 is the new upper bound for the entire approximation.

The true value for the fifth zeta zero is:

32.935061588

Already we have obtained a five digit/three decimal place approximation:

32.935091


Further subdivisions for greater accuracy.

0.00047083 - 0.000235415 = 0.000235415

0.000235415
0.000059852
0.0000353
0.0000235415
0.000011771

Substracting the bottom four values from 32.935125:

32.935065
32.935089
32.935101
32.935113


Returning to the subdivisions for the first zeta function.

0.000891 - 0.00029745 = 0.00029745

0.00029745
0.000075624

32.9348 + 0.000075624 = 32.9348756

0.00029745 - 0.000075624 = 0.000221826

0.000221826
0.0000564

32.9348756 + 0.0000564 = 32.93492

0.000165426
0.000042055

32.93492 + 0.000042055 = 32.934962

0.00012337
0.000031366

32.934962 + 0.000031366 = 32.9349934

0.000092334
0.000023475

32.9349934 + 0.000023475 = 32.93501688

0.000068859
0.0000175067

32.93501688 + 0.0000175067 = 32.9350344

0.000051353
0.000013056

32.9350344 + 0.000013056 = 32.93504746

0.000038297
0.00000973663

32.93504746 + 0.00000973663 = 32.9350572

0.000028561
0.00000726135

32.9350572 + 0.00000726135 = 32.93506446

This becomes the new upper bound of the entire approximation (a value smaller than 32.935065 obtained from the second zeta function subdivision).

0.000028561
0.00000726135
0.00000428415

32.9350572 + 0.00000428415 = 32.93506148

The true value for the fifth zeta zero is:

32.935061588

Already we have obtained an eight digit/six decimal place accuracy:

32.93506148

« Last Edit: July 31, 2018, 04:49:49 AM by sandokhan »

*

sandokhan

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Re: Advanced Flat Earth Theory
« Reply #558 on: August 01, 2018, 01:10:01 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION VIII

z5 = 32.935

L(z5) = 3.7927

36.7277

The sixth zeta zero, to three decimal places accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (basic subdivision of the first 63.63636 sacred cubit interval into five elements ratios)

14.134725 + 16.1773 + 7.1185 = 37.43

12.066 - 7.1185 = 4.9475

4.9475
1.2577
0.74205
0.49475
0.24735

37.43 + 0.24735 = 37.67735

2.64

37.8794

3.96

36.56942

37.6773 is the first upper bound.

36.56942 is the first lower bound.


0.24735
0.062886
0.0371
0.024725
0.012367

Adding the bottom four values to 37.43:

37.4929
37.467
37.455
37.442

0.24735 - 0.062886 = 0.184464

0.184464
0.0469

37.4929 + 0.0469 = 37.54

0.184464 - 0.0469 = 0.137564

0.137564
0.034974
0.02063
0.0137564
0.00688

Adding the bottom four values to 37.54:

37.575
37.5606
37.553756
37.54688

37.575 is the new lower bound.


3.96 - 2.6395 = 1.3205

1.3205
0.335724
0.198075
0.13205
0.066025

Substracting the bottom values from 37.8794 (the upper bound for the second zeta function):

37.54367
37.681
37.74735
37.81337

0.335724 - 0.198075 = 0.137649

0.137649
0.03499581
0.02064735
0.0137649
0.00688245

Substracting the bottom four values from 37.681:

37.6460042
37.66035
37.66724
37.6741

0.137649 - 0.03499581 = 0.1026532

0.1026532
0.0261

37.6460042 - 0.0261 = 37.61991

0.1026532 - 0.0261 = 0.0765532

0.0765532
0.019463

37.61991 - 0.019463 = 37.600447

0.0765532 - 0.019643 = 0.0571

0.0571
0.0145146
0.008565
0.0057
0.002855

Substracting the bottom four values from 37.600447:

37.5859324
37.592
37.5947
37.5976

37.5859324 is the new lower bound of the entire approximation.


Returning to the subdivisions for the first zeta function.

0.137564 - 0.034974 = 0.10259

0.10259
0.0260825
0.0153885
0.010259
0.0051285

Adding the bottom four values to 37.575:

37.601
37.5904
37.58526
37.58013

37.5904 is the new upper bound.

The true value for the sixth zeta zero is:

37.586178159

Already we have obtained a five digit/three decimal place approximation:

37.5859324

*

sandokhan

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Re: Advanced Flat Earth Theory
« Reply #559 on: August 01, 2018, 09:10:19 AM »
The five element subdivision algorithm is superior to the Riemann-Siegel formula: effortlessly, one can obtain the zeros of the zeta function on the 1/2 line using only elementary mathematics. B. Riemann discovered that the zeros of the zeta function are related to the primes numbers. Here, it has been shown that the five elements subdivision process generates the zeros of the zeta function in a most precise and remarkable way.

Each and every mathematician who has studied the Riemann zeta function has totally ignored the second zeta function (negative zeros located on the 1/2 line, 1/2 - it). Yet, there are two zeta functions, and their interplay is revealed by realizing that 14.134725, the value of the first zeta zero, is directly related to the sacred cubit:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301

It is the points where the two subdivisions have approximately equal values that form the set of zeta zeros.

That is, the law of five elements of proportions applied to the sacred cubit distance will reveal the approximate location of the zeta zeros.

Then, a precise algorithm featured in the previous messages will compute each and every zeta zero to the nth decimal digit.

Lehmer pairs have a definite pattern to their location on the 1/2 line:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1858153#msg1858153

The Lehmer phenomenon, a pair of zeros which are extremely close, is related to the close proximity of some of the values of the two subdivisions of the 63.6363... segment.

Only the five element subdivision can detect and at the same time explain extreme Lehmer pairs such as these:

1.30664344087942265202071895041619 x 1022

1.30664344087942265202071898265199 x 1022

The average spacing of zeros at that height is 0.128, while the above pair of zeros is separated by 0.00032 (1/400th times the average spacing).

There is something else mathematicians have overlooked: the fact that Riemann had proven that all of the zeros lie on the 1/2 line some 160 years ago, using the functional equation. There is no way that he would have embarked to derive the Riemann-Siegel asymptotic formula, had he not been totally sure of the fact that all of the zeros lie on the 1/2 line: all he wanted to do is to verify that actually the first few zeros are situated on the 1/2 line.

https://arxiv.org/ftp/arxiv/papers/0801/0801.4072.pdf

A Necessary Condition for the Existence of the Nontrivial Zeros of the Riemann Zeta Function

(a paper which shows that B. Riemann must have followed a similar kind of argument, using the newly discovered zeta functional equation, to reach the conclusion that all the nontrivial zeros are all located on the ½ line)

That is why the Clay Institute of Mathematics prize description should be modified:

http://www.claymath.org/millennium-problems/riemann-hypothesis

The prize should be awarded for a much more difficult problem.

"Answers to such questions depend on a much more detailed knowledge of the distribution of zeros of the zeta function than is given by the RH. Relatively little work has been devoted to the precise distribution of the zeros. The main reason for the lack of research in this area was undoubtedly the feeling that there was little to be gained from studying problems harder than the RH if the RH itself could not be proved."

The distribution of the zeros of the zeta function has been proven here to be directly explained in terms of the five element subdivision of the sacred cubit segment.

The values of the first six zeta zeros have been obtained in an elegant and precise manner, without using either the Riemann-Siegel formula, or the Euler-Maclaurin summation formula.

The complexity of the Riemann-Siegel coefficients:



The Riemann-Siegel formula does not deal with the distribution of zeros.

Nor can it reveal the hidden pattern/structure of the zeta zeros.

The five element subdivision algorithm creates the zeta zeros, which in turn are related to the distribution of the prime numbers.

"These zeros did not appear to be scattered at random. Riemann's calculations indicated that they were lining up as if along some mystical ley line running through the landscape."

The mystical ley line has been revealed here: it is the five element subdivision algorithm.

All of the zeta zeros have to be located on the 1/2 line, since otherwise the five element subdivision algorithm would be disrupted, reverberating all the way to the first 63.6363... segment, and thus to the very value of the first zeta zero, 14.134725.

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1859799#msg1859799

The five element subdivision algorithm is much simpler than the Riemann-Siegel formula, and it reveals the hidden pattern of the zeta zeros to the nth decimal precision. It does not require a huge sum of terms, nor extremely complex remainders, it is even an art form to derive the zeta zeros, carefully keeping track of the upper and lower bounds of the approximations.

« Last Edit: August 01, 2018, 09:20:50 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #560 on: August 05, 2018, 04:15:39 AM »
It is clearly a preliminary note and might not have been written if L. Kronecker had not
urged him to write up something about this work (letter to Weierstrass, Oct. 26 1859). It
is clear that there are holes that need to be filled in, but also clear that he had a lot more
material than what is in the note. What also seems clear : Riemann is not interested in an asymptotic formula, not in the prime number theorem, what he is after is an exact formula!

(Lecture given in Seattle in August 1996, on the occasion of the 100th anniversary
of the proof of the prime number theorem, Atle Selberg comments about Riemann’s
paper: A. Selberg, The history of the prime number theorem, A SYMPOSIUM on
the Riemann Hypothesis, Seattle, Washington)

This exact formula has been obtained here: the five element subdivisions of the interval lead directly and precisely to the values of the zeta zeros, to the nth decimal precision desired.

The Riemann-Siegel formula is a local expression, while the Five Element Subdivision algorithm is a global formula.

It involves no transcendental or algebraic functions, but only the four elementary operations of mathematics.

From a quantum physical point of view, the two counter-propagating zeta function waves represent sound waves which travel over the sacred 63.6363... distance back and forth inside a boson (or inside an antiboson).

Thus, all of the zeros of the zeta function must be located on the 1/2 critical line: if any of the Riemann zeta function ζ(s) non-trivial zeros would lay off the critical line, s = σ + it, σ = 1/2 - ε, then the values of all of the other zeros would have to be modified as well, all the way to the first zero, 14.134725.

The sum of any two sides of a triangle is greater than the third side.



The five elements sequence of proportions would be disrupted as the distance from the previous zero to the zero which is off the critical line, and from the zeta zero which finds itself on the σ = 1/2 - ε line to the next zero would be greater than the distances from that previous zero to the next two zeta zeros to be found on the critical line.

Moreover, since there are two counter-propagating zeta function waves, there would have be to TWO zeros off the critical line within the same 63.6363... segment.

To see the issues involved, here are the first five element subdivisions (first upper and lower bounds) for the second and third zeta zeros.

21.022

14.134725 + 6.3636 = 20.4975

14.134725 + 6.3636 + 0.80886 = 21.30656

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 = 22.29945

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 - 2.0757 = 20.22945

25.0108

14.134725 + 9.5445 + 1.68632 = 25.36602

14.134725 + 9.5445 + 0.99492 = 24.67

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 = 25.099425

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 1.64 = 23.459

Even the slightest deviation from the 21.022039639 and 25.010857580 values would invalidate the entire five element subdivision algorithm.

The difference between the value of the second zeta zero, 21.022, and the value of the third zero, 25.0108 is seen at a glance to be a simple substraction of 2.7834 and 2.0757 from 3.732, as opposed to substracting 1.64 from 3.732, and at the same time, adding 0.80886 to 6.3636 for the second zero, while for the third zero the figure (1.5 x 6.36363) + 1.68632 is added.

This is the great advantage of the five element subdivisions algorithm: one can observe directly the hidden pattern/structure of the distribution of the zeta zeros.

The values of the zeta zeros are a consequence of the precise five element subdivisions fractal.

Thus, if a zero should be located off the critical line, it would mean that all of the values of the previous zeros would have be modified as well, all the way to the first zero which is 14.134725.

From a quantum physical perspective, if a zero is located off the critical line, then the sound wave would no longer be aligned with the 1/2 line, and its path would be modified by an lateral angle, meaning that it would never reach again the 1/2 line. Since there are two counter-propagating waves within the same sacred 63.6363... segment, there would have to be two zeros off the critical line. That is, the second sound wave would be disrupted as well.

s = rθ

θ = 5.34, 1.361, 0.8, 0.534, 0.267

r = 68.05 (136.1/2)

s = 363.4, 92.6, 54.44, 36.36, 18.17.

92.6 - 54.44 = 38.16 = 60 sacred cubits



As our drawing clearly shows, not only the pyramid's envelope but also everything inside
it was determined with the aid of three equal circles. Theodolitic equipment placed within shaft D beamed upward a key vertical line whose function we shall soon describe. But first this equipment beamed out the horizontal rock/masonry line, on which the centers of the three circles were placed. The first of these (Point 1) was at D; Points 2 and 3, where its circle intersected the line, served as centers for the other two, overlapping circles. To draw these circles the pyramid's architects, of course, had to decide on the proper radius.

Our own calculations show that the radius adopted for the three circles envisioned by us was equal to 60 such Sacred Cubits; the number 60 being, not accidentally, the base number of the Sumerian sexagesimal mathematical system. This measure of 60 Sacred Cubits is dominant in the lengths and heights of the pyramid's inner structure as well as in the dimensions of its base.


« Last Edit: August 06, 2018, 07:04:12 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #561 on: August 06, 2018, 07:02:40 AM »
EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA

"These zeros act like telephone poles, and the special nature of Riemann’s zeta function dictates precisely how the wire — its graph — must be strung between them.”"

D. Rockmore

"The lack of a proof of the Riemann hypothesis doesn't just mean we don't know all the zeros are on the line x = 1/2 , it means that despite all the zeros we know of lying neatly and precisely smack bang on the line x = 1/2 , no one knows why any of them do, for if we had a definitive reason why the first zero 1/2 + 14.13472514 i has real value precisely 1/2 we would have a reason to know why they all do. Neither do we know why the imaginary parts have the values they do.

Answers to such questions depend on a much more detailed knowledge of the distribution of zeros of the zeta function than is given by the RH. Relatively little work has been devoted to the precise distribution of the zeros."

C. King

All of the zeros of the zeta function on the critical line are linked by the five element subdivisions.

The average spacing formula:

2π/ln(t/2π)

The five element subdivisions algorithm provides the exact spacing intervals.

21.022

14.134725 + 6.3636 = 20.4975

14.134725 + 6.3636 + 0.80886 = 21.30656

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 = 22.29945

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 - 2.0757 = 20.22945

63.6363 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 = 8.1645

63.6363 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 2.7834 - 2.0757 = 6.0888

2π/ln(14.134725/2π) = 7.7498, the average spacing

By contrast, the five element subdivisions algorithm produces the exact spacing intervals effortlessly:

6.3636 and 7.17246

6.0888 and 8.1645

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2082278#msg2082278 (these upper and lower bounds are the precise starting points for the further subdivision of these intervals according to the five elements sequence of proportions, leading directly to the value of the zeta zero, to the nth decimal precision)


25.0108

14.134725 + 9.5445 + 1.68632 = 25.36602

14.134725 + 9.5445 + 0.99492 = 24.67

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 = 25.099425

(14.134725 + 63.6363) - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 1.64 = 23.459

63.6363 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 = 10.948

63.6363 - 16.1773 - 12.066 - 8.998 - 6.7106 - 5.0045 - 3.732 - 1.64 = 9.308

2π/ln(21.022/2π) = 5.2026

21.022 - 14.134725 = 6.8873

By contrast, the five element subdivisions algorithm produces the exact spacing intervals, using only the elementary four operations of basic mathematics:

10.54 and 11.23

9.308 and 10.948

Substracting 6.8873 from each of these values:

4.077

2.437

4.344

3.648

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2082601#msg2082601


By subdividing the sacred cubit distances (63.6363...) using the five elements sequence of proportions fractal, the precise intervals (exact spacing) of the location of the zeta zeros are obtained without employing transcendental or algebraic functions, or cumbersome sums/complex form remainders.

The five element subdivisions algorithm is superior to the Riemann-Siegel formula, and reveals the hidden structure/pattern of the distribution of the zeta zeros.

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301

« Last Edit: August 06, 2018, 08:33:02 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #562 on: August 13, 2018, 05:04:44 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION IX: NEW FEATURES/RESULTS

It is my belief that RH is a genuinely arithmetic question that likely will not succumb to methods of analysis. Number theorists are on the right track to an eventual proof of RH, but we are still lacking many of the tools.

J. Brian Conrey

"...the Riemann Hypothesis will be settled without any fundamental changes in our mathematical thoughts, namely, all tools are ready to attack it but just a penetrating idea is missing."
 
Y. Motohashi

"...there have been very few attempts at proving the Riemann hypothesis, because, simply, no one has ever had any really good idea for how to go about it."

A. Selberg

"I still think that some major new idea is needed here"

E. Bombieri

The previous derivation/calculation for the sixth zeta zero:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2083623#msg2083623

The upper bound can also be used within the same derivation to obtain new estimates.

14.134725 + 16.1773 + 7.1185 = 37.43

12.066 - 7.1185 = 4.9475

4.9475
1.2577
0.74205
0.49475
0.24735

37.43 + 0.24735 = 37.67735

0.24735
0.062886
0.0371
0.024725
0.012367

Adding the bottom four values to 37.43:

37.4929
37.467
37.455
37.442


Now, the same values will be substracted from the upper bound, 37.6773:

0.062886
0.0371
0.024735
0.012367

Obtaining:

37.6144
37.6402
37.6525
37.665


The second lower bound for the second zeta function is 37.54367.

Now, this value will be used to add the corresponding values belonging to the derivation of the first zeta function.

0.062886
0.0371
0.024735
0.012367

Adding these values (belonging to the first zeta function subdivisions) to 37.54367:

37.6066
37.581
37.568
37.556


Conversely, the upper bound of the first zeta function, 37.6773, will be used to get new estimates belonging to the second zeta function.

1.3205
0.335724
0.198075
0.13205
0.066025

Substracting the bottom four values from 37.6773:

37.3416
37.479
37.545
37.6113


The lower bound can also be used within the same derivation to obtain new estimates.

0.137649
0.03499581
0.02064735
0.0137649
0.00688245

Adding the bottom four values to 37.54367:

37.5786
37.56432
37.5574
37.5505


The new estimates are: 37.545 as a lower bound, 37.6113 as an upper bound.

Then, 37.6066 becomes the new upper bound, while 37.581 is the new lower bound.

By observation, 37.600447 (the value previously calculated) becomes the new upper bound of the entire approximation.

Then, 37.5904, the value from the first zeta function subdivision is the new upper bound, while 37.58526 is the new lower bound.


Thus, these new features/results greatly simplify the entire sequence of five elements subdivisions estimates: now one can also add/substract the upper/lower bounds as needed, and use an estimate from the first zeta function (or from the second zeta function) as an upper/lower bound starting point value to use in the subdivisions calculations for the second zeta function (or for the first zeta function).


https://medium.com/@JorgenVeisdal/the-riemann-hypothesis-explained-fa01c1f75d3f

"Present an argument or formula which (even barely) predicts what the next prime number will be (in any given sequence of numbers)."

The relationship between log p and the values of the zeta zeros:

http://www.dam.brown.edu/people/mumford/blog/2014/RiemannZeta.html

The log-prime figures give oscillating terms whose discrete frequencies correspond to the true zeros of the zeta function. And this method can be extended to large primes.

Since we now know that the five element subdivision algorithm creates the actual zeta zeros values, then these values can be anticipated in a very precise fashion, thus making possible the prediction of the next prime number.


The year: 1972. The scene: Afternoon tea in Fuld Hall at the Institute for Advanced Study. The camera pans around the Common Room, passing by several Princetonians in tweeds and corduroys, then zooms in on Hugh Montgomery, boyish Midwestern number theorist with sideburns. He has just been introduced to Freeman Dyson, dapper British physicist.

Dyson: So tell me, Montgomery, what have you been up to?
Montgomery: Well, lately I've been looking into the distribution of the zeros of the Riemann zeta function.
Dyson: Yes? And?
Montgomery: It seems the two-point correlations go as.... (turning to write on a nearby blackboard):



Dyson: Extraordinary! Do you realize that's the pair-correlation function for the eigenvalues of a random Hermitian matrix? It's also a model of the energy levels in a heavy nucleus—say U-238.


""Finding this system could be the discovery of the century," [Berry] says. It would become a model system for describing chaotic systems in the same way that the simple harmonic oscillator is used as a model for all kinds of complicated oscillators. It could play a fundamental role in describing all kinds of chaos. The search for this model system could be the holy grail of chaos. Until [it is found] we cannot be sure of its properties, but Berry believes the system is likely to be rather simple, and expects it to lead to totally new physics. It is a tantalising thought."

"A variety of evidence suggests that underlying Riemann's zeta function is some unknown classical, mechanical system whose trajectories are chaotic and without [time-reversal] symmetry, with the property that, when quantised, its allowed energies are the Riemann zeros. These connections between the seemingly disparate worlds of quantum mechanics and number theory are tantalising."


https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (precise description of how the two zeta functions counter-propagate within a boson)


The five element subdivision algorithm is a global formula: the exact spacing intervals are obtained very fast, without actually having to calculate the zeta zeros values (which can be derived easily using the same subdivision algorithm), for each 63.6363... segment.



*

sandokhan

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Re: Advanced Flat Earth Theory
« Reply #563 on: August 14, 2018, 02:13:27 AM »
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 1036-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 10316 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 10316 or 1010,000 (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 1036th zero:

NEW COMPUTATIONS OF THE RIEMANN ZETA FUNCTION ON THE CRITICAL LINE

https://arxiv.org/pdf/1607.00709.pdf

As 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 1036th 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.

« Last Edit: August 14, 2018, 03:04:10 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #564 on: August 18, 2018, 03:19:51 AM »
EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA III

The number of zeta zeros whose imaginary part is less than T:

N(T) = T/2π (logT/2π - 1) + 7/8 + o(1) + Nosc(T)

Nosc(T) = S(T) = 1/π Im log ζ(1/2 + iT), the oscillatory part of the formula

<N(T)> = N(T) - Nosc(T)

Nosc(T) = S(T) is a manifestation of the apparent randomness of the actual location of the zeros.



Graphs of <N(T)> and S(T): every jump in the graph of N(T) occurs at a zero crossing of Z(t). While <N(T)> is a smooth function, S(T) oscillates at the zero crossings.

http://empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/berry-keating1.pdf

Nosc(T) = S(T) = -1/π ΣpΣm=1[exp(-mlogp/2)/m]sin(tmlogp), p = prime numbers, t = zeta zero

S(T) can be calculated exactly:

https://web.viu.ca/pughg/RiemannZeta/RiemannZetaLong.html

https://logic.pdmi.ras.ru/~yumat/talks/manchester_2012/TandNT_p.pdf

https://pdfs.semanticscholar.org/8506/b09a2cf8eb4e9cd84a88464287c23f606f4b.pdf (pg. 11-13)


No one else has observed that the <N(T)> formula is directly connected to one of the five element subdivisions values for each zeta zero, and this property can hold even for values of t > 280, independent of the values of S(t) = Nosc (as t → ∞, |S(T)| < 1 for T < 280, |S(T)| < 2 for T < 6800000, and the largest value of |S(T)| found so far is around 3.2).

For the zeta zero 21.022:

<N(21.022)> = T/2π (logT/2π - 1) + 7/8 = 1.57

<N(22.3)> = 1.822

<N(23.43)> = 2.054

The average number of zeta zeros approaches the value of an integer exactly at the five element subdivision figures.

<N(25.0108)> = 2.3933

<N(26.61)> = 2.753

<N(27.34)> = 2.922

<N(28.07)> = 3.095


<N(30.424)> = 3.671

<N(31.8494)> = 4.0337


<N(32.935)> = 4.317

<N(35.058)> = 4.8875

<N(36.58)> = 5.3091


For the first Lehmer pair:

<N(111.3)> = 33.954
<N(111.87)> = 34.338

<N(111.317)> = 34.085

For the second Lehmer pair:

<N(150.05)> = 52.771
<N(150.92)> = 53.21

<N(149.55)> = 52.5186
<N(150.94)> = 53.221

This means a pair of zeta zeros whose values are very close can be detected by computing the five element subdivision figure closest to 53.

It also means that a recurrence formula is possible.

More information on S(T):

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2007826#msg2007826

S(t) is associated with 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.

It is very possible that S(T) might be related to a different five element subdivision sequence.

53.4
106.8
136.1
160
534

63.63
19.091
16.1773
12.7272
6.363

44.5453
13.363
11.3252
8.91
4.454
« Last Edit: August 25, 2018, 04:30:43 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #565 on: August 20, 2018, 12:32:12 AM »
RADIUS OF THE SUN PARADOX II

Does the Sun have a surface? Dr. P.M. Robitaille:




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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #566 on: August 23, 2018, 01:57:55 AM »
EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA IV



"It was all very fascinating seeing the same pictures cropping up in both  areas,  but who could  point to  some  genuine  contribution  to  prime  number  theory  that  these  connections  had  made possible?  Peter Sarnak  offered  the  quantum  physicists  a  challenge:  use  the  analogy  between quantum  chaos  and  prime  numbers  to  tell  us  something  we  don't  already  know  about Riemann's landscape - something specific that couldn't be hidden behind statistics.

There  are  certain  attributes  of  the  Riemann  zeta  function,  called  its  moments,
which  it  was  known  should  give  rise  to  a  sequence  of  numbers.  The  trouble  was  that mathematicians  had very  little  clue  as  to  how  to  calculate  the sequence  itself.

Before  the  Seattle  meeting,  Conrey  had  done  a  huge  amount  of  work  on  the  problem  of the  next  number  in collaboration  with  a  colleague, Amit  Ghosh,  which  suggested  that  the third number in the sequence (after 1 and 2) was a big jump away, at 42. For Conrey, that this should be the number next in the sequence  'was  kind  of  surprising'.

In  the  meantime,  Conrey  had  joined  forces  with  another  mathematician,  Steve  Gonek. With a  huge  effort,  squeezing all  they  could  from  their  knowledge  of  number  theory,  they came  up  with  a  guess  for  the  fourth  number  in  the  sequence  -  24,024.  'So  we  had  this sequence: 1, 2, 42, 24,024, . . . We tried like the Dickens to guess what the sequence was. We knew our method couldn't go any further because it was giving a negative answer for the next number in the sequence.' It was known that all the numbers in the sequence were bigger  than  zero.  Conrey  arrived  at  Vienna  prepared  to  talk  about  why  they  thought  the next number in the sequence was 24,024.

'Keating  arrived  a  little  late.  On  the  afternoon  he  was  going  to  give  his  lecture  I  saw  him, and  I'd  seen  his  title  and  I  had  begun  to  wonder  whether  he  had  got  it.  As  soon  as  he showed  up  I  went  and  immediately  asked  him,  "Did  you  figure  it  out?"  He  said  yes,  he'd got the 42.' In fact, with his graduate student, Nina Snaith, Keating had created a formula that  would  generate  every  number  in  the  sequence.  'Then  I  told  him  about  the  24,024.' This  was  the  real  test.  Would  Keating  and  Snaith's  formula  match  Conrey  and  Gonek's guess of 24,024? After all, Keating had known that he was meant to be getting 42, so he might  have  cooked  his  formula  to  get  this  number.  This  new  number,  24,024,  was completely new to Keating and not one he could fake."

(from Music of the Primes)

The challenge for the quantum physicists then, was to use their quantum methods to check the number 42 and to calculate further moments in the series, while the number theorists tried to do the same using their methods.

Prof Jon Keating and Dr Nina Snaith at Bristol describe the energy levels in quantum systems using random matrix theory. Using RMT methods they produced a formula for calculating all of the moments of the Riemann zeta function. This formula confirmed the number 42.

Two years after Seattle, Keating and Snaith attended a follow-up conference at the Schrodinger Institute in Vienna to present their formula. Meanwhile, number theorists Conrey and Gonek had suggested the next moment in the series.

When Keating and Snaith's formula was used to calculate this moment, it coincided with the number theorists' suggestion: 24,024. The formula really works.

Usually pure mathematics supports physics, supplying the mathematical tools with which physical systems are analysed, but this is a case of the reverse: quantum physics is leading to new insights into number theory.

http://www.bristol.ac.uk/maths/research/highlights/riemann-hypothesis/

24024/286 = 84 = 42 x 2

84 x 1sc = 53.4

(286.1 is the displacement factor of the Gizeh pyramid)


https://arxiv.org/pdf/math/0602270.pdf

On the spacing distribution of the Riemann zeros: corrections to the asymptotic result

We give here arguments indicating that to leading order these deviations are the same as those of unitary random matrices of finite dimension Neff = log(E/2π)/√12Λ, where Λ = 1.57314 . . . is a well defined constant.

1.57314 = 1/sc

12/sc = 1.4134725 x 40/3


https://arxiv.org/pdf/1607.00709.pdf

New computations of the Riemann zeta function

Contains graphics of Z(t) from t = 238 to t = =88837796029624663862630219091105, also diagrams of S(t) for very large values of t.


Riemann's nachlass = manuscripts, lecture notes, calculation sheets and letters left by G.F.B Riemann

https://www.researchgate.net/publication/281403728_To_unveil_the_truth_of_the_zeta_function_in_Riemann_Nachlass

The authors assert that not all of the formulas left by Riemann in his notes have been taken into consideration, and that these neglected equations were used by Riemann to actually prove the RH.


Z(t) and S(t) for very large values of the zeta function:





Currently, the values of the zeta zeros are thought to be totally random:

http://math.sun.ac.za/wp-content/uploads/2011/03/Bruce-Bartlett-Random-matrices-and-the-Riemann-zeros.pdf

https://pdfs.semanticscholar.org/fc82/c1f7e35f23eb1695b0c78830c366e1258c88.pdf

One of the best mathematicians in the world, Dr. Yuri Matiyasevich (who solved Hilbert's tenth problem), dared to touch this red line and was refused publication for his results which prove that there is definite relationship between the values of the zeta zeros:

https://www.researchgate.net/publication/265478581_An_artless_method_for_calculating_approximate_values_of_zeros_of_Riemann's_zeta_function

https://phys.org/news/2012-11-supercomputing-superproblem-journey-pure-mathematics.html


As has been proven in this thread, the zeta zeros are generated by the five element subdivision algorithm.

This means that the values of the energy levels of all of the atoms are not random numbers: this is the reason why modern mathematics has preferred to concentrate on the Riemann hypothesis and not on research on the actual values of the zeta zeros. The question arises: how could an unconscious nature have known, well ahead of the big bang event, how to subdivide an interval into a five element sequence fractal?

http://wwwf.imperial.ac.uk/~hjjens/Riemann_talk.pdf

Subtle relations: prime numbers, complex functions, energy levels and Riemann

Prof. Henrik J. Jensen, Department of Mathematics, Imperial College London

http://www.ejtp.com/articles/ejtpv10i28p111.pdf

Riemann Zeta Function and Hydrogen Spectrum


In the previous message in this series, it was shown that the average number of zeta zeros formula nearest to an integer is directly related to the values of the five element subdivision figures closest to the respective zeta zero.

It is also assumed by modern mathematics that the fluctuations (oscillatory nature) of S(t) are totally random.

It is my belief that S(t) also is directly related to the five element subdivision figures.

The oscillatory nature could be described as follows: S(t) is a ratio.

6.3636 and 7.17246

6.0888 and 8.1645

6.3636 + 7.17246 + 6.0888 + 8.1645/(4 x 8.1645) = 0.851, a value very close to the known figures of S(t):  |S(T)| < 1 for T < 280, |S(T)| < 2 for T < 6800000, and the largest value of |S(T)| found so far is around 3.2; however, G. França and A. LeClair offer very interesting arguments to the fact that actually -1 < S(T) < 1:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2104086#msg2104086

10.54 and 11.23

9.308 and 10.948

10.54 + 11.23 + 9.308 + 10.948/(4 x 11.23) = 0.935

The random oscillatory nature of S(t) is generated by the four values of the five element subdivision algorithm. Of course, a general formula has to include the very slow increase of S(t), a sign function, but its most important feature, the apparent random oscillatory attribute, is definitely related to the values of the subdivision algorithm.

That is, <N(t)> (average number of zeta zeros) + S(t) (oscillatory part) = N(t) (total number of zeta zeros formula) = one of the values of the five element subdivision algorithm close to the respective zeta zero (which is different than the five element partition figure which is closest to an integer in the average number of zeta zeros formula). Then, since <N(t)> is easily calculated, we can find out the value of S(t) by a simple substraction.

Still, the fact that <N(t)> is very close to the value of an integer exactly at one of the four subdivision points of the five element algorithm, means that there is precise criterion by which to evaluate the multiple five element partition figures.

« Last Edit: November 04, 2018, 02:08:49 PM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #567 on: August 24, 2018, 12:31:01 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION IX

z6 = 37.586

L(z6) = 3.5126

41.098

The seventh zeta zero, to three significant digits accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (basic subdivision of the first 63.63636 sacred cubit interval into five elements ratios)

14.134725 + 16.1773 + 7.1185 = 37.43

12.066 - 7.1185 = 4.9475

4.9475
1.2577
0.74205
0.49475
0.24735

37.43 + 1.2577 = 38.6877

4.9475 - 1.2577 = 3.6898

3.6898
0.9381
0.55347
0.36898
0.1845

Adding the bottom four values to 38.6877:

39.6258
39.24117
39.05668
38.8722


5.309

44.2

8.998

40.52

40.52 is the first lower bound.

44.2 is the first upper bound (even though 44.2 is greater than the eighth zeta zero, 43.327)


Now, the new features/results from the previous message on this page will be used.

8.998 - 5.309 = 3.689

3.689
0.9379
0.55335
0.3689
0.18445

Adding the bottom four values to 40.52:

41.458
41.073
40.889
40.704

40.889 is the new lower bound.

41.073 is the new upper bound.

38.6877 + 3.6898 = 42.378

1.2577
0.74205
0.4947
0.24735

Substracting 1.2577 from 42.378 we obtain 41.1203.

3.6898
0.9381
0.55347
0.36898
0.1845

Substracting the bottom four values from 41.1203:

40.1823
40.567
40.7513
40.9358

Since 40.9358 is a smaller value than 41.073, 40.9358 is the new upper bound of the entire approximation.

The true value for the seventh zeta zero is: 40.918719012

Already we have obtained a three significant digit approximation:

40.9358


A more difficult approach, without using the new features/results, would be use the five element subdivision algorithm, starting with 44.2 for the second zeta function (44.2, 43.262, 42.5632, 42.04172, 41.6526, 41.3624, 40.5112 and 41.146, 40.9846, 40.9136 and 40.93726) and continuing with the value of 38.6877 for the first zeta function (39.6258, 40.3254, 40.847126 and 40.9236, 40.86658, 40.8811, 40.892, 40.90007).

Once the four subdivision figures are obtained, there is no need to even bother to find the value of the corresponding zeta zero: all that matters are the five element subdivision points, then the zeta zero can be computed effortlessly if so desired.

« Last Edit: August 28, 2018, 12:57:06 AM by sandokhan »

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #568 on: August 27, 2018, 12:15:46 AM »
RADIUS OF THE SUN PARADOX III

“Young people, especially young women, often ask me
for advice. Here it is, valeat quantum. Do not undertake
a scientific career in quest of fame or money. There
are easier and better ways to reach them. Undertake it
only if nothing else will satisfy you; for nothing else is
probably what you will receive. Your reward will be
the widening of the horizon as you climb. And if you
achieve that reward you will ask no other.”

Cecilia Payne-Gaposchkin


http://vixra.org/pdf/1310.0134v1.pdf

Commentary Relative to the Distribution of Gamma-Ray Flares on the Sun:
Further Evidence for a Distinct Solar Surface

http://vixra.org/pdf/1310.0108v1.pdf

The Solar Photosphere: Evidence for Condensed Matter

http://vixra.org/pdf/1310.0110v1.pdf

Forty Lines of Evidence for Condensed Matter — The Sun

Dr. P.M. Robitaille, Department of Radiology, The Ohio State University

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sandokhan

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Re: Advanced Flat Earth Theory
« Reply #569 on: August 28, 2018, 12:54:59 AM »
SACRED CUBIT EXACT SPACING FORMULA FOR THE ZETA FUNCTION X

z7 = 40.9187

L(z7) = 3.353

44.272

The eighth zeta zero, to three significant digits accuracy, using only the five elements subdivision applied to both zeta functions as a guide.

63.636363
16.1773
9.5445
6.36363
3.1815

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (basic subdivision of the first 63.63636 sacred cubit interval into five elements ratios)

14.134725 + 16.1773 + 12.066 = 42.378

14.134725 + 16.1773 + 12.066 + 1.77 = 44.148


5.309

44.2

8.998

40.51


1.77
0.45
0.2655
0.177
0.0885

Adding the bottom four values to 42.378 (which is the first lower bound):

42.83
42.64
42.555
42.46


Substracting the bottom four values from 44.148 (the first upper bound), thus using the new results/features posted on this page:

43.7
43.88
43.97
44.06


For the second zeta function:

8.998 - 5.309 = 3.689

3.689
0.9379
0.55335
0.3689
0.18445

Substracting the bottom four values from 44.2:

43.262
43.646
43.8311
44.015

Adding the bottom four values to 42.378:

43.316
42.93
42.747
42.562

42.83 is the new lower bound; since 43.316 has a greater value than 42.83, 43.316 is the new lower bound for the entire approximation.

43.7 is the new upper bound.

At this point, a three digit approximation has already been obtained (true value of the eighth zeta zero is 43.327073281); however, the five element subdivision algorithm will be continued, in order to show the precise calculations.


Since 43.262 is the lower bound for the second zeta function (while 43.646 is the upper bound), we already know that the true value of the eighth zeta zero is to be found in the 0.9379 - 0.55335 interval.


3.689 - 0.9379 = 2.7511

2.7511
0.699
0.4126
0.27511
0.1375

Substracting the bottom four values from 43.262:

42.563
42.85
42.987
43.124

Adding the bottom four values to 43.316:

44.015
43.73
43.6
43.45


0.9379 - 0.55335 = 0.38455

0.38455
0.0977
0.0577
0.038455
0.01923

43.646 - 0.0977 = 43.55

0.38455 - 0.0977 = 0.28685

0.28685
0.07293

43.55 - 0.07293 = 43.477

0.28685 - 0.07293 = 0.21392

0.21392
0.0544

43.477 - 0.0544 = 43.42

0.21392 - 0.0544 = 0.15952

0.15952
0.0405

43.42 - 0.0405 = 43.38

0.15952 - 0.0405 = 0.11902

0.11902
0.03026

43.38 - 0.03026 = 43.35

0.11902 - 0.03026 = 0.08876

0.08876
0.022566

43.35 - 0.022566 = 43.327 (a five digit approximation)


Returning to the calculations for the first zeta function.

1.77 - 0.45 = 1.32

1.32
0.3356
0.198
0.132
0.066

Adding the bottom four values to 42.83:

43.1656
43.028
42.962
42.896

Substracting the bottom four values from 43.7:

43.364
43.502
43.568
43.634

1.32 - 0.3356 = 0.9844

0.9844
0.2503
0.14706
0.09844
0.04922

Adding the bottom four values to 43.1656:

43.416
43.315
43.264
43.215

Substracting the bottom four values from 43.364:

43.114
43.216
43.26
43.315

0.9844 - 0.2503 = 0.7341

0.7341
0.1866
0.11
0.07341
0.0367

Adding the bottom four values to 43.416:

43.6
43.526
43.49
43.45


Returning to the calculations for the second zeta function:

0.1375
0.034958
0.0206
0.01375
0.006875

Adding the bottom four values to 43.316:

43.351
43.2826
43.276
43.268


Successively, the new upper bounds are: 43.634, 43.6, 43.568, 43.526, 43.502, 43.45, 43.416, 43.364.

The new lower bounds are: 42.83, 42.93, 42.987, 43.028, 43.215, 43.26, 43.316.

The true value for the eighth zeta zero is: 43.327073281

Already we have obtained a three significant digit approximation:

43.316

« Last Edit: August 28, 2018, 09:55:52 AM by sandokhan »