Advanced Flat Earth Theory

TWO ZETA FUNCTIONS SOUND WAVES = MASS OF A BOSON



“What we call mass would seem to be nothing but an appearance, and all inertia to be of electromagnetic origin”

Henri Poincare

“Light cannot be anything else but a longitudinal disturbance in the ether, involving alternate compressions and rarefactions. In other words, light can be nothing else than a sound wave in the ether”

“It being a fact that radio waves are essentially like sound waves in the air"

Nikola Tesla

"The limiting velocity is c, but a limit has two sides"

Gerald Feinberg

“If a special geometry has to be invented in order to account for a falling apple, even Newton might be appalled at the complications which would ensue when really difficult problems are tackled”

"If we could understand the structure of the particle, in terms of the medium of which it is composed, and if we knew the structure of the rest of the medium also, so as to account for the potential stress at every point—that would be a splendid step, beyond anything accomplished yet”

Oliver Lodge

“We are about to enter the 21st century but our understanding of the origin of inertia, mass, and gravitation still remains what has been for centuries – an outstanding puzzle”

Vesselin Petkov

“The more we study gravitation, the more there grows upon us the feeling that there is something peculiarly fundamental about this phenomenon to a degree that is unequalled among other natural phenomena. Its independence of the factors that affect other phenomena and its dependence only upon mass and distance suggest that its roots avoid things superficial and go down deep into the unseen, to the very essence of matter and space”

Paul Heyl

”Mass is a very important property of matter, and we have nothing in our current theory that says even a word about it”

Claude Detraz, one of the two research directors at CERN

"Instead of asking himself what caused the apple to fall to the ground, Sir Isaac Newton should have asked how it got up there in the first place! What else if not levitation enables a tree to grow upwards against the action of gravity?"

Viktor Schauberger


https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2006301#msg2006301 (sacred cubit configuration of a boson)

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1774536#msg1774536 (journey inside a boson)

Yang = sound = ether

Yin = silence = aether

The interplay of yin and yang inside a boson takes place in conformity with the five elements subdivision of the sacred cubit distance which generate the two zeta function sound waves.

http://www.subtleenergies.com/ormus/oc/chaptr01.htm

The sound waves of the two zeta functions create light inside a boson.

Aether = medium through which the boson strings propagate

Boson = photon = neutrino

A boson is a cavity resonator.

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1998110#msg1998110 (longitudinal boson strings within transverse subquark waves)

Chris Hill, theorist at Fermilab, indicated the view in “New Scientist”, 11 May 1996, page 29, “It would suggest that whatever lies inside the quarks is incredibly tightly bound, in a way that theory can’t yet accommodate.”


The sound waves inside a boson (light) create the mass of the boson itself: mass is generated by the Riemann zeta function waves.

Whittaker scalar subquark potential waves:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1994059#msg1994059


https://arxiv.org/pdf/physics/0012025.pdf

Did 20th century physics have the means to reveal the nature of inertia and gravitation?

EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA V

<N(t)> is very close to the value of an integer exactly at one of the four subdivision points of the five element algorithm.

This fact is of great aid in finding the "regular" zeta zeros values.

The ninth zeta zero is 48.005150881.

The four initial five element partition figures:

3.54

45.92

5.309

47.687


3.54

45.98

1.77

47.75

12.066

49.52

Since <N(45.8 )> ~= 8.0 and <N(48.7)> = 8.996, this means that 47.75 is the first lower bound of the entire approximation (and not the upper bound).

3.689
0.9379
0.55335
03689
0.18445

47.687 + 0.9379 = 48.625

<N(48.625)> = 8.972


For Lehmer pairs, the calculations are more involved (corresponding <N(t)> values in the parentheses).

7005.0629 and 7005.10056

7004.0437 (6707.487)
7005.0629 (6708.626)
7005.10056 (6708.667)
7006.74 (6710.498)

That is, the Lehmer pair will be located between the average number of zeta zeros values of 6708 and 6709.

Sacred cubit interval: 63.6363636363...

6999.999
7063.6363

For the first zeta function the values are:

7003.1814
7003.9903
7004.593
7005.0435
7005.17544
7005.379

The calculations for the second zeta function:

7004.541
7005.175
7005.48

Since 7005.175 and 7005.17544 are very close figures, a Lehmer pair must be located in the vicinity of these values.


Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive: the nearest values are 7004.6825 and 7004.689, 7005.2652 and 7005.23.


63137.2115 and 63137.2324

63136.537 (82551.023)
63137.2115 (82552.013)
63137.2324 (82552.0434)
63138.2238 (82553.4973)

Sacred cubit interval: 63.6363636363...

63127.21
63190.846

For the first zeta function the values are:

63136.755
63137.42
63137.0866

(+9.5445, +0.66328, +0.33164)

The calculations for the second zeta function lead to these values:

63138.1574
63137.61
63137.063

(...-3.7322, -0.5474, -1.09478)

Since 63137.0866 and 63137.063 are very close figures, a Lehmer pair must be located in the vicinity of these values.

Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive; the nearest values are: 63137.32 and 63137.185.


71732.9012 and 71732.91591

71732.02 (95246.674)
71732.9012 (95247.984)
71732.91591 (95248.00608)
71734.097 (95249.76)

Sacred cubit interval: 63.6363636363...

71718.11
71781.75

For the first zeta function the values are:

71734.287
71727.655
71729.341
71730.6
71731.536
71732.235
71732.757

The calculations for the second zeta function:

71732.7936
71732.06
71731.326
71730.6

Two pairs of zeta zeros which are very close: 71730.6 and 71732.757 and 71732.7936.

To distinguish between these choices the second five element subdivision algorithm will be applied.

53.4
106.8
136.1
160
534

63.636363
19.091
16.1773
12.7272
6.363

63.63 - 19.091 = 44.5453

44.5453
13.363
11.3252
8.91
4.454

44.5453 - 13.363 = 31.1823

31.1823
9.3547
7.9278
6.23646
3.11823

31.1823 - 9.3547 = 21.8276

21.8276
6.5483
5.5494
4.36552
2.18276

21.8276 - 6.5483 = 15.2793

15.2793
4.5838
3.88461
3.05586
1.52793

For the first zeta function, the values are:

71737.201
71730.8372
71731.8722
71734.2873
71732.6
71732.938
71732.77

The calculations for the second zeta function:

71733.4
71731.865
71730.337
71732.935
71732.663
71732.721

Since 71730.8372 and 71730.337 are not as close to one another as the corresponding pair using the first five element subdivision algorithm, it means we are not dealing with a Lehmer pair; the same analysis applies to 71731.8722 and 71731.865 (the corresponding pair using the five element subdivision algorithm are not this close to one another).

Amazingly, the two five element subdivision algorithms have located the precise interval of the Lehmer pair: 71732.757 and 71732.7936, 71732.938 and 71732.935.

The actual values are: 71732.9012 and 71732.91591.

Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive; the nearest values are: 71732.9171 and 71732.909, after a long series of calculations (more involved than using 63.6363636 as a sacred cubit interval).


220538.853 and 220538.8702

220537.0585 (332251.37)
220537.4266 (332251.98)
220538.853 (332254.36)
220538.8702 (332254.39)
220539.8528 (332256.0258)

220538.853 is the 332254th zero, 220538.8702 is the 332255th zero, where the average spacing is 0.6.

Sacred cubit interval: 63.6363636363...

220499.78
220563.416

For the first zeta function the values are:

220537.0213
220538.341
220538.926
220538.676
220538.824

The calculations for the second zeta function:

220537.925
220538.863


Since 220538.824 and 220538.863 are very close figures, a Lehmer pair must be located in the vicinity of these values. 

Using the second five element subdivision algorithm, the following results are obtained:

220538.471
220538.901
220538.81
220538.64

220535.415
220539.871
220538.534
220538.738
220538.98

Again, the Lehmer pair must be located very close to the value of 220538.9.

Using 200/π = 63.66197724 as a sacred cubit interval the results are not as impressive: the nearest values are: 220538.8085 and 220538.8266.


435852.8393 and 435852.8572

435851.967 (703890.467)
435852.8393 (703892.015)
435852.8572 (703892.046)
435853.455 (703893.107)

Sacred cubit interval: 63.6363636363...

435845.45
435909.0865

435851.814
435852.623
435852.743

435853.6145
435852.7981

The Lehmer pair must be located around the value of 435852.78.


555136.9163 and 555136.9315

555136.284 (917905.02)
555136.9163 (917906.17)
555136.9315 (917906.195)
555137.412 (917907.066)

Sacred cubit interval: 63.63636363...

555099.9944
555163.631

555133.547
555137.2357
555136.385
555136.6
555136.763
555136.8831

555135.3877
555140.3352
555137.44
555136.92
555136.767


773657.1461 and 773657.1559

773656.6413
773657.1461
773657.1559
773658.041

Sacred cubit interval: 63.63636363

773627.265
773.690.9014

773655.51
773.657.278
773657.3665

773657.35


947107.8201 and 947107.8325

947107.2485
947107.8201
947107.8325
947108.2566

Sacred cubit interval: 63.63636363

947099.9905
947163.627

947106.354
947107.163
947107.766

947108.155
947107.7468

Both sets of five element subdivision algorithms are needed to detect the Lehmer pairs, the zeta zeros values which are most difficult to find.

EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA VI

http://www.dtc.umn.edu/~odlyzko/doc/zeta.derivative.pdf

1.30664344087942265202071895041619 x 10²²

1.30664344087942265202071898265199 x 10²²

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

Using a very large number calculator:

205329683587299385104.8399385104 = 13066434408794226520207/63.63636363

12935770065999861261552 =  205329683587299385104 x 63

130664342794365258601.54936752 = 205329683587299385104 x .63636363
 
53.45063194549 = 0.8399835 x 63.63636363


13,066,434,408,794,226,520,153.54936752
13,066,434,408,794,226,520,217.18573115

The calculations for the first zeta function:

13,066,434,408,794,226,520,169.72670085
(+16.17733333)

13,066,434,408,794,226,520,181.79268471
(+12.06598386)

13,066,434,408,794,226,520,190.791012836
(+8.998328126)

13,066,434,408,794,226,520,197.501606019
(+6.710593183)

13,066,434,408,794,226,520,202.506097991
(+5.004491972)

13,066,434,408,794,226,520,206.238247924
(+3.732149933)

13,066,434,408,794,226,520,207.332996246
(+1.094748322)

13,066,434,408,794,226,520,206.785622085
(+0.547374161)

13,066,434,408,794,226,520,206.924786491
(+0.139164406)

13,066,434,408,794,226,520,207.028569738
(+0.103783247)

13,066,434,408,794,226,520,207.105967138
(+0.0773974)

13,066,434,408,794,226,520,207.163687018
(+0.05771988)

13,066,434,408,794,226,520,207.189083398
(+0.02539638)


The computations for the second zeta function:

13,066,434,408,794,226,520,207.64027661
(-9.54545454)

13,066,434,408,794,226,520,207.308682645
(-0.331593965)

13,066,434,408,794,226,520,207.224378195
(-0.08430445)

0.247289515 interval

13,066,434,408,794,226,520,207.187284768
(-0.037093427)


The second five element subdivision algorithm:

13,066,434,408,794,226,520,172.64027661
(+19.090909)

13,066,434,408,794,226,520,186.00391297
(+13.363636)

13,066,434,408,794,226,520,195.35845842
(+9.35454545)

13,066,434,408,794,226,520,201.906640238
(+6.548181818)

13,066,434,408,794,226,520,206.490367508
(+4.58372727)

1.06953636
0.320860909
0.271919
0.21390727
0.106953636

13,066,434,408,794,226,520,206.811228417
(+0.320860909)

13,066,434,408,794,226,520,207.035831017
(+0.2246026)

13,066,434,408,794,226,520,207.193052861
(+0.1572218844)


13,066,434,408,794,226,520,210.82209485
(-6.363636)

13,066,434,408,794,226,520,208.91300395
(-1.9090909)

13,066,434,408,794,226,520,207.576640314
(-1.336363636)

3.11818181

13,066,434,408,794,226,520,207.264822133
(-0.3118181)

13,066,434,408,794,226,520,207.202458497
(-0.062363636)

0.07927665
0.0623636

0.01691303
0.00507391

13,066,434,408,794,226,520,207.197384587
(-0.00507391)

0.0118391
0.003551737

13,066,434,408,794,226,520,207.19383285
(-0.0035517370

EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA VII



7954022502373.43289015387
7954022502373.43289494012

t₂ - t₁ = 4.7863 x 10^-6 = 0.0000047863

7,954,022,502,331.87047696
7,954,022,502,395.50684059

7,954,022,502,348.04781029
(+16.17733333)

7,954,022,502,360.11379415
(+12.06598386)

7,954,022,502,369.112122276
(+8.998328126)

7,954,022,502,373.071330025
(+3.959207749)

2.751385438
0.69951223
0.412707815
0.2751385438
0.13756927

7,954,022,502,373.3464685688
(+0.2751385438)

0.13756927
0.03497561

7,954,022,502,373.3814441788
(+0.03497561)

0.03497561 = 1/28.5914, where 286.1 is the displacement factor of the Gizeh pyramid

0.10259366
0.0260834

7,954,022,502,373.4075275788
(+0.0260834)

0.07651025
0.019451965

7,954,022,502,373.4269795438
(+0.019451965)

0.057058282

0.0057058282

7,954,022,502,373.432685372
(+0.0057058282)


7,954,022,502,395.50684059

7,954,022,502,379.32950729
(-16.1773333)

7,954,022,502,374.58360426
(-4.74590303)

2.372951517
0.60329919

7,954,022,502,373.98030507
(-0.60329919)

7,954,022,502,373.530388664
(-0.449916406)

1.319735916
0.131973591
0.065986795

7,954,022,502,373.464401869
(-0.065986795)

0.065986795
0.016776482

7,954,022,502,373.447625387
(-0.016776482)

0.04921031
0.01251123

7,954,022,502,373.435114157
(-0.01251123)

0.036699082

7,954,022,502,373.43327920286
(-0.00183495414)

7,954,022,502,373.43281268412
(-0.00046651874)


The second five element subdivision algorithm calculations:

7,954,022,502,350.96138605
(+19.09090909)

7,954,022,502,364.32502241
(+13.36363636)

7,954,022,502,373.67956786
(+9.35454545)


7,954,022,502,376.4159315
(-19.090909)

4.45454545
1.3363636

7,954,022,502,375.0795679
(-1.3363636)

7,954,022,502,374.1441134
(-0.9354545)

2.18272727
0.65481818
0.5549366
0.43654545
0.2182727

7,954,022,502,373.48929522
(-0.65481818)


Using the Riemann-Siegel asymptotic formula, the sum will feature at least O(4.56 x 10¹⁰) terms (for t = 1.30664344087942265202071895041619 x 10²²):



Using the five element subdivision algorithms, we only need to translate/shift the 63.63636363 interval by a factor of k: k = [t/63.6363636363] x 63.6363636363, where [ x ] denotes the integer part, and then simply apply the five element partition process for the two zeta functions to detect both regular zeta zeros and Lehmer pairs, carefully keeping a check on the average number of zeros values which are close to an integer (these are the values which are equivalent to a five element subdivision figure).

UPPER/LOWER BOUNDS FOR THE SIX GATES II



https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1726000#msg1726000 (part I)

The 28° angle measurement was obtained using the figures published by wikipedia.

http://blog.world-mysteries.com/wp-content/uploads/2015/08/AG_Giza3B.jpg

One of the readers of the article "Giza: The Time Machine" commented:

What if the 117,5 dimensions are instead 116.26? Therefore, pi*116.26 = 365.242 (the exact length of a solar year)….that is, the circumference of a circle with the diameter = 116.26, that can be formed within the square (117.5 horizontal & vertical dimensions you’ve shown above). This would also make the distance between the small pyramid apexes = 58.13 m, instead of what Wikipedia has stated.

An angle of 27.49° makes more sense: 1/27.49 = 0.036376864 = 0.1 - 0.063623.

2 x 27.49 = 54.98

54.98° = 0.959582 rad

s = r x θ

6106.4248 = 6363.63 x 0.959582

6106.4248/6 = 1017.73747 km, the distance alloted for each gate

The arclength for each gate (space alloted for the each of the six periods running from the winter solstice to the summer solstice, and from the summer solstice to the winter solstice) is 1017.737 km.

There are several possibilities relating to describing the solar precession within the context of the 1017.737 km alloted for the each gate.

Obviously, the upper bound must be 508.87 km (1017.737/2): solar precession = 1.5 km, otherwise at the end of the precessional cycle the solar orbit would intersect the space alloted for the next gate (or the space beyond the tropic of Cancer/Capricorn latitudes).

That is, we would have a mobile section (which would be used by the Sun for 30 days, the maximum interval alloted for each gate, measuring 508.87 km) which now moves westward until it reaches the limit/boundary (that is, it can only move/travel for 508.87 km; its starting point will travel 508.87 km, across a time period of 339.25 years, and its endpoint, already on the 508.87 km mark, will travel another 508.87 km to reach the outer limit/boundary, after 339.25 years, at a precession 1.5 km per year).

The lower bound has to be 254.43 km (1017.737/4): we cannot imagine a precessional cycle which would occupy less than 254.43 km (of the total 1017.737 km).

First upper bound: 339.25 years

First lower bound: 169.62 years

In the new radical chronology of history, the last major planetary cataclysm occurred at least before 1770 AD.

Then, the new lower bound is: 254.43 years (254.43 x 1.5 = 381.645 km, where 381.78 = 600 sc)

1017.73 - (2 x 381.645) = 254.447 (the displacement factor for a single gate)

The only other significant sacred cubit figure between 339.25 and 254.43 is 286.1 (the displacement factor of the Gizeh pyramid).

286.1 x 1.5 = 429.15

1017.73 - (2 x 429.15) = 159.437 (159.09 = 1000sc/4 = 250 sc)

1733 AD + 286 = 2019

1764 AD + 286 = 2050 (105 years after the end of WWII)

Likely upper bound: 286.1 years

Most probable upper bound: 254.43 years

EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA VIII: FRANÇA-LECLAIR POINTS

Transcendental equations satisfied by the individual zeros of
Riemann ζ, Dirichlet and modular L-functions

https://arxiv.org/pdf/1502.06003.pdf

Statistical and other properties of Riemann zeros based on an
explicit equation for the n-th zero on the critical line

https://arxiv.org/pdf/1307.8395.pdf

<N(T)> = T/2π(logT/2πe) + 7/8 + O(log T)

N(T) = T/2π(logT/2πe) + 7/8 + 1/π(arg ζ(1/2 + iT))+ O(1/T)

Then, a simple observation (that the right hand side of N(T) jumps by one at each zero, with values −1/2 to the left and +1/2 to the right of the zero so that one can replace n → N(T) + 1/2)) leads to this equation:

T/2π(logT/2πe) = n - 11/8

http://mathworld.wolfram.com/LambertW-Function.html

The Lambert W function is the inverse of f(W) = We^W.

Using the transformation T = 2π(n - 11/8)/x,

xe^x = (n - 11/8)/e

França-LeClair points = 2πe ⋅ e^{W[(n - 11/8)/e]}

Mathematica software for the Lambert W function:

http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ProductLog

14.5213, 20.6557, 25.4927, 29.7394, 33.6245, 37.2574, 40.7006,
43.994, 47.1651, 50.2337, 53.2144, 56.1189, 58.9563, 61.7338,
64.4577, 67.133, 69.764, 72.3544, 74.9073, 77.4257, 79.9118, 82.3678,
84.7957, 87.1972, 89.5737, 91.9268, 94.2576, 96.5674, 98.8571,
101.128, 103.38, 105.615, 107.833, 110.036, 112.223, 114.395,
116.554, 118.698, 120.83, 122.949

These points are much closer to the true zeta zeros values than the Gram points.

https://sites.google.com/site/riemannzetazeros/grampoints





(good and bad Gram points)

However, the authors of the articles have not noticed that the 11/8 value is directly related to the sacred cubit.

The 11/8 value, 1.375, is a sacred cubit average of 1.361 and 1.4134725.

1.375 - 1.361 = 0.014

1.4134725 - 1.375 = 0.0384725

0.0384725/0.014 = 2.748 = 0.3638963 = 1 - sc

That is why these estimates, the França-LeClair points, do not capture the exact values of the Lehmer pairs (110.036 and 112.223, where the first Lehmer pair is 111.03 and 111.87); however, they can be used as a first approximation in the equation which features S(t), [1/π(arg ζ(1/2 + iT))], to derive an algorithm (see the papers) for finding the zeta zeros which is easier than the Riemann-Siegel formula, but still features transcendental functions and an evalution of either integral representations of S(t) or convergent series which approximate S(t).

S(t), a sign function and k which captures the oscillatory part of S(t) (k = (f₁ + f₂ + f₃ + f₄)/4f₄, where f_n are the four five element subdivisions figures, and f₄ is the largest figure).

However, this would mean one has to find the f_n values first, which is equivalent to finding the correct location of each Lehmer pair (the most difficult zeta zeros values to be calculated).

As exemplified in the previous messages, the Lehmer pair values are connected with a combination of the two five element subdivisions algorithms, which can capture directly these figures, without the need to compute S(t).

https://arxiv.org/ftp/arxiv/papers/1510/1510.06333.pdf

Exploring Riemann’s functional equation (including the França-LeClair points)

https://projecteuclid.org/download/pdf_1/euclid.acta/1485892173

On the roots of the Riemann zeta-function (the classic paper published in 1956 by D.H. Lehmer)

RIEMANN'S HYPOTHESIS AND THE SHAPE OF THE EARTH

If Riemann's hypothesis is true, the Earth is flat.

If the de Bruijn-Newman constant equals zero (Λ = 0), the Earth is flat.

If there exists an infinity of Lehmer pairs, the Earth is flat.



https://arxiv.org/pdf/1508.05870.pdf

Lehmer pairs revisited

The Riemann hypothesis means that the de Bruijn-Newman constant is zero.

Unusually close pairs of zeros of the Riemann zeta function, the Lehmer pairs, can be used to give lower bounds on Λ.

Soundararajan’s Conjecture B implies the existence of infinitely many strong Lehmer pairs, and thus, that the de Bruijn-Newman constant Λ is 0.


http://www.math.kent.edu/~varga/pub/paper_209.pdf

Lehmer pairs of zeros and the Riemann ξ-function

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.30.9492&rep=rep1&type=pdf

A new Lehmer pair of zeros and a new lower bound for the de Bruijn-Newman constant Λ

http://www.academia.edu/19018042/Lehmer_pairs_of_zeros_the_de_Bruijn-Newman_constant_and_the_Riemann_Hypothesis

Lehmer pairs of zeros, the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

http://www.dtc.umn.edu/~odlyzko/doc/debruijn.newman.pdf

An improved bound for the de Bruijn-Newman constant

https://www.ams.org/journals/mcom/2011-80-276/S0025-5718-2011-02472-5/S0025-5718-2011-02472-5.pdf

An improved lower bound for the de Bruijn-Newman constant


Recently, it was proven that the de Bruijn-Newman constant is non-negative:

https://arxiv.org/pdf/1801.05914.pdf

This means that an infinite sequence of Lehmer pairs of arbitrarily high quality will prove that the de Bruijn-Newman constant is equal to zero (Λ = 0).

https://terrytao.wordpress.com/2018/01/20/lehmer-pairs-and-gue/


692,736.741 and 692,736.7631 (Lehmer pair, zeta zeros 1169838 and 1169839)

692736/63.63 =~ 10887

10888 = 136.1 x 80

692736/136.1 =~ 5090

5.09/2 = 2.545

8595 x 136.1 = 1169779.5
8596 x 136.1 = 1169915.6

135 x 63.63 = 8590

18385 x 63.63 = 1169837.55

135 x 136.1 = 18373.5

These calculations show that there is a formula, expressed in terms of 136.1 and 63.636363, which predicts the sacred cubit interval where a Lehmer pair is located.

EXACT DISTRIBUTION OF THE ZETA ZEROS FORMULA IX: LEHMER PAIRS

A proton consists of two laevorotatory quarks and one dextrorotatory quark. A neutron has two dextrorotatory quarks and one laevorotatory quark in its composition.

https://web.archive.org/web/20141027125332/http://www.scientificexploration.org/journal/jse_09_4_phillips.pdf (page 502)

A quark has three subquarks. The subquark is formed of some 14 billion bosons distributed in double torsion fashion.

A boson is created by an even smaller particle:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1774536#msg1774536

The smallest particle, a cavity resonator, has Riemann zeta function waves travelling in both directions:

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

Lehmer pairs are unusually close pairs of zeros of the Riemann zeta function:



These pairs have symmetrical sacred cubit formulas.



7005.0629 and 7005.10056, zeta zeros 6709 and 6710

51 x 136.1 = 6941.11

6941.11 + 63.63 = 7004.74

49 x 136.1 = 6668.9
50 x 136.1 = 6805

39.333 + 6668.9 = 6708.233

110 x 63.63 = 6999.3

105 x 63.63 = 6681.15

6999.3 - 6681.15 = 318.15 = 5 x 63.63



17143.786 and 17143.8218, zeta zeros 18859 and 18860

126 x 136.1 = 17148.6

269 x 63.63 = 17116.47
270 x 63.63 = 17180

138 x 136.1 = 18781.8
139 x 136.1 = 18918

296 x 63.63 = 18834.5

18834.5 - 17116.47 = 286.1 x 6


63137.2115 and 63137. 2324, zeta zeros 82552 and 82553

464 x 136.1 = 63150

606 x 136.1 = 82476.6

82552 - 82476.6 = 75.4 = 24 x π

63137/63.63 = 992.25

992 = 7.2887 x 136.1

82552/63.63 = 1297.37

1361 - 1297 = 64


The five element subdivision points = the location of the zeta zeros, using sacred cubit intervals of 63.63636363 units.

The Lehmer pairs values = the five element subdivision intervals/points, using a larger sacred cubit interval, 6363.636363 units.

Zeta zeros distribution: regular zeros, large gaps between zeros, Lehmer pairs and strong Lehmer pairs.

A strong Lehmer pair is to the Lehmer pairs, what the Lehmer pairs represent in terms of the zeta zeros.

The larger sacred cubit interval, 6363.636363 units, is used to detect both the Lehmer pair interval, and its values.

That is, I believe that these pairs are not randomly located within the distribution of the zeta zeros, but have a precise location.

There are even double Lehmer pairs, two pairs which are very closely located to each other:

1579400.943 and 1579400.968
1579721.076 and 1579721.097

2378769.005 and 2378769.021
2378798.466 and 2378798.483


A 6363.636363 interval has ten 636.36363 subintervals, and one hundred 63.63636 subintervals.

The first Lehmer pair, 111.029 and 111.874 is a regular Lehmer pair.

415.0188 and 415.455 is the first strong Lehmer pair.

The 636.3636 interval can be subdivided according to the five element subdivisions algorithm: one adds/substracts 161.773, 120.66, 89.98, 67.106 ... to locate the Lehmer pairs.

The 6363.6363 interval can be subdivided as follows: one adds/substracts 636.36, 954.54, 1617.733, 1206.6, 899.8, 670.16 ... to locate further Lehmer pairs.

The pairs 630.47 and 630.8, 637.397 and 637.93 are located exactly very close to the 636.36 value.

636.36 + 318.18 = 954.54, while 954.13 and 954.83 is another Lehmer pair.

Each of the values 636.63, 954, 1617.7, 2824, 3724, 4395.2, 4895.6 is located very close to a Lehmer pair. 6363.63 is located exactly next to a large gap.

6999, 7317, 7981, 9187.96, 10087.7, 10758.8, 11259.3, 11632.5, 11910.7, 12118.3 are positioned right next to Lehmer pair values.

As an example, we have the interval from 4836.36 to 5154.5454.

The Lehmer pairs are located at these values (* denotes strong Lehmer pairs):

4862
4870
4888
4890
4893
4900 *
4917
4928 *
4951 *
4960 *
4966 *
4990 *
5010 *
5022
5035 *
5045 *
5052
5064 *
5069
5074
5081
5092 *
5096 *
5108
5115 *
5128 *
5154

4836.36
4899.99
4963.6363
5027.2727
5090.909
5154.5454

Each of these values either coincides with a Lehmer pair figure or is very close to it.

https://web.archive.org/web/20110610163654/http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1


There are even certain sacred cubit values connections between some Lehmer pairs:

17143/7005 = 1/sc² + 63.6363

5124/415 = 5/sc²

1025/415 = 1/sc²

12658/1025 = 5/sc²

17335277232221.245/7005 = 2474700532.793

2.474 = 1/sc²

1336685304932.843/7005 = 190818744.458

1.908 = 3 x 1sc

35615956517.47854/7005 = 5084362.10099

5.084/2 = 2.542, where 0.6363 x 4 = 2.5452

161886592540.9931/17143 = 9443305.87

9.4433 =~ 3 x π

RIEMANN'S HYPOTHESIS AND THE SHAPE OF THE EARTH II

Highest zeta zero ever computed:
t ≈ 81029194732694548890047854481676712.9879 ( n = 10³⁶ + 4242063737401796198).

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

1273315917355388788579148020712834 x 63 = 80218902793389493680486325304908542

1273315917355388788579148020712834 x 0.63636363 = 810291939305055209561529176768134.23182742


81,029,194,732,694,548,890,047,854,481,676,676.23182742
81,029,194,732,694,548,890,047,854,481,676,739.86819105

81,029,194,732,694,548,890,047,854,481,676,692.40916072
(+16.1773333)

81,029,194,732,694,548,890,047,854,481,676,704.47514472
(+12.065984)

81,029,194,732,694,548,890,047,854,481,676,713.47347282
(+8.9983281)

81,029,194,732,694,548,890,047,854,481,676,709.78410162
(+5.3089569)

81,029,194,732,694,548,890,047,854,481,676,710.72208732
(+0.9379857)

81,029,194,732,694,548,890,047,854,481,676,711.42159955
(+0.69951223)

81,029,194,732,694,548,890,047,854,481,676,711.943267789
(+0.521668239)

81,029,194,732,694,548,890,047,854,481,676,712.332307089
(+0.3890393)

81,029,194,732,694,548,890,047,854,481,676,712.622437039
(+0.29012995)

81,029,194,732,694,548,890,047,854,481,676,712.838804339
(+0.2163673)



81,029,194,732,694,548,890,047,854,481,676,723.69085805
(-16.177333)

81,029,194,732,694,548,890,047,854,481,676,711.62487405
(-12.065984)

81,029,194,732,694,548,890,047,854,481,676,716.5720035
(-7.11885455)

81,029,194,732,694,548,890,047,854,481,676,715.3142453
(-1.2577582)

81,029,194,732,694,548,890,047,854,481,676,714.37625955
(-0.93798575)

81,029,194,732,694,548,890,047,854,481,676,713.6767473
(-0.69951225)

81,029,194,732,694,548,890,047,854,481,676,713.15507905
(-0.52166825)


Very interesting comments on the S(t) function:

https://arxiv.org/pdf/1407.4358.pdf (page 46)


To test any hypotheses regarding S(t), the five element subdivision algorithms should be used around the zeta zero t = 10^10,000, since the 63.636363636363... interval has to be shifted/translated only using arbitrary-precision arithmetic ([10^10,000/63.6363636] x 63.63636363, where [ x ] denotes the integral part of x). To detect the correct number of zeros in the interval [10^10,000/63.63636363...] x 63.63636363, [10^10,000/63.63636363...] x 63.63636363 + 63.636363636363, Gram points, França-LeClair points, Backlund's method should be utilized, and then simply use the five element subdivision algorithms to compute the zeta zeros within that interval, and discover how S(t) behaves at that height (for 10^10,000, the average spacing is 0.0002729).

DEPALMA SPINNING EFFECT ON LONG DISTANCE ARTILLERY PROJECTILES III



https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2029817#msg2029817 (part I)

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2032069#msg2032069 (part II, formula)

This is the RE formula for a ballistic trajectory:

R = [v_o²sin(2θ_o)]/g x {1 + [v_o²/gR_e][cos²θ_o]}

This is the FE formula for a ballistic trajectory (limit as R_e goes to infinity):

R = [v_o²sin(2θ_o)]/g

The difference is considerable: it amounts to kilometers.

That is why no other FE for the past 150 years has been able to address this very important matter. The most that some of them have done, is to deny the actual calculations.

Here are the trajectory range tables used by the US Navy during WWII:

http://www.eugeneleeslover.com/USN-GUNS-AND-RANGE-TABLES/OP-770-1.html

Had the FE formula with a fixed g been used, each and every target during WWI and WWII would have been missed by a large margin (mobile targets - other ships, fixed targets - ports/cities).

However, this is the correct FE formula:

R = [v_o²sin(2θ_o)]/f(k)

k is the variable electrogravitational value, which depends on the altitude, the atmospheric ether tide, the density of ether at a certain altitude, and the spin rate

The curvature factor is ~EQUAL to the antigravitational effect produced by the spin rate of the projectile which forms a torsion field which partially cancels out the g force.

HANS COLER'S MAGNETSTROMAPPARAT/STROMZEUGER DEVICES



"Coler's innovative research on an over-unity ether field generator was interrupted by Allied bombings during World War II. But after the war Coler's project showed enough promise that the British Secret Service interrogated Coler, as well as his colleague and financial backer, Dr. F. Modersohn. On January 7, 1946, British Intelligence produced a 32-page Final Report No. 1043. The document was classified CONFIDENTIAL (it was declassified in 1962). The object of their investigation were two devices that Coler had invented between 1926 and 1945, which produced electrical energy without a chemical or mechanical source of power."

Hans Coler did not apply any external source of power to his magnets. He set up the apparatus and its circuit, adjusted the spacing between the magnets he had arranged in a ring and waited.

https://web.archive.org/web/20080218060731/http://www.rexresearch.com/coler/colerb~1.htm (British Secret Service report on the Hans Coler devices, 1946)

Accordingly Coler was visited and interrogated. He proved to be cooperative and willing to disclose all details of his devices, and consented to build up and put into operation a small model of the so-called "Magnetstromapparat" [Magnet Power Apparatus] using material supplied to him by us, and working only in our presence. With this device, consisting only of permanent magnets, copper coils, and condensers in a static arrangement he showed that he could obtain a tension of 450 millivolts for a period of some hours; and in a repetition of the experiment the next day 60 millivolts was recorded for a short period. The apparatus has been brought back and is now being further investigated.

In 1933 Coler and von Unruh made up a slightly larger model with an output of 70 watts. This was demonstrated to Dr. F. Modersohn, who obtained from Schumann and Kloss confirmation of their tests in 1926.

Entitled "The Invention of Hans Coler, Relating to an Alleged New Source ol Power," B.I.O.S. Final Report no 1043, Item No. 31, Summer 1946, this report consisted of tests and findings on two strange circuits conducted at the University of Berlin between the World Wars under the auspices of none other than Dr. Winfried Otto Schumann, discoverer of the Schumann resonance of the earth. A mere glance will explain why the device attracted the immediate attention of the German Navy, which classified it as a possible source of quiet and limitless energy for submarine propulsion.

It will be noted that this hexagonal construction of coils and magnets and two "rotating" sub-circuits has absolutely no source of power. Yet, to the mystified Coler and Dr. Schumann, it nevertheless managed to produce, or better, transduce power seemingly... from nowhere.

In 1937, Coler built a 6 KW version of the Stromzeuger. In 1942, Modersohn demonstrated the device to the Research Dept. of the German Navy, which intervened and supplied them with materials, meters and tools. Thereafter the research was directed by Oberbaurat (Naval Construction Chief) Seysen, who assigned Dr. H. Frolich to assist Coler for several months. The operation of the newly developed apparatus turned out to be more complicated than they had first thought, but progress was made. The large Stromzeuger was destroyed by a bomb which struck Coler's house in Kolberg (Pomerania) in 1945. Coler had been powering his house with the unit for three years.

Hans Coler German patent (1939):

https://web.archive.org/web/20071114064818/http://www.rexresearch.com/coler/de680761.pdf


http://www.intalek.com/Index/Projects/Research/HansColer/HansColer.htm

http://discaircraft.greyfalcon.us/coler.html

http://www.rexresearch.com/coler/coler2.htm


Professor W.O. Schumann (Munich) also tested the Stromzeuger in 1926; his 6-page analysis was included in an appendix of the BIOS Report:

"The apparatus in question principally consists of two parallel connected spools, which being bifilarly wound in a special way, are magnetically linked together. One of these spools is composed of copper sheets (the spool is called the plate spool), the other one of a number of thin parallel connected isolated wires (called: spool winding), running parallel at small intervals to the plates. Both spools can be fed by separate batteries; at least two batteries are necessary to put the spools to work.

"The spools are arranged in two halves each, according to the bifilar winding system. The batteries are attached to the starting points, and the current-receivers to the parallel connected ends. Inter-communications are connected between parallel windings of the two halves of the plate spool which contain iron rods with silver connections. These rods are magnetized by a special battery through applied windings (called: exciter windings).

 "According to the statement of the inventor, the production of energy principally takes place in these iron rods, and the winding of the spools plays an important part in it (The form of the spool is a long small rectangle).

"The inventor stated that the apparatus in its installation was very sensitive, especially with regard to the magnetic conditions of the iron cores, and that a wrong treatment [internal measurements] would cause interferences which would be wearisome and very difficult to be eliminated.

"The exciter winding is electrically completely separated from the other windings...

"Installed in the apparatus were 3 current meters for the currents from the 3 batteries, and furthermore current and volt meters for the current receivers. One and two bulbs respectively were employed for this purpose.

"As a striking fact it should be mentioned that the spool circuit having been at first always switched on alone, received a current of 104 mA. As soon as plates and exciter circuit additionally and simultaneously were turned on, as, according to the inventor, the apparatus demands it, the current in the spool circuit comes down to about 27 mA.

"After the present examination, carried through as carefully as [possible], I must surmise that we have to face the exploitation of a new source of energy whose further developments can be of an immense importance. I believe that a further development of the apparatus will prove justified and of great importance."

The results of the tests are compiled in the annexed table.

The figures show very well that the consumption of energy in the external circuit is greater than the energy taken from the batteries. According to the circuit, produced by Captain Coler, which within this short time I could not check in all its parts, the magnet-exciting circuit is fed by a special battery, completely separated from the other two circuits. Consequently, a direct comparison of efficiency and consumption of the apparatus would mean that only the sum of current of the plate circuit and of the spool circuit would count. After the established estimates with my own instrument and on a load of 3 bulbs, there was resulting a current from the two mentioned batteries of 0.215 + 0.070 = 0.285 ampere. At the same time the three bulbs consumed ca. 3.7 ampere, according to the built-in instruments, which is about 0.2 ampere too much as was proved later on by a control of this instrument, so that the real consumption has been about 3.5 ampere at a tension of about 2.3 volt.

The reception of current from the two batteries in this case consequently was 1.7 watt while the consumption of the bulbs amounted to about 8 watt. Especially striking in this connection is the considerably higher current-power in the bulb-circuit being about 12 times bigger than the current coming from the two batteries.

We have also absolutely made sure that from the batteries no other conductors led to the apparatus than those into which my instrument was built-in. The fact that an increase of power from the battery to the terminal clamps of the effective circuit in the plate-system takes place, could, indeed, not be tested on the different parts of the apparatus by a direct measuring of the power itself, because Mr. Coler declared that when switching on an instrument in the interior of the system, probably the "adjustment" would be disturbed.

I have therefore tested the decrease of tension in the single plates on a load of three lamps by means of a millivolt-meter, make of Hartmann & Braun, Nr. 462375, in order to get at least in an indirect way an explanation for the increase of current. This examination showed a remarkable increase of tension-losses with a distinct maximum on the third-last plate of the one row. These estimates too are compiled in the table at the end of this judgment.

Results ~ The result of the investigation showed an astonishing working of the apparatus, which, without further researches cannot be explained or compared with the hitherto known characteristics.


Admiral Byrd 1947 South Pole expedition: Operation Highjump

https://web.archive.org/web/20090319144420/http://www.militaryphotos.net/forums/archive/index.php/t-33370.html

http://www.south-pole.com/p0000152.htm

http://www.south-pole.com/p0000150.htm

http://www.germanufochatter.com/Nazi-South-Polar-Base/index.html

https://www.bibliotecapleyades.net/tierra_hueca/esp_tierra_hueca_6c.htm

https://web.archive.org/web/20100108140745/http://www.eyepod.org/Nazi-Disc-Photos.html


Beatles update (August 7 - October 8 )

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1082425#msg1082425

EXACT FORMULA FOR ONE OF THE FIVE ELEMENTS MAIN SUBDIVISION POINTS

<N(T)> = T/2π(logT/2πe) + 7/8

Let <N(T)> = n (value of an integer)

n - 7/8 = T/2π(logT/2πe)

http://mathworld.wolfram.com/LambertW-Function.html

The Lambert W function is the inverse of f(W) = We^W.

Main subdivision point =~ 2πe ⋅ e^{W[(n - 7/8)/e]}

Mathematica software for the Lambert W function:

http://functions.wolfram.com/webMathematica/FunctionEvaluation.jsp?name=ProductLog

The value of one of the four main subdivision points will always be extremely close to the figure obtained using the above formula, in fact it can be recognized instantly, once the sacred cubit interval is subdivided using both zeta functions. Then, we can identify the other three subdivision points. Once we have the four subdivision points there is no need to even bother to find the value of the corresponding zeta zero, since now it can be found easily using the algorithm indicated in the previous messages on this subject.

These main partition points are not approximate Gram points, or França-LeClair points: the four main subdivision points are the exact values which lead directly to the corresponding zeta zero to the nth decimal required, using the algorithm outlined before.

As we have seen, the Lehmer points are not located randomly on the critical line.

Example

7005.0629 and 7005.10056, zeta zeros 6709 and 6710

Sacred cubit interval: 63.6363636363...

6999.999
7063.6363

For the first zeta function the values are:

7003.1814
7003.9903
7004.593
7005.0435
7005.17544
7005.379

The calculations for the second zeta function:

7004.541
7005.175
7005.48

Using the above formula, with n = 6708 and 6707:

7004.493
7003.582

7004.541 is one of the principal subdivision points for the zeta zero corresponding to n = 6708 (which is 7004.0437).

7003.1814 is one of the principal subdivision points for the zeta zero corresponding to n = 6707 (which is 7002.6915).

EXACT LEHMER PAIR SEQUENCE FORMULA

(636.3 x 3 - 16.9)(1.361 x 6.666666) = 17166.746

17166.746 - 17143.8 = 22.9
17166.746 - 17143.6 = 22.7

http://www.dtc.umn.edu/~odlyzko/zeta_tables/zeros1

17143.786536184
17143.821843505

22.7 x 6 = 1.362

22.9/36 = 0.636

136.1 x 2.5 = 340.25

340.25/0.6363 = 534.732

340.25 - (169 x 2) = 0.5 x 4.5


(636.3 x 6 - 2x16.9)(1.361 x 6.666666) = 34333.49

2 x 16.9 = 33.8

34333.49 - 34295.37 = 38.12
34333.49 - 34295.03 = 38.46

34295.104944255
34295.371984027

38.12 = 6 x 6.3533333

38.46/22.7 = 1.6943


(636.3 x 9 - 3x16.9)(1.361 x 6.666666) = 51500.23485

3 x 16.9 = 50.7

51448.076349964
51448.729475327
51449.153911623

51500.23485 - 51448.05485 = 52.18

(22.7 + 38.46 + 52.18) x 3 = 340.02


(636.3 x 12 - 4x16.9)(1.361 x 6.666666) = 68666.918

4 x 16.9 = 67.6

68597.636479797
68597.971396943

68666.918 - 69.08 = 68597.838

69.08 - 52.18 = 16.9


(636.3 x 15 - 5x16.9)(1.361 x 6.666666) = 85833.72475

5 x 16.9 = 84.5

85752.427507194
85752.870238027

85833.724 - 81.72 = 85752.00475

22.7 x 3.6 = 81.72

85748.621773488
85748.861163006

85833.72475 - 85.8 = 85747.9248

8.58 = 3 x 2.86


(636.3 x 18 - 6x16.9)(1.361 x 6.6666666) = 103000.4697

6 x 16.9 = 101.4

102907.166732245
102907.475751344

103000.4697 - 97.779 = 102902.6907

2.5423 x 38.46 = 97.777

97.779 - 52.18 = 2 x 22.7

103000.4697 - 102907.5 = 92.9697

92.9697/52.18 = 5.345/3


(636.3 x 21 - 7x16.9)(1.361 x 6.666666) = 120167.2146

7 x 16.9 = 118.3

120165.009100584
120165.181710116

120167.2146 - 120053.7146 = 113.5

113.5 = 22.7 x 5

120055.446373211
120055.565321075


Lehmer pair sequence = (636.3 x 3k - 16.9 x k)(1.361 x 6.666666) - f(16.9)

The terms (1.361 x 6.666666) and f(16.9) might be replaced by a single constant, whose value is yet to be determined:

Lehmer pair sequence = (636.3 x 3k - 16.9 x k) x C

(where C, of course, is related to the sacred cubit constants)


7005.062866175
7005.100564674

(636.3 + 136.1)(1.361 x 6.666666) = 7008.235

7008.235 - 7005 = 3.235

(636.3 x 3 + 136.1)(1.361 x 6.66666) = 18554.96

18539.140716112
18539.436652430

18554.96 - 16.9 = 18538.06

(636.3 x 6 + 136.1)(1.361 x 6.6666666) = 35875.05

35839.415210178
35839.746238617

35875.05 - 35839.5 = 35.549

3.235 x 11 = 35.585

I believe that each Lehmer pair is part of a certain sequence, similar to the ones derived above.


A formula which gives slightly better results for Lehmer pairs, than the França-LeClair points:

2πe ⋅ e^{W[(n - 1.1444)/e]}

1.1444 = 2.861 x 4

In order to precisely locate n, other than the formula above, within the n x 63.63636363 interval, a careful study of the two counterpropagating zeta functions has to be undertaken.

7005.062866175
7005.100564674

For the second zeta function, the zeros to be found in the same place of the interval are:

7057.64086 (7005.9874)
7058.63 (7004.9963)

7004.9963 (6708.55132)

<N>(7057.64086) = 6767.3715
<N>(7058.63) = 6768.477

<N>(7004.493) = 6708
<N>(7005.403) = 6709

<N>(7058.2122) = 6768
<N>(7059.13) = 6769

59.13 + 4.493 = 63.623

58.212 + 5.403 = 63.615

6708.55132 + 6768.477 = 13477.028, very close to a whole integer value

7005.0629 (6708.626)

6767.3715 + 6708.626 = 13475.99975, very close to a whole integer value

That is, there is a very interesting mathematical relationship between the Lehmer pair zeros and the corresponding regular zeros of the other zeta function on the same 63.636363 interval.


17143.7865
17143.8218

17143.7865 (17156.2136)

The corresponding zero of the other zeta function is:

17156.4314 (18874.9906)

<N>(17142.9383) = 18858
<N>(17143.733) = 18859
<N>(17144.527) = 18860

<N>(17155.645) = 18874
<N>(17156.439) = 18875

17144.527 + 17156.439 = 34300.966, very close to a whole integer value

There might also be certain mathematical relationships between the four main subdivision figures for both zeta functions.

EXACT LEHMER PAIR SEQUENCE FORMULA II

If Riemann's hypothesis is true, the Earth is flat.

If the de Bruijn-Newman constant equals zero (Λ = 0), the Earth is flat.

If there exists an infinity of Lehmer pairs, the Earth is flat.

Infinite sequence of Lehmer pairs formula

(636.3 x 3n - 16.9 x n)2π/ln2

(n = 1,2,3...)


s = r x θ

r = 68.1 (136.2/2, 22.7 = 1.362 x 16.66666, 38.136 = 1.362 x 28, 51.756 = 1.362 x 38, 68.1 = 1.362 x 50, 81.72 = 1.362 x 60, 98.064 = 1.362 x 72, 118.494 = 1.362 x 87)

θ = 136.12°

sin 136.12° = ln2

136.12° = 2.375742 radians

s = 161.78804

63.6363/16.1773 = 1/0.25422

2π/ln2 = (10s - 1000)/r

10 x 136.12° radians - 1000/r = 2π/ln2

136.12° radians x 3.819072 = 2π/ln2

That is, 2π/ln2 is the arclength corresponding to the 136.12° expressed in radians multiplied by 6 sacred cubits.

The Gizeh pyramid was built using a geometrical design based on three circles each featuring a radius of 60 sacred cubits (38.15 meters).

Let us recall the value of the angle of the Gizeh pyramid: 51.8554°

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1834389#msg1834389

90 - 38.1446 = 51.8554

51.85 x ln2/2π = 2 x 2.86


ln2/68.1 = 0.010178372

1.0178372 = 4 x 0.25445

2.5445/4 = 0.636

2π/0.010178372 = 617.58976

ln2/2π - 2π = -6.172867

2π/ln2 - 2π - 2.377138 = 0.40439625 = 0.6359216²

136.2° = 2.37713844 radians

2.377138/68.1 = 0.0349066

1/2.861 = 0.349528

2.544° = 0.0444 radians

0.0444 x 68.1 = (2π/ln2)(1/3)

List of zeta zeros:

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

2π/ln2 is the most important constant of the eta zeta function (alternating series zeta function):

https://arxiv.org/pdf/math/0209393.pdf

https://arxiv.org/pdf/0706.2840.pdf

EXACT LEHMER PAIR SEQUENCE FORMULA III

The spacing of the zeros exhibits the same statistical pattern as the spectra of atomic energy levels.

A prime case of chaos

https://www.ams.org/publicoutreach/math-history/prime-chaos.pdf

Zeta zeros and quantum chaos

http://web.math.ucsb.edu/~jcs/zeta.pdf


Infinite sequence of Lehmer pairs formula (2)

(636.3 x 3k + 136.1)2π/ln2

(k = 1/3, 1, 2, 3...)

k = 1/3
7001.5 (computed value using the formula)
7005.1 (zeta zero, Lehmer pair)

k = 1
18537.3
18539.141

k = 2
35840.9
35839.41

k = 3
53144.64
53145.521

k = 4
70448.286
70447.585

k = 5
87751.93
87753.47

k = 6
105055.57
105056.714

k = 7
122359.2
122358.45

k = 8
139662.86
139662.274

k = 9
156966.5
156969.277

k = 10
174270.154
174275.356

k = 11
191573.8
191573.094

k = 12
208877.4
208880.027

k = 24
416521.2
416520.54

k = 60
1039452.4
1039454.62


Another infinite sequence formula (3):

136.1 x 3n

(n = 1,2,3)

n = 1
408.3
415.0188

n = 60
24498
24495.47
24498.26

n = 90
36747
36750.187

n = 91
37155.3
37152.48

n = 95
38788.5
38790.248

n = 170
69411
69410.08




There can’t be a zero of ζ'(s) between every pair of zeros of ζ(s) because the density of zeros of ζ(s) is log(T/2π)/2π while the density of zeros of ζ'(s) is log(T/4π)/2π. So on average there is a “missing” zero of ζ'(s) in each T interval of width 2π/log 2 ≈ 9.06.

ROOTS OF THE DERIVATIVE OF THE RIEMANN ZETA FUNCTION

https://arxiv.org/pdf/1002.0372.pdf

On Small Distances Between Ordinates of Zeros of ζ(s) and ζ'(s)

http://math.boun.edu.tr/instructors/yildirim/paper/OnSmallDistancesBtwOrdinates.pdf

LEHMER PAIRS AND DERIVATIVES OF HARDY’S Z-FUNCTION

https://arxiv.org/pdf/1612.08627.pdf

The author has calculated that the first two million zeros include 4637 pairs of zeros which satisfy the first assertion, while 1901 pairs actually belong to the set L.


LEHMER PAIRS REVISITED

https://arxiv.org/pdf/1508.05870.pdf

In other words, strong Lehmer pairs tend to arise from a small gap between zeros of ζ(s), and from the zeros of ζ'(s) very near the critical line.



Figure 2 shows the argument of ζ'(s)/ζ(s), interpreted as a color, in a region which includes Lehmer’s example. The Riemann zeros 1/2 + iγ₆₇₀₉ and 1/2 + iγ₆₇₁₀ are now poles, while in between we see a zero of ζ'(s) at 0.50062354 + 7005.08185555i, very close to the critical line, even on the scale of this close pair of Riemann zeros.


7 x 2π/ln2 = 63.453042

63.636363/7 = 9.090909

9.0909 - 2π/ln2 = 0.02618

2.618 = phi²

Therefore, 2π/ln2 fits perfectly as a seven note + two intervals (FA-MI and SI-DO) pattern for the 63.636363 segment.

7 x 0.02618 = 0.18326

1.8332/2 = 0.9166 = 2.7498/3

2.7498 = 1 - 1sc

7 x 1.2727 + 2x0.778 = 2π/ln2

1.778 x 3 = 5.334


http://www.math.mcgill.ca/radziwill/farmer15.pdf

GAPS BETWEEN ZEROS OF ζ(s) AND THE DISTRIBUTION OF ZEROS OF ζ'(s)

EXACT LEHMER PAIR SEQUENCE FORMULA IV

Large gaps formula for the zeta function zeros


32 + 2⁵ x n

16 + 2⁵ x n

8 + 2⁴ x n


There will always be large gaps right next to these values of the critical line.

2π/ln2 x 15 = 135.9708043

     14.134725142
     21.022039639
     25.010857580
     30.424876126
     32.935061588
     37.586178159
     40.918719012
     43.327073281
     48.005150881
     49.773832478
     52.970321478
     56.446247697
     59.347044003
     60.831778525
     65.112544048
     67.079810529
     69.546401711
     72.067157674
     75.704690699
     77.144840069
     79.337375020
     82.910380854
     84.735492981
     87.425274613
     88.809111208
     92.491899271
     94.651344041
     95.870634228
     98.831194218
    101.317851006
    103.725538040
    105.446623052
    107.168611184
    111.029535543
    111.874659177
    114.320220915

Large gaps at:

16
24
32
40
48
56
64
72
80
88
96
104
112
120
128

    399.985119876
    401.839228601
    402.861917764
    404.236441800
    405.134387460
    407.581460387
    408.947245502
    410.513869193
    411.972267804
    413.262736070
    415.018809755
    415.455214996
    418.387705790

Large gaps at 400, 408 and 416.

It could be that the very precise location of these large gaps in the values of the critical line is related to the decimal part of 2π/ln2 when multiplied by k (1, 2, 3... to 16):

2π/ln2 x 1 = 9.064720284
2π/ln2 x 2 = 18.129440568
2π/ln2 x 3 = 27.194160852
2π/ln2 x 4 = 36.258881136
2π/ln2 x 5 = 45.32360142
2π/ln2 x 6 = 54.388321704
2π/ln2 x 7 = 63.453041988
2π/ln2 x 8 = 72.517762272
2π/ln2 x 9 = 81.582482556
2π/ln2 x 10 = 90.64720284
2π/ln2 x 11 = 99.711923124
2π/ln2 x 12 = 108.776643408
2π/ln2 x 13 = 117.841363692
2π/ln2 x 14 = 126.906083976
2π/ln2 x 15 = 135.97080426
2π/ln2 x 16 = 145.035524544

(an addition of 1 to the decimal part after the multiplication by 16)

The large gaps are connected to the values of the first derivative of the zeta function, as are the values of the Lehmer pairs.

The value of the large gaps is now known to a precision of 8 units on the critical line.

That is, any Lehmer pair will be located within this 8 unit interval.

2π/ln2/8 = 3.39927/3 = 1.133090036 = 1/0.8825424

3.39927/2.5 = 1.35971

3.39927/1sc = 5.342

Since there is a definite pattern to the large gaps, there must be a similar structure of the Lehmer pairs.

Already, in the two previous messages, three infinite sequences of such Lehmer pairs values have been derived.

The final configuration of the location of the Lehmer pairs/strong Lehmer pairs could be related to the imbedded/nested seven notes/two intervals distribution:

EXACT LEHMER PAIR SEQUENCE FORMULA V

The correct definition for a Lehmer pair: the distance between two consecutive zeta zeros is less than the average spacing.

This way, one has a chance to discover the overall pattern of these special pairs of zeros.


Global formula for Lehmer pairs/close values of the pairs of zeta zeros


T =~ {n ⋅ 2π/ln2 + n ⋅ 2π/ln2 + π/ln2}/2

T =~ {n ⋅ 2π/ln2 + π/ln2 + (n + 1) ⋅ 2π/ln2 }/2


n > 2

T will always be part of an infinite sequence of Lehmer pairs (which includes also strong Lehmer pairs); special values of T have been determined in the previous messages.

Each 8 unit interval will include a pair of zeros whose distance is less than the average spacing.

First this 8 unit interval is determined.

Then, we find the 2π/ln2 and the π/ln2 intervals which overlap the 8 unit segment.

The average of these values will generate the value of one of the close pairs of zeta zeros/Lehmer pairs.


Examples:

415.018809755
415.455214996

8 unit interval: 408 to 416

2π/ln2 + π/ln2 interval

403.38 to 412.445

π/ln2 interval

407.912 to 416.977

(416.977 + 412.445)/2 = 414.711


7005.062866175
7005.100564674

8 unit interval: 7000 to 7008

2π/ln2 interval

6997.964 to 7007.029

2π/ln2 + π/ln2 interval

7002.496 to 7011.56

(7007.029 + 7002.496)/2 = 7004.763


17143.786536184
17143.821843505

8 unit interval

17136 to 17144

2π/ln2 interval

17141.386 to 17150.451

2π/ln2 + π/ln2 interval

17145.918 to 17154.98

(17145.918 + 17141.386)/2 = 17143.65


35839.415210178
35839.746238617

8 unit interval

35832 to 35840

2π/ln2 interval

35832.8 to 35841.9

2π/ln2 + π/ln2 interval

35837.36 to 35846.42

(35837.36 + 35841.9)/2 = 35839.6


How to generate the 2π/ln2 intervals

9.06472
18.1294
27.194
36.258
45.323...

(we simply multiply 2π/ln2 by n, n = 1,2,3...)

How to generate the 2π/ln2 + π/ln2 intervals

(9.06472 + 18.1294)/2 =  13.5971

13.5971 - 2π/ln2 = 4.532360142

4.53236
13.5971
22.6618
31.726
41.4197...

(we simply shift the 2π/ln2 intervals by a factor of π/ln2)


415.018809755, 7005.062866175 and 17143.786536184 are true Lehmer pairs.

What are the starting points of the sequences which generate these values?

415/2π/ln2 = 45.78

9.06472 x 5 = 45.3236

4.53236 + 9.06472 x 5 = 49.856

The average value is 47.818.

The nearest Lehmer pair, from the very first 63.636363 sacred cubit interval, is:

48.005150881
49.773832478

Therefore, the generating value for the Lehmer pair located at 415.018809755 is 48.00515, the interval where 9.06472 x 5 = 45.3236.

48 = 16 x 3 = 8 x 6


7005.062866175/2π/ln2 = 772.7828

The decimal part of 2π/ln2 x n (n = 1,2,3...16) repeats itself.

2π/ln2 x 16 = 145.0355

772.7828/144 = 5.366

[5.366] = 5, so the interval 2π/ln2 x 5 is related to finding the generating point.

Also, 7005/48 = 145.9375.

This means that the generating value for the Lehmer pair located at 7005.062866175 is again 48.00515, the interval where 9.06472 x 5 = 45.3236.


17143.786536184/2π/ln2 = 1891.265

1891.265/144 = 13.13378

[13.13378] = 13

2π/ln2 x 13 = 117.84

108.77 to 117.84
104.24 to 113.31

(113.31 + 108.77)/2 = 111.04

Therefore, the generating point for the Lehmer pair located at 17143.786536184 is the following Lehmer pair:

111.03
111.87


A strong Lehmer pair will be part of a sequence, or grand cycle, which consists of other cycles of Lehmer pairs.

These cycles are generated by both the 8 unit interval units and the 2π/ln2 x 16 cycles (the decimal part of 2π/ln2 x 16 x k becomes part of an even greater sequence).

The values of n (see first two formulas) which include the Lehmer pairs/strong Lehmer pairs can be deduced by discovering the hidden pattern of the intersection of the corresponding 8 unit intervals and the 2π/ln2, 2π/ln2 + π/ln2 segments.

GLOBAL FORMULA FOR STRONG LEHMER PAIRS


2π/ln2 ⋅ 144(n + ε)

144 ⋅ ε = k, where k = 1,2,3...,143

2π/ln2 ⋅ (144n + k)


The previous formulas featured 2π/ln2 multiplied by n; this global formula incorporates the decimal parts as well, which have special values.

This formula can be used to find the strong Lehmer pairs.

1187 pairs:

http://www.slideshare.net/MatthewKehoe1/riemanntex (pg. 64-87)




1.30664344087942265202071895041619 x 10²²
1.30664344087942265202071898265199 x 10²²

Examples:

17143.7865

17143.7865/2π/ln2 = 1891.2648

1891.2648/144 = 13.1338

144 x 0.1338 =~ 19

2π/ln2 x (144 x 13 + 19) = 17141.385

The equations derived previously are special cases of this global formula.


169872.853

2π/ln2 x (144 x 130 + 20) = 169872.858


45505.59

2π/ln2 x (144 x 34 + 124) = 45504.8944


45436.65

2π/ln2 x (144 x 34 + 117) = 45441.44


412597.295

2π/ln2 x (144 x 316 + 13) = 412598.86


555136.9163

2π/ln2 x (144 x 425 + 41) = 555132.52
2π/ln2 x (144 x 425 + 42) = 555141.58

Average = 555137.0511


7954022502373.43289015387

2π/ln2 x (144 x 6093543501 + 75) = 7954022502369.544
2π/ln2 x (144 x 6093543501 + 76) = 7954022502378.53

Average = 7954022502374.037


2414113624163.41943



2π/ln2 x (144 x 1849442389 + 136) = 2414113624163.446


13066434408794226520207.1895041619





Since now T is very large (the average spacing is 0.128), the decimal parts of k also can be used (k = v + 1/2, v+ 1/4, v+ 3/4); in this case 144 x 0.1441 = 20.75).

With 20.75, we get:

2π/ln2 x (144 x 10010140964026289815 + 20.75) = 13,066,434,408,794,226,520,207.14279




8847150598019.22359827



2π/ln2 x (144 x 6777765214 + 101) = 8847150598015.23188
2π/ln2 x (144 x 6777765214 + 102 = 8847150598024.2966

Average = 8847150598019.764243


2π/ln2 x 144 = 415.496 x π


7005.1

2π/ln2 x (144 x 5 + 53) = 7007.028

HOW TO PROVE THE RIEMANN HYPOTHESIS

If the de Bruijn-Newman constant is equal to zero, Λ = 0, then Riemann's hypothesis (all zeta zeros lie on the 1/2 critical line) is true.

The search for the lower bounds of this constant:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2102664#msg2102664

However, in order to prove that -10^-20 < Λ, at least 10³⁰ zeros would have to be examined. The total number of simple arithmetic mathematical operations that have been performed by all digital computers in history is only on the order of 10²³.

Not even with improvements in hardware, it cannot be hoped to compute 10³⁰ zeta zeros using existing methods.

http://www.dtc.umn.edu/~odlyzko/doc/debruijn.newman.pdf

An improved bound for the de Bruijn-Newman constant

https://arxiv.org/pdf/1508.05870.pdf

Lehmer pairs revisited

Strong/high quality Lehmer pairs can be used to give lower bounds for Λ.

The existence of infinitely many Lehmer pairs implies that the de Bruijn-Newman constant Λ is equal to 0.

Therefore, a constructive/computer-assisted proof of the Riemann hypothesis would be possible, if further Lehmer pairs can be produced with little computational effort.

2π/ln2 ⋅ 144(n + ε)

144 ⋅ ε = k, where k = 1,2,3...,143

2π/ln2 ⋅ (144n + k)

k can also equal v + 1/2, v + 1/4, v + 3/4, if T is O(10²²), since then the average spacing will measure O(0.128).

This infinite sequence of numbers includes ALL of the close pairs of zeta zeros, Lehmer pairs and strong/high-quality Lehmer pairs.

A special case of this formula is:

(636.3 x 3n - 16.9 x n)2π/ln2

(n = 1,2,3...)

144 x 13 + 20 corresponds to 636.3 x 3 - 16.9 x 1

144 x 26 + 40
144 x 39 + 60
144 x 52 + 80
144 x 65 + 100
144 x 78 + 120
144 x 91 + 140
144 x 105 + 16
144 x 118 + 36
144 x 131 + 56 corresponds to 636.3 x 3 x 10 - 16.9 x 10
144 x 182 + 126 corresponds to 636.3 x 3 x 14 - 16.9 x 14
144 x 197 + 12 corresponds to 636.3 x 3 x 15 - 16.9 x 15
144 x 275 +132 (3n = 63)
144 x 289 + 8    (3n = 66)
144 x 302 + 28  (3n = 69)

The Lehmer pairs are located at very precise points on the critical line, and so are the strong Lehmer pairs.

The above infinite sequence of values can be used to find further Lehmer pairs.

I believe that the strong Lehmer pairs have shortcut formulas/special infinite sequences from which they can be generated with little effort.

The (636.3 x 3n - 16.9 x n)2π/ln2 infinite sequence certainly suggests that other similar sequences do exist.

Since now we no longer have to rely on the Riemann-Siegel formula to produce the zeta zeros, the calculation of zeros around the 10⁵⁰, 10³⁰⁰, 10¹⁰⁰⁰ values on the 1/2 critical line become possible using the four subdivisions algorithm, the França-LeClair equation (ϑ(t_n) + lim_δ→0+ arg ζ(1/2 + δ + it_n) = (n - 3/2)π), used in conjunction with Backlund's method and Gram points.

That is, further Lehmer pairs can be produced with very little effort, using the two infinite sequences above: further sequences exist, which can capture the strong Lehmer pairs even better.

These Lehmer pairs then can be used to produce lower and lower bounds for the de Bruijn-Newman constant, finally proving that Λ is equal to zero.

GLOBAL FORMULA FOR STRONG LEHMER PAIRS II



15202440116027338092/9.0647202836543876525 = 1677099749392219025.627807945

1677099749392219025.627807945/144 = 11646526037445965.455748666

1677099749392219025.627807945 = 144 x 11646526037445965 + 65

1677099749392219025 x 9.0647202836543876525 = 15202440116027338086.30909

1677099749392219026 x 9.0647202836543876525 = 15202440116027338094.53526

Average = 15202440116027338090.422178


The peak values of the Lehmer pairs (strong Lehmer pairs) must be related to sacred units figures.

100sc + 100sc/45 = 63.63636363 + 14.134725 = 100sc (1 + 1/4.5) = 100sc x (1.2222222...) = 100/6 x 1/135.9708°in radians x 2π/ln2 x 1.222222...

100/6 x 1/135.9708°in radians x 1.222222... = 3 x 2.8612

100sc + 100sc/45 = 2π/ln2 x 3 x 2.8612

Therefore, 2.8612 (the displacement factor of the Gizeh pyramid) is the conversion factor between the zeta zeros and the first derivative of the zeta function.


2π/ln2 = 2.66666... x 3.399 = 2.666666... x 5.34sc

2π/ln2 = 5.34 x 8/3 x sc

Having expressed 2π/ln2 in terms of sacred cubits units, now we have much more information and values at our disposal which can be used to understand the precise location of the strong Lehmer pairs.

Also, 534 x 10si = 2π/ln2 x 15, so that 100sc = 2π/ln2 x 15/5340 x 2500

2206356 x 2π/ln2 = 2 x 10⁷ (to seven decimal places)


However, 534 =~ 2 x 174.53 + 4 x 14.134725 + 2 x 63.636363.


45 x 2π/ln2 = 45 (2 x 174.53 + 4 x 14.134725 + 2 x 63.636363)/100 x 8/3 x sc

45 x 1.4134725 = 100sc

45 x 2π/ln2 = 407.9, where 415.0188 and 415.45 is the first strong Lehmer pair (46 x 2π/ln2 = 416.977).

1892 x 1.4134725 = 2674.29, where 2.67 = 5.34/2 and 1892 x 2π/ln2 = 17150.45 (1891 x 2π/ln2 = 17141.38), 17143.7865 and 17143.8218 is another strong Lehmer pair.

773 x 8 x sc = 3935.268 = 1/0.000254112, where 773 x 2π/ln2 = 7007.03, 7005.0628
and 7005.1005 is another strong Lehmer pair.

The precise location of the strong Lehmer pairs must be related to these sacred cubit values (expressing 2π/ln2 in terms of sacred cubits).

GLOBAL FORMULA FOR STRONG LEHMER PAIRS III

The values of the strong Lehmer pairs behave like regular zeta zeros in a way: they exhibit large gaps and double Lehmer pairs (two pairs which are located very close to each other).

To understand the behavior of the regular zeta zeros, a certain interval was used (100 sacred cubits), and we used the five elements subdivision algorithm to capture perfectly the value of each zeta zero.

To infer the pattern of the strong Lehmer pairs, we also need a certain interval to start with.

On this very page, precise formulas for the location of the large gaps, as well as for the close pairs (Lehmer pairs + strong Lehmer pairs) were derived.

The Lehmer pairs (see the definition used earlier) occur each and every 2π/ln2 units or at an average of the n x 2π/ln2 and (n + 1) x 2π/ln2 values.

Even this information can be used with great advantage, together with the five elements subdivision algorithm or with the França-LeClair equation to find the values of each and every Lehmer pair at very high figures on the 1/2 critical line, a feat which could not be accomplished before.

Strong Lehmer pairs tend to arise from a small gap between zeros of ζ(s), and from the zeros of ζ'(s) very near the critical line.


Interval for the strong Lehmer pairs

2π/ln2 x 100 sacred cubits

That is, we treat each 2π/ln2 value as a single unit of measure (a distance of 9.064720284... = one unit).

2π/ln2 x 100 sacred cubits = 576.84583...

Then, we subidivide this interval just like before using the 26.7, 53.4, 80, 136.1, 534 subdivisions, looking for the location of the strong Lehmer pairs.

576.84583
146.657
86.52676
57.6845
28.842

576.84583 - 146.657 = 430.88

430.88
109.371
64.5282

430.88 - 109.371 = 320.817

320.817
81.5645
48.1225

320.817 - 81.5645 = 239.2525

239.2525
60.827

239.2525 - 60.827 = 178.4255

178.4255
45.388
26.71
17.84
8.92

178.4255 - 45.388 = 133.0375

133.0375
33.823
19.955
13.303
6.65

133.0375 - 33.823 = 99.2145

99.2145
25.224

With these values, we obtain very nice approximations for the Lehmer pairs located at 111.03 and 415.45: 416.26 (first zeta function) and 419.37 (second zeta function) and 113.08 (first zeta function).

To capture the values of the higher strong Lehmer pairs, 7005.1 and 17143.78, the interval will be increased to 57684.52413 (2π/ln2 x 10000 sacred cubits).

Using the same subdivision, we get 7048.6 and 16950.5, as first values of entire sequence of approximations.

For the strong Lehmer pairs which have 12 digits, the interval becomes:

57684583623255.194152266 (2π/ln2 x 1 x 10¹² sacred cubits)

or an even better approximation,

57684583623255.1941522669588143649472078575

This would be the only way to get approximate values of very large strong Lehmer pairs, and to gain an understanding of their location, which is not random, but has a very precise pattern, based on the 2π/ln2 x 100 sacred cubits interval.

The second five elements subdivision algorithm could also be used in parallel with the first subdivision algorithm.

GENERALIZED SAGNAC EFFECT IV



Point A is located at the detector
Point B is in the bottom right corner
Point C is in the upper right corner
Point D is in the upper left corner

l₁ is the upper arm.
l₂ is the lower arm.

Here is the most important part of the derivation of the full/global Sagnac effect for an interferometer located away from the center of rotation.

A > B > C > D > A is a continuous counterclockwise path, a negative sign -

A > D > C > B > A is a continuous clockwise path, a positive sign +

The Sagnac phase difference for the clockwise path has a positive sign.

The Sagnac phase difference for the counterclockwise has a negative sign.


Sagnac phase components for the A > D > C > B > A path (clockwise path):

l₁/(c - v₁)

-l₂/(c + v₂)

Sagnac phase components for the A > B > C > D > A path (counterclockwise path):

l₂/(c - v₂)

-l₁/(c + v₁)


For the single continuous clockwise path we add the components:

l₁/(c - v₁) - l₂/(c + v₂)

For the single continuous counterclockwise path we add the components:

l₂/(c - v₂) - l₁/(c + v₁)


The net phase difference will be (let us remember that the counterclockwise phase difference has a negative sign attached to it, that is why the substraction of the phase differences becomes an addition):

{l₁/(c - v₁) - l₂/(c + v₂)} - (-){l₂/(c - v₂) - l₁/(c + v₁)} = {l₁/(c - v₁) - l₂/(c + v₂)} + {l₂/(c - v₂) - l₁/(c + v₁)}

Rearranging terms:

l₁/(c - v₁) - l₁/(c + v₁) + {l₂/(c - v₂) - l₂/(c + v₂)} =

2(v₁l₁ + v₂l₂)/c²

Exactly the formula obtained by Professor Yeh:

φ = -2(φ₂ - φ₁) = 4π(R₁L₁ + R₂L₂)Ω/λc = 4π(V₁L₁ + V₂L₂)/λc

Since Δφ = 2πc/λ x Δt, Δt = 2(R₁L₁ + R₂L₂)Ω/c² = 2(V₁L₁ + V₂L₂)/c²

CORRECT SAGNAC FORMULA:

2(V₁L₁ + V₂L₂)/c²

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)


This is how the correct Sagnac formula is derived: we have single continuous clockwise path, and a single continuous counterclockwise path.

If we desire the Coriolis effect, we simply substract as follows:

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.


However, for the SAGNAC EFFECT, we have a single CONTINUOUS CLOCKWISE PATH, and a single CONTINUOUS COUNTERCLOCKWISE PATH, as the definition of the Sagnac effect entails.

HERE IS THE DEFINITION OF THE SAGNAC EFFECT:

Two pulses of light sent in opposite direction around a closed loop (either circular or a single uniform path), while the interferometer is being rotated.

Loop = a structure, series, or process, the end of which is connected to the beginning.

A single continuous pulse A > B > C > D > A, while the other one, A > D > C > B > A is in the opposite direction, and has the negative sign.


We can see at a glance each and every important detail.


For the Coriolis effect, one has a formula which is proportional to the area; only the phase differences of EACH SIDE are being compared, and not the continuous paths.

For the Sagnac effect, one has a formula which is proportional to the velocity of the light beam; the entire continuous clockwise path is being compared to the other continuous counterclockwise path exactly as required by the definition of the Sagnac effect.

Experimentally, the Michelson-Gale test was a closed loop, but not mathematically. Michelson treated mathematically each of the longer sides/arms of the interferometer as a separate entity: no closed loop was formed at all. Therefore the mathematical description put forth by Michelson has nothing to do with the correct definition of the Sagnac effect (two pulses of light are sent in opposite direction around a closed loop) (either circular or a single uniform path). By treating each side/arm separately, Michelson was describing and analyzing the Coriolis effect, not the Sagnac effect.

Loop = a structure, series, or process, the end of which is connected to the beginning.

Connecting the two sides through a single mathematical description closes the loop; treating each side separately does not. The Sagnac effect requires, by definition, a structure, the end of which is connected to the beginning.

GENERALIZED SAGNAC EFFECT V

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



The promise made by A. Michelson, "the difference in time required for the two pencils to return to the starting point will be...", never materialized mathematically.

Instead of applying the correct definition of the Sagnac effect, Michelson compared TWO OPEN SEGMENTS/ARMS of the interferometer, and not the TWO LOOPS, as required by the exact meaning of the Sagnac experiment.

As such, his formula captured the Coriolis effect upon the light beams.

Not even the formal derivation of the Sagnac effect formula is not entirely correct.





This is the correct way to derive the Sagnac formula:

Sagnac phase component for the clockwise path:

2πR(1/(c - v))

Sagnac phase component for the counterclockwise path:

-2πR(1/(c + v))

The continuous clockwise loop has a positive sign +

The continuous counterclockwise loop has a negative sign -

The net phase difference will be (let us remember that the counterclockwise phase difference has a negative sign attached to it):

2πR(1/(c - v)) - (-){-2πR(1/(c + v))} = 2πR(1/(c - v)) - (+)2πR(1/(c + v)) = 2πR(1/(c - v)) - 2πR(1/(c + v)) = 2vL/c²


The definition of the Sagnac effect is applied to a closed loop (either circular or a uniform path).

Loop = a structure, series, or process, the end of which is connected to the beginning.

Thus, from a mathematical point of view, Michelson did not derive the Sagnac effect formula at all, since he compared two open segments, and not two loops.

Using the correct definition, we recover not only the error-free formula, but also the precise velocity addition terms.



Practically, A. Michelson received the Nobel prize (1907) for the wrong formula (published in 1904 and 1887; E.J. Post proved in 1999 that the Michelson-Morley interferometer is actually a Sagnac interferometer).

No other physicist has been able to derive the correct Sagnac formula: for the past 100 years they have been using the wrong formula (the Coriolis effect equation) to describe a very different physical situation.

Here, for the first time, the correct Sagnac formula for an interferometer located away from the center of rotation has been derived in a precise manner.

OUMUAMUA INTERSTELLAR PROBE



"The object, nicknamed 'Oumuamua, meaning "a messenger that reaches out from the distant past" in Hawaiian, was discovered in October 2017 by the Pan-STARRS 1 telescope in Hawaii.

Since its discovery, scientists have been at odds to explain its unusual features and precise origins, with researchers first calling it a comet and then an asteroid before finally deeming it the first of its kind: a new class of "interstellar objects."

A new paper by researchers at the Harvard Smithsonian Center for Astrophysics raises the possibility that the elongated dark-red object, which is 10 times as long as it is wide and traveling at speeds of 196,000 mph, might have an "artificial origin." "

"The theory is based on the object's "excess acceleration," or its unexpected boost in speed as it traveled through and ultimately out of our solar system in January.

"Considering an artificial origin, one possibility is that 'Oumuamua is a light sail, floating in interstellar space as a debris from an advanced technological equipment," wrote the paper's authors, suggesting that the object could be propelled by solar radiation."

https://arxiv.org/pdf/1810.11490.pdf

COULD SOLAR RADIATION PRESSURE EXPLAIN ‘OUMUAMUA’S PECULIAR ACCELERATION?

Harvard Smithsonian Center for Astrophysics

Published in the Astrophysical Journal Letters

"Recently, Micheli et al. (2018) reported that ‘Oumuamua showed deviations from a Keplerian orbit at a high statistical significance."

https://blogs.scientificamerican.com/observations/6-strange-facts-about-the-interstellar-visitor-oumuamua/ (a superb analysis by A. Loeb)

The hypothesis that Oumuamua originated as a planetesimal from a binary system is wrong:

https://academic.oup.com/mnras/article/476/3/3031/4909830 (section 5)

(See http://www.mpia.de/homes/calj/gdr2_oumuamua/oumuamua_gdr2.pdf

Plausible home stars of the interstellar object ‘Oumuamua found in Gaia DR2)

https://www.ncbi.nlm.nih.gov/pubmed/29950718

Non-gravitational acceleration in the trajectory of 1I/2017 U1 ('Oumuamua)


https://arxiv.org/pdf/1809.06389.pdf

However, a recent measurement by Micheli et al (2018) of a substantial non-gravitational acceleration affecting the orbit of this object has been interpreted as resulting from its cometary activity, which must be rather vigorous. Here we critically re-assess this interpretation by exploring the implications of measured non-gravitational acceleration for the ’Oumuamua’s rotational state. We show that outgassing torques should drive rapid evolution of ’Oumuamua’s spin (on a timescale of a few days), assuming torque asymmetry typical for the Solar System comets. However, given the highly elongated shape of the object, its torque asymmetry is likely higher, implying even faster evolution. This would have resulted in rapid rotational fission of ’Oumuamua during its journey through the Solar System and is clearly incompatible with the relative stability of its rotational state inferred from photometric variability.
Based on these arguments, as well as the lack of direct signs of outgassing, we conclude that the classification of ’Oumuamua as a comet (invoked to explain its claimed anomalous acceleration) is questionable.


https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1086205#msg1086205 (Nov. 13, Nov. 30, Dec. 9, 2018, Beatles series continues: the origin of Lady Madonna, We Are The Champions was a Beatles song)

STATIONARY EARTH: AYAHUASCA AND CURARE



"In 1995 a remarkable book was published in Switzerland entitled Le serpent cosmique, l’ADN et les origines du savior (The Cosmic Serpent, DNA and the Origins of Knowledge) by Swiss anthropologist Jeremy Narby. (It was first published in English in 1998.) It presents the results of Narby’s personal study of Amazonian shamans, and reveals the remarkable scope of the information shamans glean during the ecstatic trances they induce by taking natural hallucinogenic substances, primarily one called ayahuasca.

In the mid-1980s Narby was studying for his doctorate among the indigenous people of the Peruvian Amazon, working on an environmental project. Like many before him, he soon became fascinated by the astounding botanical knowledge of these so-called ‘primitive’ people, specifically their medicinal use of certain rare plants. He was impressed by the range of plant-derived medicines used by the tribal shamans - ayahuasqueros - and by their effectiveness, especially after they cured a long-standing back problem which European doctors had proved completely incapable of treating. The more he learned, the more intrigued he became about the ways in which the Amazonian natives had developed or acquired this knowledge. The odds against them coming up with even one of these recipes by chance or even by experimentation are simply overwhelming. There are some 80,000 species of plants in the Amazonian rain forest, so to discover an effective remedy using a mixture of just two of them would theoretically require the testing of every possible combination - about 3,700,000,000. It does not end there: many of their medicines involve several plants, and even then such a calculation does not allow for experimentation with the often extremely complex procedures necessary to extract the active ingredients and produce a potent mixture.

One good example of this mysterious medicinal knowledge is ayahuasca itself, a combination of just two plants. The first comes from the leaves of a shrub and contains a hormone naturally secreted in the human brain, dimethyltryptamine, a powerful hallucinogen only discovered by Western science in 1979. If taken orally, though, it is broken down by an enzyme in the stomach and becomes totally ineffective, so the second component of ayahuasca, extracted from a creeper, contains several substances that protect the dimethyltryptamine from that specific enzyme.

In effect, ayahuasca is a designer drug, made to order. It is as if the exact requirements of the mixture were specified in advance, then the correct ingredients chosen to meet the requirements. But how? How could anyone, even sophisticated Western botanists, have found the perfect ingredients without spending decades - perhaps even centuries - on trial and error? How can the ‘primitive’ Amazonian natives have known the properties of these two plants? After all, the odds against them coming up with this combination by accident are truly astronomical.

As Narby writes:

So here are people without electron microscopes who choose, among some 80,000 Amazonian plant species, the leaves of a bush containing a hallucinogenic brain hormone, which they combine with a vine containing substances that inactivate an enzyme of the digestive tract, which would otherwise block the hallucinogenic effect. And they do this to modify their consciousness. It is as if they knew about the molecular properties of plants and the art of combining them, and when one asks them they know these things, they say their knowledge comes directly from hallucinogenic plants.

Another example given by Narby is that of curare. This powerful nerve poison is another ‘made-to-order’ drug, whose ingredients this time come from several different plants and fit a very precise set of requirements. As Narby points out, the natives needed a substance that, when smeared on the tips of blowpipe darts, would not only kill the animal but also ensure that it would fall to the ground. Tree monkeys, for example, if shot with an unpoisoned arrow, often tighten their grip on the branches with a reflex action and so die out of reach of the hunter. The meat itself would, of course, have to be free from poison and safe to eat. It seemed like a very tall order, but curare fits all these requirements: it is a muscle relaxant (killing by arresting the respiratory muscles); it is only effective when injected into the bloodstream - hence its delivery by blowpipe; and it has no effect when taken orally.

The invention of curare is a truly astounding thing. The most common type requires a complex method of preparation in which several plants are boiled for three days, during which lethal fumes are produced. And the final result needs a specific piece of technology - the blowpipe - to deliver it. How was all this discovered in the first place?

The problem becomes even more baffling when it is realised that forty different types of curare are used across the Amazon rain forest, all with the same properties but each using slightly different ingredients as the same plants do not grow in every region. Therefore, in effect, curare was invented forty times. The Western world only learned of it in the 1940s, when it first began to be used as a muscle relaxant during surgery.

The Amazonians themselves do not claim to have invented curare, but that it was given to them by the spirits, through their shamans. These are just two examples from a vast range of vegetable mixtures used by the peoples of the Amazon, the full extent of which has not yet been catalogued by modern botanists. Realising that it was nonsense to suggest that these complex recipes could have been achieved by experimentation, Narby began to ask local people and shamans how they had acquired this knowledge. They told him that the properties of plants and the recipes for combining them are given directly to the shaman by very powerful spirit entities while he is in ecstatic trance under the influence of hallucinogens such as ayahuasca."

The astral plane is not subject to the law of terrestrial gravitation (dextrorotatory waves); as such, it would disappear in a fraction of second, given the 30km/s speed of the supposed heliocentrical path of the Earth, not to mention the daily rotation around its own axis.

The dangers of accessing the astral plane:

https://www.prosveta.fr/looking-into-the-invisible-intuition-clairvoyance-dreams

Jeremy Narby's Cosmic Serpent:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1343816#msg1343816

Don Juan Matus' dextrorotatory waves:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1825278#msg1825278

LUNAR ECLIPSE ALLAIS EFFECT




Dr. Paul Marmet
Assistant professor in the physics department of the University of Ottawa
Senior researcher at the Herzberg Institute of Astrophysics of the National Research Council of Canada
Director of the laboratory for Atomic and Molecular Physics at Laval University in Québec City
Member of the executive committee of the Atomic Energy Control Board of Canada from 1979 to 1984
Fellow of the Royal Society of Canada
Officer of the Order of Canada

https://newtonphysics.on.ca/astronomy/index.html

Enlargement of the Earth's Shadow on the Moon: An Optical Illusion

Dr. Marmet proves that the usual explanation accepted by modern science for the 2% Earth's larger umbra during a lunar eclipse, namely atmospheric absorption, cannot be true.

During a lunar eclipse, it has been observed that the Earth's shadow (official science theory) is 2% larger than what is expected from geometrical considerations and it is believed that the Earth's atmosphere is responsible for the extent of the enlargement, but it is realized that the atmospheric absorption cannot explain light absorption at a height as high as 90 km above the Earth, as required by this hypothesis (as several authors have noted).

http://vixra.org/pdf/1311.0156v1.pdf

Lunar eclipses and the Allais effect

A beautiful exposition of the history of the anomalies observed through the centuries during the lunar eclipse.

"It was also argued that the irradiation of the Moon in the Earth's shadow during the eclipse is caused by the refraction of sunlight in the upper regions of the Earth's atmosphere. However, the shade toward the center is too bright to be accounted for by refraction of visible sunlight.

That is, the pronounced red colour in the inner portions of the umbra during an eclipse of the Moon is caused by refraction of sunlight through the upper regions of the Earth's atmosphere, but the umbral shadow towards the centre is too bright to be accounted for by refraction of visible sunlight."

It has systematically been found that the shadow of the Earth seems to be 2% larger
than what is expected from geometrical predictions.

For his part, in an analysis of 57 eclipses over a period of 150 years, Link (1969) found an enlargement of the shadow of 2.3% on average. Furthermore, schedules inputs and outputs of the crater through the umbra for four lunar eclipses from 1972 to 1982 strongly support the Chauvenet value of 2%.

The increase of the Earth`s umbral shadow during eclipses of the Moon is the
classical value of 2% (the rule of the fiftieth) used in most calculations of lunar eclipses.

J. Meeus, Nouvelles brèves : L’accroissement du diamètre de l’ombre de la Terre lors des éclipses de Lune, Ciel et Terre, Vol. 88, p. 491 (1972)

As the author demonstrates in his paper, the only possible explanation is a variation of the gravitational potential, a lunar eclipse Allais effect.

Just like in the case of the solar Allais effect, this variation of the gravitational potential means that the heavenly body which causes the lunar eclipse cannot be the Earth.

The existence of the Shadow Moon (the same diameter as that of the Sun, Black Sun, Moon, Jupiter, Aurora) was anticipated by the best astronomers of the 19th century.

That many such bodies exist in the firmament is almost a matter of certainty; and that one such as that which eclipses the moon exists at no great distance above the earth's surface, is a matter admitted by many of the leading astronomers of the day. In the report of the council of the Royal Astronomical Society, for June 1850, it is said:

"We may well doubt whether that body which we call the moon is the only satellite of the earth."

In the report of the Academy of Sciences for October 12th, 1846, and again for August, 1847, the director of one of the French observatories gives a number of observations and calculations which have led him to conclude that,

"There is at least one non-luminous body of considerable magnitude which is attached as a satellite to this earth."

Sir John Herschel admits that:

"Invisible moons exist in the firmament."

Sir John Lubbock is of the same opinion, and gives rules and formulæ for calculating their distances, periods.

Lambert in his cosmological letters admits the existence of "dark cosmical bodies of great size."


The subquarks constantly being supplied to form the telluric currents come in two flavors, as already discussed within this thread.

One of the dark bodies which orbit above the Earth emits the laevorotatory subquarks, the antigravitational subquarks, as proven by the Allais effect.

Logically, the invisible moon emits the dextrorotatory subquarks.

http://www.blazelabs.com/f-g-rpress.asp

In fact, cosmic waves have far greater penetrating power than the man-made gamma radiation, and can even pass through a thickness of two metres of lead. The highest frequency possible, that is, the shortest wavelength limit is equal to the dimension of the unit element making up space-time itself, equal to Planck length, radiating at a frequency of 7.4E42Hz.

As you might be thinking already, the radiation pressure exerted by such high frequency radiation, in the top part of the EM spectrum, would be a perfect candidate for the gravity effect, since such radiation would penetrate ANY matter and act all over its constituent particles, not just its surface. The radiation can be visualised as a shower of high energy EM waves imparting impulses of momentum to all bodies in space. It also explains the great difficulty we have to shield anything from such force. The energy of each individual photon is a crucial component of the momentum necessary to create pressure for gravity to be possible. The shadow of incoming high energy EM wave packets can be pictured as the carriers of the gravitational force, the normal role assigned to the theoretical graviton. Hence, gravitons have been theorised due to the lack of knowledge of radiation pressure and radiation shadowing, and that's why they will never be detected. If photons represent the luminance of electromagnetic radiation, then, gravitons represent the shadowing and can be considered as negative energy waves, lack of photons or photon-holes.

This radiation shadowing is being emitted by the heavenly body which does cause the lunar eclipse: read the phrase - that is why they will never be detected.

"Gravitons represent the shadowing and can be considered as negative energy waves, lack of photons or photon-holes".

The Shadow Moon, the source of the dextrorotatory subquarks causes the lunar eclipse.

We know for sure that the Moon does not cause the solar eclipse, here is the Allais effect:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg760382#msg760382

HOW TO PROVE THE RIEMANN HYPOTHESIS II

Two mathematicians from the Lomonosov Moscow State University have used the mollifier function introduced by N. Levinson in a novel way:

https://arxiv.org/pdf/1805.07741.pdf

100% OF THE ZEROS OF THE RIEMANN ZETA-FUNCTION ARE ON THE CRITICAL LINE

Earlier, they published another paper in which they showed that at least 47% of the zeros of the Riemann zeta function lie on the critical line (the previous records were Feng (41%), Conrey (40%) and Levinson (34%)).

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.758.4457&rep=rep1&type=pdf

https://arxiv.org/pdf/1403.5786.pdf (the original paper on the novel way of using mollifier functions)

https://arxiv.org/pdf/1207.6583.pdf

Limitations to mollifying ζ(s)

http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=F7C33227D1D6635FFBC27972BA54E5A8?doi=10.1.1.36.9777&rep=rep1&type=pdf

Long mollifiers of the Riemann zeta function

https://arxiv.org/pdf/1604.02740.pdf

THE θ = ∞ CONJECTURE IMPLIES THE RIEMANN HYPOTHESIS

https://rjlipton.wordpress.com/2018/09/26/reading-into-atiyahs-proof/ (on Sir M. Atiyah's use of the Todd function for the Riemann hypothesis)

Mathematicians complain that 99% of the proofs submitted to the Annals are rejected because they make use of the zeta functional equation.

"The reason for this is (as has been known since the work of Davenport and Heilbronn) that there are many examples of zeta-like functions (e.g., linear combinations of L-functions) which enjoy a functional equation and similar analyticity and growth properties to zeta, but which have zeroes off of the critical line. Thus, any proof of RH must somehow use a property of zeta which has no usable analogue for the Davenport-Heilbronn examples."

However, the arguments used in the following papers are very well presented and make a lot of sense.

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.

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)


https://arxiv.org/pdf/1704.05834.pdf

On large gaps between zeros of L-functions from branches

Andre LeClair (Cornell University) proves that the normalized gaps between consecutive ordinates t_n of the zeros of the Riemann zeta function on the critical line cannot be arbitrarily large.

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg1126290#msg1126290 (Beatles series, Dec. 24, 2018, Jan. 15, 2019, Jan. 22, 2019 episodes; we will find out how Pink Floyd's best known songs, especially Shine On You Crazy Diamond, are actually modifications of other Beatles songs, also an analysis of RHCP's best songs in the context of Beatles super hits)

GENERALIZED SAGNAC EFFECT VI

A second reference which confirms my global/generalized Sagnac effect formula.

https://apps.dtic.mil/dtic/tr/fulltext/u2/a206219.pdf

Studies of phase-conjugate optical devices concepts

US OF NAVAL RESEARCH, Physics Division

Dr. P. Yeh
PhD, Caltech, Nonlinear Optics
Principal Scientist of the Optics Department at Rockwell International Science Center
Professor, UCSB
"Engineer of the Year," at Rockwell Science Center
Leonardo da Vinci Award in 1985
Fellow of the Optical Society of America, the Institute of Electrical and Electronics Engineers



page 152 of the pdf document, section Recent Advances in Photorefractive Nonlinear Optics page 4

The MPPC acts like a normal mirror and Sagnac interferometry is obtained.



Phase-Conjugate Multimode Fiber Gyro

Published in the Journal of Optics Letters, vol. 12, page 1023, 1987

page 69 of the pdf document, page 1 of the article


A second confirmation of the fact that my formula is correct.

Here is the first confirmation:



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)


Exactly the formula obtained by Professor Yeh:

φ = -2(φ₂ - φ₁) = 4π(R₁L₁ + R₂L₂)Ω/λc = 4π(V₁L₁ + V₂L₂)/λc

Since Δφ = 2πc/λ x Δt, Δt = 2(R₁L₁ + R₂L₂)Ω/c² = 2(V₁L₁ + V₂L₂)/c²

CORRECT SAGNAC FORMULA:

2(V₁L₁ + V₂L₂)/c²

The very same formula obtained for a Sagnac interferometer which features two different lengths and two different velocities.

http://www.dtic.mil/dtic/tr/fulltext/u2/a170203.pdf

ANNUAL TECHNICAL REPORT PREPARED FOR THE US OF NAVAL RESEARCH.

Page 18 of the pdf document, Section 3.0 Progress:

Our first objective was to demonstrate that the phase-conjugate fiberoptic gyro (PCFOG) described in Section 2.3 is sensitive to rotation. This phase shift plays an important role in the detection of the Sagnac phase shift due to rotation.

Page 38 of the pdf document, page 6 of Appendix 3.1


it does demonstrate the measurement of the Sagnac phase shift Eq. (3)


HERE IS EQUATION (3) OF THE PAPER, PAGE 3 OF APPENDIX 3.1:

φ = -2(φ₂ - φ₁) = 4π(R₁L₁ + R₂L₂)Ω/λc = 4π(V₁L₁ + V₂L₂)/λc

Since Δφ = 2πc/λ x Δt, Δt = 2(R₁L₁ + R₂L₂)Ω/c² = 2(V₁L₁ + V₂L₂)/c²

CORRECT SAGNAC FORMULA:

2(V₁L₁ + V₂L₂)/c²

GENERALIZED SAGNAC EFFECT PHASE DIFFERENCE AND FREQUENCY FORMULAS FOR A SQUARE RING LASER GYROSCOPE



First, the Sagnac effect formula for a square interferometer which rotates around its own geometrical center.

Let L = r√2 (r = distance from point O to one of the corners)

Time travel along side AB:

dt_ab = L/(c - v/√2)

(distance from point O to one of the sides is r/√2, and since v = r x ω, velocity for the light beam traveling along a side is v/√2)

dt_{counterclockwise} = 8r/(√2c + v)

dt_clockwise = 8r/(√2c - v)

Δt = 8rv/c²

Δφ = Δt x c/λ

Δf = Δφ x c/P

(P = perimeter = 4L)


Now, the much more difficult case for the same square ring laser interferometer located away from the center of rotation.

Let us now rotate the square interferometer by 135° in the clockwise direction: point A will be located in the uppermost position (the source of light will be placed at point A as well).

Distance from the center of rotation to point C is k₂, while the distance from the center of rotation to point A is k₁.

v₁ = k₁ x ω

v₂ = k₂ x ω

Proceeding exactly as in the case of the interferometer in the shape of a rectangle, we have two loops, one counterclockwise, one clockwise.

A > B > C > D > A is the clockwise path

A > D > C > B > A is the counterclockwise path

Sagnac phase components for the counterclockwise path (only the v_x components of the velocity vector are subject to a different time phase difference in rotation, not the v_y components):

L/(c - v₁)

-L/(c + v₂)

-L/(c + v₂)

L/(c - v₁)

Sagnac phase components for the clockwise path:

-L/(c + v₁)

L/(c - v₂)

L/(c - v₂)

-L/(c + v₁)

For the single continuous counterclockwise path we add the components:

L/(c - v₁) - L/(c + v₂) - L/(c + v₂) + L/(c - v₁) = 2L/(c - v₁) - 2L/(c + v₂)

For the single continuous clockwise path we add the components:

-L/(c + v₁) + L/(c - v₂) + L/(c - v₂) - L/(c + v₁) = -2L/(c + v₁) + 2L/(c - v₂)

The net time phase difference will be (let us remember that the counterclockwise phase difference has a negative sign attached to it, that is why the substraction of the phase differences becomes an addition):

2L/(c - v₁) - 2L/(c + v₂) -(-)[-2L/(c + v₁) + 2L/(c - v₂)] = 2L(2v₁/c²) + 2L(2v₂/c²) = 4L(v₁ + v₂)/c²

This is the correct global/generalized SAGNAC EFFECT formula for a square shaped ring laser interferometer:

4L(v₁ + v₂)/c²

For the same interferometer, the CORIOLIS EFFECT formula is:

4Aω/c²


The phase difference for the SAGNAC EFFECT is:

Δφ = Δt x c/λ = [4L(v₁ + v₂)]/c² x c/λ = [4L(v₁ + v₂)]/cλ

The frequency formula for the SAGNAC EFFECT is:

Δf = Δφ x c/P = [4L(v₁ + v₂)]/λP


There have been some questions regarding the loops of the stationary Sagnac interferometer located away from the center of rotation.

For a stationary interferometer, we simply let v = 0 in the formula:



So, there will be no time difference, l/c - l/c = 0.

In the same way, we let v₁ and v₂ = 0 in the generalized Sagnac effect formula:

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

No time difference, (l₁ + l₂)/c - (l₁ + l₂)/c = 0.

This much is evident from the derivation of the global/generalized Sagnac effect formula:

https://www.theflatearthsociety.org/forum/index.php?topic=30499.msg2117351#msg2117351

GENERALIZED SAGNAC EFFECT PHASE DIFFERENCE AND FREQUENCY FORMULAS FOR A SQUARE RING LASER GYROSCOPE II



Gran Sasso, Italy - GINGERino experiment

Latitude: 42.4166°

λ_He:Ne = 632 nm

L = 3.6 m

The formula for the square ring laser interferometer located away from the center of rotation derived in the previous message, could have been obtained directly from the global/generalized Sagnac formula, by letting l₁ = l₂ = 2L:

2(V₁L₁ + V₂L₂)/c²

In fact, we can derive the formula for a triangular shaped ring laser gyroscope from the same generalized Sagnac effect equation:

2L(2v₂ + v₁)/c²

For a triangular shaped interferometer whose center of rotation coincides with the geometrical center (equilateral triangle), all three sides will contribute to the time phase difference. If one of the vertices (A) is located right on the center of rotation, then only the side BC will be a factor in deriving the final Sagnac formula. If the center of rotation is located away from the center of rotation, then all three sides will give rise to Sagnac time phase differences.


Frequency formula for the CORIOLIS EFFECT at Gran Sasso, Italy, ring laser gyroscope:

4Aω/λP = Lω/λ

(A = L², P = 4L)

Frequency formula for the SAGNAC EFFECT at Gran Sasso, Italy, ring laser gyroscope:

[4L(v₁ + v₂)]/λP = 2v/λ

(v = Rω, since the sides of the square interferometer measure 3.6 meters in length, v₁ practically equals v₂)

2v/λ / Lω/λ = 2R/L

At the Gran Sasso latitude, R = 4,710 km = 4,710,000 meters

(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 )

L = 3.6 meters

2R/L = 2,616,666.666

The SAGNAC EFFECT frequency is larger by a factor of 2,616,666.666 times than the CORIOLIS EFFECT frequency.

As we have seen earlier, for the Michelson-Gale experiment, the SAGNAC EFFECT time phase difference is 21,000 times greater than the CORIOLIS EFFECT phase difference.

The CORIOLIS EFFECT frequency formula is not always written in its full form, which must include the conversion factor from rad/s to Hz:

https://pos.sissa.it/318/181/pdf (the 2π factor is featured in the formula)

https://www.scitepress.org/papers/2015/54380/54380.pdf (the authors do not include the 2π conversion factor)

https://bura.brunel.ac.uk/bitstream/2438/7277/1/FulltextThesis.pdf (it includes the correct derivation for the CORIOLIS EFFECT frequency formula, pg. 39-40  and 60)

The huge error introduced by Albert Michelson in 1925 has not been observed by all of the distinguished physicists who have published works on the SAGNAC EFFECT, including E.J. Post who had no idea in 1967 that he was deriving and describing the CORIOLIS EFFECT formula.

http://www.orgonelab.org/EtherDrift/Post1967.pdf

http://signallake.com/innovation/andersonNov94.pdf

https://phys.org/news/2017-03-deep-earth-rotational-effects.html

https://agenda.infn.it/event/7524/contributions/68390/attachments/49528/58554/Schreiber.pdf

https://pdfs.semanticscholar.org/47ea/33bdc7d0247772658b1e29c3e9e2a4578d17.pdf

http://inspirehep.net/record/1468904/files/JPCS_718_7_072003.pdf