Proof of Riemann Hypothesis

[Guest]

A Recap On Why The Riemann Hypothesis Is True:

There are infinitely many nontrivial and simple zeros of the complex Riemann Zeta Function,

[tex]\zeta(z) = \sum_{k=1}^{\infty }\frac{1}{k^{z}}[/tex],

whose part equals one-half or Re(z)= 1/2, and there are infinitely many primes. Every nontrivial and simple zero has a unique simple prime associated with it according to the following important equation:

[tex]f(n, p_n, z_n) =[/tex] [tex]\sum_{∀m\ge 1\ s.t. \ gcd(p_n, m)=1}^{\infty} m^{-z_n} = 0[/tex]

where [tex]z_n[/tex] is the nth nontrivial and simple zero of the Riemann Zeta Function, and [tex]p_n[/tex] is the nth simple prime.

Moreover, the wonderful Harmonic Series is the prime source of analysis for the great results, the Riemann Hypothesis and the Prime Number Theorem...

David Cole.

Note: The important complex equation,

[tex]f(n, p_n, z_n) =[/tex] [tex]\sum_{∀m\ge 1\ s.t. \ gcd(p_n, m)=1}^{\infty} m^{-z_n} = 0[/tex],

is derived from the complex Riemann Zeta Function.

[Guest]

There is a great deal of important mathematical literature which depends on the assumption of the Riemann Hypothesis. And that is the right assumption to make since the Riemann Hypothesis is true!

In Lord GOD and in the Riemann Hypothesis, we trust! Amen!

[Guest]

There is a great deal of important mathematical literature which depends on the assumption of the Riemann Hypothesis. And that is the right assumption to make since the Riemann Hypothesis is true!

In Lord GOD and in the Riemann Hypothesis, we trust! Amen!

Relevant Reference Link:

'Why is RH optimum?',

https://www.math10.com/forum/viewtopic.php?f=63&t=8042

[Guest]

Here's some for thought, possible proof of the Riemann Hypothesis:

'A Potential Elementary Proof Of The Riemann Hypothesis' by Prof. A.Garcés Doz.

Please see attached file for its interesting details.

[Guest]

Here's some for thought, possible proof of the Riemann Hypothesis:

'A Potential Elementary Proof Of The Riemann Hypothesis' by Prof. A.Garcés Doz.

Please see attached file for its interesting physical details.

Some more food for thought, possible proof of the Riemann Hypothesis:

'The Riemann Zeros as Spectrum and the Riemann Hypothesis' by Prof. Germán Sierra.

Please see attached file for its interesting physical details.

[Guest]

Proof is already here.

https://www.linkedin.com/pulse/finding- ... hung-tran/

In quantum mechanics, the eigenvalues of a Hamiltonian represent the stable energy
levels of a system. The values for the energy levels are always real numbers.
This has lead researchers to believe that studying this Hamiltonian, could lead
to a proof of the Riemann hypothesis, because for these eigenvalues can
only be real when Re(s) = 1/2.

One of the problems in moving forward from here, is that this Hamiltonian
is not self-adjoint. If H were self-adjoint, then that would ensure the reality of
the eigenvalues. If ϕ is a normalizable eigenfunction of a self-adjoint Hamilto-
nian H with corresponding eigenvalue λ, then the eigenvalue has to be real.

[Guest]

FYI: 'Old and new arithmetic and analytic equivalences of the Riemann hypothesis' by Prof. Kevin Broughan are excellent.

Relevant Reference Links:

https://www.math.waikato.ac.nz/~kab/ERH/icm2018poster.pdf;

https://www.math.waikato.ac.nz/~kab/ERH/EquivRH.html

[Guest]

FYI: 'What does proving the Riemann Hypothesis accomplish?',

https://math.stackexchange.com/questions/404624/what-does-proving-the-riemann-hypothesis-accomplish?rq=1.

[Guest]

FYI: 'Here’s why we care about attempts to prove the Riemann hypothesis',

https://www.sciencenews.org/article/why-we-care-riemann-hypothesis-math-prime-numbers.

[Guest]

At last I proved Riemann hypothesis. Please have a look.

[Guest]

Relevant Reference Link:

'Riemann Zeta Function Implies the Harmonic Series....',

https://www.math10.com/forum/viewtopic.php?f=63&t=8577.

[Guest]

An explicit function is found, that accurately shows the non-trivial roots of the Riemann Zeta function and their alternation with the roots of its derivative - see the scientific forum "Riemann Hypothesis": https://dxdy.ru/topic20981-615.html and the continuation in the telegram channel t.me/ZetaRH !

[Guest]

FYI: 'The Riemann Hypothesis and distribution of the primes' by Prof. N. Mantzakouras,

https://www.researchgate.net/publication/355032015_The_Riemann_Hypothesis_and_distribution_of_the_primes.

Enjoy!

[Guest]

I guess this is the Proof.
https://www.hrpub.org/journals/article_ ... ?aid=10710

[hatem]

Please check the proof at:
https://www.youtube.com/watch?v=pXmAbiTcvFM&t=12s
https://www.researchgate.net/publicatio ... Hypothesis

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an exp[*]ression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...
A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex] with (0 < error < 1)

if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{N}\frac{1}{(kp)^{ \frac{1}{2} ± bi}} + \gamma(\frac{1}{2} \mp bi) + \sum_{k=1}^{M}\frac{1}{(kp)^{ \frac{1}{2} \mp bi}} + R( \frac{1}{2} ± bi) = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Relevant Reference Link:

[url=https://en.m.wikipedia.org/wiki/Riemann%E2%80%Siegel_formula]Riemann Siegel formula]

Remark: The second equation is tentative since I do not completely understand all its details and since I fooled myself before. That could happen again. And I am very likely to make minor or important mistakes, too.

We must admit that our second equation is extraordinary, beautiful, and very promising. It must be correct!

Dave.

Relevant Reference Link:
[url]http://www.math10.com/forum/viewtopic.phpf=63&t=8042&start=80[/url]
Why is RH optimum?

[Guest]

Relevant Reference Links:

Riemann-Siegel formula

Why is RH optimum?

[Guest]

Relevant Reference Links:

Riemann-Siegel formula

Why is RH optimum?

Remark:

The great RH is true! Why?

The Riemann Hypothesis (RH) is perfect for detecting both, all simple nontrivial zeros, 1/2 ± bi, of the Riemann zeta function and all positive prime numbers (p),
ζ(1/2 ± bi, p) = 0.

[Guest]

Relevant Reference Links:

...

Why is RH optimum?

Remark:

The great RH is true! Why?

The Riemann Hypothesis (RH) is perfect for detecting both, all simple nontrivial zeros, 1/2 ± bi, of the Riemann zeta function and all positive prime numbers (p),
ζ(1/2 ± bi, p) = 0.

Remark: For every b value that solves that equation, there's an unique prime number (p) associated with it.