Electroweak Theory (parser)

Establishment theoretical framework as the standard theory of electroweak interactions: Higgs searches, quark mixing, neutrino oscillations.

Tip

This section is referring to wiki page-25 of main section-3 that is inherited from the spin section-137 by prime spin-34 and span- with the partitions as below.

/parser

  1. Power of Magnitude
  2. Magnitude Order (spin 11)
  3. Exponentiation Zones (30-36)
    1. Electrodynamics (maps)
    2. Quantum Gravity (feed)
    3. Chromodynamics (lexer)
    4. Electroweak Theory (parser)
    5. Grand Unified Theory (syntax)
  4. Identition Zones (36-102)
    1. Theory of Everything (span 12)
    2. Everything is Connected (span 11)
    3. Truncated Perturbation (span 10)
    4. Quadratic Polynomials (span 9)
    5. Fundamental Forces (span 8)
    6. Elementary Particles (span 7)
    7. Basic Transformation (span 6)
    8. Hidden Dimensions (span 5)
    9. Parallel Universes (span 4)
    10. Vibrating Strings (span 3)
    11. Series Expansion (span 2)
    12. Wormhole Theory (span 1)

Gauge invariance is a powerful tool to determine the dynamical forces. Particle content, structure and symmetries of Lagrangian are discussed.

Standard Theory

Note

The Higgs and the electromagnetic field have no effect on each other, at the level of the fundamental forces (“tree level”), while any other combination of the hypercharge and the weak isospin must interact with the Higgs. This causes an apparent separation between the weak force, which interacts with the Higgs, and electromagnetism, which does not. (Wikipedia)

image

f22b28c976a4980061b601872e2faac8039dd7d8

images (2)

images (4)

images (3)

Experiments have verified that the weak and electromagnetic force become identical at very small distances and provide the GUT description of the carrier particles for the forces.

Interactions

images (1)

boson-particle-decay-virtual-particle-w-and-z-bosons-lepton-synchrotron-hadron-particle-physics-annihilation-scattering-thumbnail

TjQdBoIUDG

image

1

EWT3b-600x400

Figure_34_06_01

w-boson-kaon-w-and-z-bosons-weak-interaction-meson-standard-model-feynman-diagram-elementary-particle-pion-boson

weak-nuclear-force-1

  Fermion  | spinors | charged | neutrinos |   quark   | components | parameter
   Field   |   (s)   |   (c)   |    (n)    | (q=s.c.n) |  Σ(c+n+q   | (complex)
===========+=========+=========+===========+===========+============+===========
boson-1    |    ..   |    ..   |     ..    |     ..    |      5     |    i5
-----------+---------+---------+-----------+-----------+------------+-----------
boson-2    |    ..   |    ..   |     ..    |     ..    |      7     |    i7
-----------+---------+---------+-----------+-----------+------------+-----------
boson-3    |    ..   |    ..   |     ..    |     ..    |     11     |   i11
-----------+---------+---------+-----------+-----------+------------+-----------
boson-4    |    ..   |    ..   |     ..    |     ..    |     13     |   i13
-----------+---------+---------+-----------+-----------+------------+-----------
boson-5    |    ..   |    ..   |     ..    |     ..    |     17     |   i17
===========+=========+=========+===========+===========+============+===========
  SubTotal |    ..   |    ..   |     ..    |     ..    |     53     |   i53
===========+=========+=========+===========+===========+============+===========
bispinor-1 |    2    |    3    |     3     |    18     |     24     |   19
-----------+---------+---------+-----------+-----------+------------+-- 17
bispinor-2 |    2    |    3    |     3     |    18     |     24     |   i12
===========+=========+=========+===========+===========+============+===========
bispinor-3 |    2    |    3    |     3     |    18     |     24     |   11
-----------+---------+---------+-----------+-----------+------------+-- 19
bispinor-4 |    2    |    3    |     3     |    18     |     24     |   i18
===========+=========+=========+===========+===========+============+===========
  SubTotal |    8    |   12    |    12     |    72     |     96     |  66+i30
===========+=========+=========+===========+===========+============+===========
majorana-1 |   2x2   |    -    |    18     |     -     |     18     |   18
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-2 |   2x2   |    -    |    12     |     -     |     12     |   12
-----------+---------+---------+-----------+-----------+------------+-----------
majorana-3 |   2x2   |    -    |    13     |     -     |     13     |   i13
===========+=========+=========+===========+===========+============+===========
  SubTotal |    12   |    -    |    43     |     -     |     43     |  30+i13
===========+=========+=========+===========+===========+============+===========
     Total |    20   |   12    |    55     |    72     |    192     |  96+i96 ✔️

Symmetry Breaking

Note

The pattern of weak isospin, T3, and weak hypercharge, YW, of the known elementary particles, showing electric charge along the weak mixing angle. The four components of the Higgs field (squares) break the electroweak symmetry and interact with other particles to give them mass, with three components becoming part of the massive W and Z bosons. Allowed decays of the neutral Higgs boson, H, (circled) satisfy electroweak charge conservation. (Wikipedia)

Electroweak svg (2)

The Lagrangian for the electroweak interactions is divided into four parts before electroweak symmetry breaking becomes manifest,

$True Prime Pairs:
(5,7$True Prime Pairs:
(5,7), (11,13), (17,19)

     |    168    |    618    |
-----+-----+-----+-----+-----+                                             ---
 19¨ |  3¨ |  4¨ |  6¨ |  6¨ | 4¤  ----->  assigned to "id:30"             19¨
-----+-----+-----+-----+-----+                                             ---
 17¨ | {5¨}| {3¨}|  2¨ |  7¨ | 4¤  ----->  assigned to "id:31"              |
     +-----+-----+-----+-----+                                              |
{12¨}|  6¨ |  6¨ |  2¤ (M & F)     ----->  assigned to "id:32"              |
     +-----+-----+-----+                                                    |
 11¨ |  3¨ | {3¨}| {5¨}| 3¤  --->  Np(33)  assigned to "id:33"  ----->  👉 77¨
-----+-----+-----+-----+-----+                                              |
 19¨ |  4¨ |  4¨ |  ❓ |  ❓ | 4¤ ✔️ --->  assigned to "id:34"              |
     +-----+-----+-----+-----+                                              |
{18¨}|  .. |  .. |  .. | 3¤        ----->  assigned to "id:35"              |
     +-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
 43¨ |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. | 9¤ (C1 & C2)  43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
139¨ |  1     2     3  |  4     5     6  |  7     8     9  |
                    Δ                 Δ                 Δ       

Beta-minus_Decay svg

Unlike the strong and electromagnetic forces, the intermediary particles of the weak interaction, the W+, the W-, and the Z0 have rather large masses.

Note

A key aspect of the theory is the explanation of why three out of four of the intermediary particles of the electroweak force are massive. Illustration of two weak reactions.

  • The left panel shows beta decay while the middle panel shows how electron antineutrinos can be detected by conversion to a positron.
  • The right panel shows how W- emission works according to the quark model, resulting in the conversion of a down quark to an up quark and the resulting transformation of a neutron into a proton.

The real reason for the apparent weakness of the weak force is the large mass of the intermediary particles. As we have seen, large mass translates into short range for a virtual particle at low momentum transfers. This short range is what causes the weak force to appear weak for momentum transfers much less than the masses of the W and Z particles. (libre texts.org)

Beta decay

$True Prime Pairs:
(5,7$True Prime Pairs:
(5,7), (11,13), (17,19)

     |    168    |    618    |
-----+-----+-----+-----+-----+                                             ---
 19¨ |  3¨ |  4¨ |  6¨ |  6¨ | 4¤  ----->  assigned to "id:30"             19¨
-----+-----+-----+-----+-----+                                             ---
 17¨ | {5¨}| {3¨}|  2¨ |  7¨ | 4¤  ----->  assigned to "id:31"              |
     +-----+-----+-----+-----+                                              |
{12¨}|  6¨ |  6¨ |  2¤ (M & F)     ----->  assigned to "id:32"              |
     +-----+-----+-----+                                                    |
 11¨ |  3¨ | {3¨}| {5¨}| 3¤  --->  Np(33)  assigned to "id:33"  ----->  👉 77¨
-----+-----+-----+----+-----+                                              |
 19¨ |  4¨ |  4¨ |  5¨ |  6¨ | 4¤ ✔️ --->  assigned to "id:34"              |
     +-----+-----+-----+-----+                                              |
{18¨}|  .. |  .. |  .. | 3¤        ----->  assigned to "id:35"              |
     +-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
 43¨ |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. | 9¤ (C1 & C2)  43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
139¨ |  1     2     3  |  4     5     6  |  7     8     9  |
                    Δ                 Δ                 Δ       

Problem

Note

Consider the following contradiction in the electroweak theory of the Standard Model.

The electroweak theory of neutrino interaction uses factors like in order to account for a complete parity violation. This factor implies a massless neutrino [1]: “Nature had the choice of an aesthetically satisfying, but a left-right, symmetry violating theory, with a neutrino which travels exactly with the same velocity of light; or alternatively a theory where left-right symmetry is preserved, but the neutrino has a tiny mass – some ten thousand times smaller than the mass of the electron.”The neutrino masslessness is also stated by other authors. A review article on neutrino properties states that “two-components left-handed massless neutrino fields play a crucial role in the determination of the charged current structure of the Standard Model” (see the Abstract of [2]). Similarly, a Quantum Field Theory textbook states: “Thus, massless neutrinos are a prediction of the Standard Model” (see [4], p. 555). Indeed, a massless neutrino is the basis for the two-component Weyl neutrino, which shows parity violation (see e.g. section 2.2 of [2]). The same argument appears on p. 139 of [3].

On the other hand, a recent review article negates the foregoing ides and states that it is now admitted “that neutrinos can no longer be considered as massless particles” (see [5], p. 1307). This statement is adopted by the Particle Data Group [6], which is the authorized organization for the definition of reliable particle data. The recognition of this fact by the community was demonstrated by the 2015 Nobel Prize, awarded to the persons who have discovered this property [7].It follows that the experimentally confirmed massive neutrino undermines the basis of the Standard Model electroweak theory, since the massless neutrino is a crucial element in this theory.

Research topic: Can the validity of the electroweak theory be restored?

Remark: Further contradictions are discussed in [8]. (Research Topics)

A Problem with the Electroweak Theory

The True Prime Pairs
(5,7), (11,13), (17,19)

Tabulate Prime by Power of 10
loop(10) = π(10)-π(1) = 4-0 = 4
loop(100) = π(100)-π(10)-1th = 25-4-2 = 19
loop(1000) = π(1000) - π(100) - 10th = 168-25-29 = 114

--------------------------+----+----+----+----+----+----+----+----+----+-----
 True Prime Pairs → Δ→π  |  1 |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 | Sum 
==========================+====+====+====+====+====+====+====+====+====+=====
 19 → π(∆10) → π(10)     |  2 |  3 |  5 |  7 |  - |  - |  - |  - |  - | 4th  4 x Root
--------------------------+----+----+----+----+----+----+----+----+----+-----
 17 → π(10+∆9) → π(19)   | 11 | 13 | 17 | 19 |  - |  - |  - |  - |  - | 8th  4 x Twin
==========================+====+====+====+====+====+====+====+====+====+===== 1st Twin
 13 → π(19+∆10) → π(29)  | 23 | 29 |  - |  - |  - |  - |  - |  - |  - |10th
--------------------------+----+----+----+----+----+----+----+----+----+-----
 11 → π(29+∆12) → π(41)  | 31 | 37 | 41 |  - |  - |  - |  - |  - |  - |13th
==========================+====+====+====+====+====+====+====+====+====+===== 1st Twin
  7 → π(41+∆18) → π(59)  | 43 | 47 | 53 | 59 |  - |  - |  - |  - |  - |17th
--------------------------+----+----+----+----+----+----+----+----+----+----- 3rd Twin
  5 → π(59+∆13) → π(72)  | 61 | 67 | 71 |  - |  - |  - |  - |  - |  - |20th
==========================+====+====+====+====+====+====+====+====+====+===== 4th Twin
  3,2 → 18+13+12 → 43    | 73 | 79 | 83 | 89 | 97 | 101| 103| 107| 109|29th 
==========================+====+====+====+====+====+====+====+====+====+=====
         Δ                                                            Δ
12+13+(18+18)+13+12   ← 36th-Δ1=151-1=150=100+2x(13+12)   ←   30th = 113 = 114-

How do you resolve Maxwell equations as euler-lagrange equation without electromagnetic electromagnetism, lagrangian formalism, field theory, Maxwell equations, variational principle potential.

Note

Axial (e-e rES repulsions blue aggregating to black axial outward, vs weak axial inward) to generate the Bose “cylinder surface” proof of statistical mechanics.

  • Axial View of one hemisphere set of one subshell (N,1,many,-1/2) quantum number example below:
  • That gives the path from Planck strength to the Maxwell strengths. Those are not independent, but all based upon h (or h-hat*c version in this case).
  • Yes, I used Euler to get there! The weakness of the Lagrangian is that introduces errors in (a0/re)N scaling ^2 vs ^3 (extra 1/r wrongly called angular momentum by Bohr) that introduces an error correction. Hence, circling back to QED methods of error-correction (loops, re-normalization).

So, in the end, you do need. But the path can get similar when you move off arbitration x,y,z or X1,X2,X3 frame-of-reference to the quantitized hemispherical coordinates of the quantum numbers understood as (r#,theta#,phi#,z#).

main-qimg-521a032d4132a419487624564dd201b2-pjlq

main-qimg-5f05266cfdc63d60f86ad0852076ee00

1729 = 7 x 13 x 19
1729 / 7 = 13 x 19 = 247

1729 = 7 x 13 x 19
       7 + 13 = 20 = d(2)
                     └──  2 x 19 = 38

+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
| {1}|  2 |  3 |  4 |  5 | {6}| {7}|  8 |  9 | 10 | 11 | 12 | 13 | 14 |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
| {3}| {4}|  3 |  4 |  5 |  2 |  3 |  2 |  2 |  1 |  2 |  5 |  1 |  1 |{38}
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+---- } 285
|  3 |  8 |  9 | 16 | 25 |{12}|{21}| 16 | 18 | 10 | 22 | 60 |{13}|{14}|{247}
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
|-- 38 ---|              |-- 33 ---|                        |-- {27}--|

1591890434759 (1)

$True Prime Pairs:
(5,7$True Prime Pairs:
(5,7$True Prime Pairs:
(5,7), (11,13), (17,19)

     |    168    |    618    |
-----+-----+-----+-----+-----+                                             ---
 19¨ |  3¨ |  4¨ |  6¨ |  6¨ | 4¤  ----->  assigned to "id:30"             19¨
-----+-----+-----+-----+-----+                                             ---
 17¨ | {5¨}| {3¨}|  2¨ |  7¨ | 4¤  ----->  assigned to "id:31"              |
     +-----+-----+-----+-----+                                              |
{12¨}|  6¨ |  6¨ |  2¤ (M & F)     ----->  assigned to "id:32"              |
     +-----+-----+-----+                                                    |
 11¨ |  3¨ | {3¨}| {5¨}| 3¤  --->  Np(33)  assigned to "id:33"  ----->  👉 77¨
-----+-----+-----+-----+-----+                                              |
 19¨ |  4¨ |  4¨ |  5¨ |  6¨ | 4¤ ✔️ --->  assigned to "id:34"              |
     +-----+-----+-----+-----+                                              |
{18¨}|  .. |  .. |  .. | 3¤        ----->  assigned to "id:35"              |
     +-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
 43¨ |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. |  .. | 9¤ (C1 & C2)  43¨
-----+-----+-----+-----+-----+-----+-----+-----+-----+-----+               ---
139¨ |  1     2     3  |  4     5     6  |  7     8     9  |
                    Δ                 Δ                 Δ       

electron orbit

True Prime Pairs:
(5,7), (11,13), (17,19)

     |    168    |    618    | ✔️
-----+-----+-----+-----+-----+     -----------------------------------------------
{786}| 1,2 |  2  | 2,3 | 3,4 | {19}                                          |
-----+-----+-----+-----+-----+                                               |
 {86}|  4  | 4,5 | 5,6 |{6,7}| 17                                        Base Zone
     +-----+-----+-----+-----+                                               |
 {78}|{7,8}| 8,9 | 12 (M dan F) ----> Δ                                      |
     +-----+-----+-----+                                               -----------
 {67}| 9,11|11,12|12,14| 11 <----------- Mid Zone                            |
 ----+-----+-----+-----+-----+                                               |
  {6}|15,16|17,18|18,20|21,22| 19                                      Mirror Zone
     +-----+-----+-----+-----+                                               |
  {8}|23,25|25,27|27,29| 18                                                  |
     +-----+-----+-----+-----+-----+-----+-----+-----+-------+         -----------
  {7}|29,33|33,36|36,39|39,41|41,45|46,51|51,57|58,66|{67,77}| 43 (C1 dan C2)<---Δ
-----+-----+-----+-----+-----+-----+-----+-----+-----+-------+         -----------
     |  1     2     3  |   4     5     6 |   7     8      9  |
     |------ 29' ------|--------------- 139' ----------------|
     |------ 618¨ -----|--------------- 168¨ ----------------|

IMG_20240118_121014

Prime Identity

We are going to assign prime identity as a standard model that attempts to stimulate a quantum field model called eQuantum for the four (4) known fundamental forces.

Tip

This section is referring to wiki page-25 of main section-3 that is inherited from the spin section-137 by prime spin-34 and span- with the partitions as below.

/parser

  1. Power of Magnitude
  2. Magnitude Order (spin 11)
  3. Exponentiation Zones (30-36)
    1. Electrodynamics (maps)
    2. Quantum Gravity (feed)
    3. Chromodynamics (lexer)
    4. Electroweak Theory (parser)
    5. Grand Unified Theory (syntax)
  4. Identition Zones (36-102)
    1. Theory of Everything (span 12)
    2. Everything is Connected (span 11)
    3. Truncated Perturbation (span 10)
    4. Quadratic Polynomials (span 9)
    5. Fundamental Forces (span 8)
    6. Elementary Particles (span 7)
    7. Basic Transformation (span 6)
    8. Hidden Dimensions (span 5)
    9. Parallel Universes (span 4)
    10. Vibrating Strings (span 3)
    11. Series Expansion (span 2)
    12. Wormhole Theory (span 1)

This presentation was inspired by theoretical works from Hideki Yukawa who in 1935 had predicted the existence of mesons as the carrier particles of strong nuclear force.

Addition Zones

Here we would like to recompile the way we take on getting the arithmetic expresion of an individual unit expression (identity) such as a taxicab number below.

Note

It is a taxicab number, and is variously known as Ramanujan’s number and the Ramanujan-Hardy number, after an anecdote of the British mathematician GH Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital (Wikipedia).

Ramanujan-Hardy number

These three (3) number are twin primes. We called the pairs as True Prime Pairs. Our scenario is mapping the distribution out of these pairs by taking the symmetrical behaviour of 36 as the smallest power (greater than 1) which is not a prime power.

Tip

The smallest square number expressible as the sum of four (4) consecutive primes in two ways (5 + 7 + 11 + 13 and 17 + 19) which are also two (2) couples of prime twins! (Prime Curios!).

$True Prime Pairs:
 (5,7), (11,13), (17,19)
 
 layer|  i  |   f
 -----+-----+---------
      |  1  | 5
   1  +-----+
      |  2  | 7
 -----+-----+---  } 36 » 6®
      |  3  | 11
   2  +-----+
      |  4  | 13
 -----+-----+---------
      |  5  | 17
   3  +-----+     } 36 » 6®
      |  6  | 19
 -----+-----+---------

Thus in short this is all about the method that we called as the 19 vs 18 Scenario of mapping the quantum way within a huge of primes objects (5 to 19) by lexering (11) the ungrammared feed (7) and parsering (13) across syntax (17).

Φ(1,2,3) = Φ(6,12,18) = Φ(13,37,61)

$True Prime Pairs:
(5,7), (11,13), (17,19)
 
layer | node | sub |  i  |  f
------+------+-----+----------
      |      |     |  1  | 
      |      |  1  +-----+          
      |  1   |     |  2  | (5)
      |      |-----+-----+
      |      |     |  3  |
  1   +------+  2  +-----+----
      |      |     |  4  |
      |      +-----+-----+
      |  2   |     |  5  | (7)
      |      |  3  +-----+
      |      |     |  6  |
------+------+-----+-----+------      } (36)
      |      |     |  7  |
      |      |  4  +-----+
      |  3   |     |  8  | (11)
      |      +-----+-----+
      |      |     |  9  |
  2   +------|  5  +-----+-----
      |      |     |  10 |
      |      |-----+-----+
      |  4   |     |  11 | (13)
      |      |  6  +-----+
      |      |     |  12 |
------+------+-----+-----+------------------
      |      |     |  13 |
      |      |  7  +-----+
      |  5   |     |  14 | (17)
      |      |-----+-----+
      |      |     |  15 |
  3   +------+  8  +-----+-----       } (36)
      |      |     |  16 |
      |      |-----+-----+
      |  6   |     |  17 | (19)
      |      |  9  +-----+
      |      |     |  18 |
------|------|-----+-----+------

The main background is that, as you may aware, the prime number theorem describes the asymptotic distribution of prime numbers which is still a major problem in mathematic.

Multiplication Zones

Instead of a proved formula we came to a unique expression called zeta function. This expression first appeared in a paper in 1737 entitled Variae observationes circa series infinitas.

Tip

This expression states that the sum of the zeta function is equal to the product of the reciprocal of one minus the reciprocal of primes to the powers. But what has this got to do with the primes? The answer is in the following product taken over the primes p (discovered by Leonhard Euler):

zeta function

This issue is actually come from Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that is considered to be the most important of unsolved problems in pure mathematics.

Note

In addition to the trivial roots, there also exist complex roots for real t. We find that the he first ten (10) non-trivial roots of the Riemann zeta function is occured when the values of t below 50. A plot of the values of ζ(1/2 + it) for t ranging from –50 to +50 is shown below. The roots occur each time the locus passes through the origin. (mathpages).

trivial roots

Meanwhile obtaining the non complex numbers it is easier to look at a graph like the one below which shows Li(x) (blue), R(x) (black), π(x) (red) and x/ln x (green); and then proclaim "R(x) is the best estimate of π(x)." Indeed it is for that range, but as we mentioned above, Li(x)-π(x) changes sign infinitely often, and near where it does, Li(x) would be the best value.

non complex numbers

And we can see in the same way that the function Li(x)-(1/2)Li(x1/2) is ‘on the average' a better approximation than Li(x) to π(x); but no importance can be attached to the latter terms in Riemann's formula even by repeated averaging.

Exponentiation Zones

The problem is that the contributions from the non-trivial zeros at times swamps that of any but the main terms in these expansions.

Warning

A. E. Ingham says it this way: Considerable importance was attached formerly to a function suggested by Riemann as an approximation to π(x)… This function represents π(x) with astonishing accuracy for all values of x for which π(x) has been calculated, but we now see that its superiority over Li(x) is illusory… and for special values of x (as large as we please) the one approximation will deviate as widely as the other from the true value (primes.utm.edu).

howmany primes

Moreover in it was verified numerically, in a rigorous way using interval arithmetic, that The Riemann hypothesis is true up to 3 · 10^12. That is, all zeroes β+iγ of the Riemann zeta-function with 0<γ≤3⋅1012 have β=1/2.

Danger

We have Λ ≤ 0.2. The next entry is conditional on taking H a little higher than 10*13, which of course, is not achieved by Theorem 1. This would enable one to prove Λ < 0.19. Given that our value of H falls between the entries in this table, it is possible that some extra decimals could be wrought out of the calculation. We have not pursued this (arXiv:2004.09765).

functional equation

This Euler formula represents the distribution of a group of numbers that are positioned at regular intervals on a straight line to each other. Riemann later extended the definition of zeta(s) to all complex numbers (except the simple pole at s=1 with residue one). Euler's product still holds if the real part of s is greater than one. Riemann derived the functional equation of zeta function.

Danger

The Riemann zeta function has the trivial zeros at -2, -4, -6, … (the poles of gamma(s/2)). Using the Euler product (with the functional equation) it is easy to show that all the other zeros are in the critical strip of non-real complex numbers with 0 < Re(s) < 1, and that they are symmetric about the critical line Re(s)=1/2. The unproved Riemann hypothesis is that all of the nontrivial zeros are actually on the critical line (primes.utm.edu).

zeta function

If both of the above statements are true then mathematically this Riemann Hypothesis is proven to be incorrect because it only applies to certain cases or limitations. So first of all the basis of the Riemann Hypothesis has to be considered.

Warning

The solution is not only to prove Re(z)= 1/2 but also to calculate ways for the imaginary part of the complex root of ζ(z)=0 and also to solve the functional equations. (Riemann Zeta - pdf)

Riemann hypothesis

On the other hand, the possibility of obtaining the function of the distribution of prime numbers shall go backwards since it needs significant studies to be traced.

Or may be start again from the Euleur Function.

Identition Zones

Freeman Dyson discovered an intriguing connection between quantum physics and Montgomery's pair correlation conjecture about the zeros of the zeta function which dealts with the distribution of primes.

Note

The Mathematical Elementary Cell 30 (MEC30) standard unites the mathematical and physical results of 1972 by the mathematician Hugh Montgomery and the physicist Freeman Dyson and thus reproduces energy distribution in systems as a path plan more accurately than a measurement. (Google Patent DE102011101032A9)

The Mathematical Elementary Cell 30

The path plan assume that a symmetric distribution of prime numbers with equal axial lengths from a middle zero axis = 15 is able to determine the distribution of primes in a given number space. This assumption finally bring us to the equation of Euler's identity.

Note

Euler’s identity is considered to be an exemplar of deep mathematical beauty as it shows a profound connection between the most fundamental numbers. Three (3) of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation (Wikipedia).

Euler's identity

The finiteness position of Euler's identity by the said MEC30 opens up the possibility of accurately representing the self-similarity based on the distribution of True Prime Pairs so that all number would belongs together with their own identitities.

Tip

Several aspects of torsion in string-inspired cosmologies are reviewed. In particular, its connection with fundamental, string-model independent, axion fields associated with the massless gravitational multiplet of the string are discussed.

  • It is argued in favour of the role of primordial gravitational anomalies coupled to such axions in inducing inflation of a type encountered in the Running-Vacuum-Model (RVM) cosmological framework, without fundamental inflaton fields.15-Figure1-1
  • The gravitational-anomaly terms owe their existence to the Green–Schwarz mechanism for the (extra-dimensional) anomaly cancellation, and may be non-trivial in such theories in the presence of (primordial) gravitational waves at early stages of the four (4) dimensional string universe (after compactification).Torsion in String Cosmologies
  • The paper also discusses how the torsion-induced stringy axions can acquire a mass in the post inflationary era, due to non-perturbative effects, thus having the potential to play the role of (a component of) dark matter in such models.triangular wave

Finally, the current-era phenomenology of this model is briefly described with emphasis placed on the possibility of alleviating tensions observed in the current-era cosmological data. A brief phenomenological comparison with other cosmological models in contorted geometries is also made. (Torsion in String Cosmologies - pdf)

DE102011101032A9.pdf

Nothing is going to be easly about the nature of prime numbers but they demonstrably congruent to something organized. Let's discuss starting with the addition zones.

eQuantum Project
Copyright © 2023-2024

Reference: