Proof of Riemann Hypothesis

[Guest]

Read "The Proof of the Riemann Hypothesis on a Relativistic
Turing Machine" published in the International Journal of Theoretical and Applied Mathematics
2017; 3(6): 219-224
http://www.sciencepublishinggroup.com/j/ijtam
doi: 10.11648/j.ijtam.20170306.17

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Does the truth of the Riemann hypothesis confirm the existence of prime numbers in an optimum way (via a minimum or maximum value)?

Please see the answer given by David Cole at links below:

https://www.quora.com/profile/David-Cole-146

https://www.quora.com/profile/David-Cole-146/Posts/What-does-the-Riemann-Hypothesis-mean

[Guest]

On the Riemann Hypothesis:

Relevant Reference Link (Q & A):

https://www.quora.com/profile/David-Cole-146/answers/Riemann-Hypothesis

[Guest]

Does [tex]f(n, p_l, z_n) =[/tex] [tex]\sum_{∀m∈N s.t. gcd(p_n,m)=1}m^{-z_n} = 0?[/tex]

Does [tex]h(l, n\ne l, p_l, z_n) =[/tex] [tex]\sum_{∀m∈N s.t. gcd(p_l,m)=1}m^{-z_n} = 0?[/tex]

Note: [tex]p_n[/tex], [tex]p_l[/tex] and [tex]z_n[/tex] are the nth prime, the lth prime, and the nth nontrivial zeta zero of the Riemann Zeta Function, respectively. We assume the Riemann Hypothesis is true. And of course, m is any nonzero positive integer. And N is the set of all positive nonzero integers.

Reference Link to a List of Nontrivial Zeta Zeros:

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

[Guest]

Does [tex]f(n, p_l, z_n) =[/tex] [tex]\sum_{∀m∈N s.t. gcd(p_n,m)=1}m^{-z_n} = 0?[/tex]

Does [tex]h(l, n\ne l, p_l, z_n) =[/tex] [tex]\sum_{∀m∈N s.t. gcd(p_l,m)=1}m^{-z_n} = 0?[/tex]

Note: [tex]p_n[/tex], [tex]p_l[/tex] and [tex]z_n[/tex] are the nth prime, the lth prime, and the nth nontrivial zeta zero of the Riemann Zeta Function, respectively. We assume the Riemann Hypothesis is true. And of course, m is any nonzero positive integer. And N is the set of all positive nonzero integers.

Reference Link to a List of Nontrivial Zeta Zeros:

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

Correction!

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

[Guest]

Some Important Observations about functions f and g where

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

[tex]g(n, x \in N[/tex] and x is not prime, [tex]z_n[/tex]) = [tex]\sum_{∀m∈N s.t. x \nmid m }m^{-z_n}[/tex]

The set [tex]M_f[/tex] = {[tex]m\in N |gcd(p_n, m) =1[/tex]}. And therefore,

[tex]M_f[/tex]= {1, 2, ..., [tex]p_n[/tex]-1, [tex]p_n[/tex]+1, ..., 2[tex]p_n[/tex]-1, 2[tex]p_n[/tex]+1, ..., k[tex]p_n[/tex] -1, k[tex]p_n[/tex]+1, ...} for some positive integer k > 2.

[tex]M_f[/tex] is of course an infinite set.

The set [tex]M_g[/tex] = {m∈N and x is not prime|[tex]x \nmid m[/tex]}. And therefore,

[tex]M_g[/tex] = {1, 2, ..., x-1, x+1, ..., 2x-1, 2x+1, ..., kx -1, kx+1, ...} for some positive integer k > 2.

[tex]M_g[/tex] is of course an infinite set too.

However, gcd(x, [tex]m\in M_g[/tex]) [tex]\ne[/tex] 1 for infinitely many elements, m, in [tex]M_g[/tex] since x is a product of two or more primes.

Thus, the set [tex]M_f[/tex] is more relatively prime than the set [tex]M_g[/tex] with respective to gcd([tex]p_n[/tex], [tex]m\in M_f[/tex]) and gcd(x, [tex]m\in M_g[/tex]), respectively.

[Guest]

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

then could that result be because of the uniqueness property, [tex]p_n[/tex] exists if and only if [tex]z_n[/tex] exists, and because of the greatest degree of relative primeness of set [tex]M_f[/tex] ...?

And according to Hardy's Theorem: There are infinitely many nontrivial zeta zeros of the Riemann Zeta Function whose real part equals 1/2.

Can we conclude that the Riemann Hypothesis, Re([tex]z_n[/tex]) = 1/2, is true?

Relevant Reference Link:

https://en.wikipedia.org/wiki/Riemann_hypothesis

[Guest]

The Riemann Hypothesis is true! Thank Lord God! Amen!

[Guest]

Hmm. In our 'number theory of relativity', the infinite and ordered set, [tex]M_g[/tex], has a very rich and endless complexity beyond the 'ground state' of the infinite and ordered set, [tex]M_f[/tex]. Please ponder the very important Fundamental Theorem of Arithmetic...

[Guest]

Happy Birthday (September 17)! G. F. B. Riemann!

Romans 8:28 (KJV):

"And we know that all things work together for good to them that love God, to them who are the called according to his purpose." Amen!


Relevant Reference Link:

https://en.wikipedia.org/wiki/Bernhard_Riemann

[Guest]

International Journal of Theoretical and Applied Mathematics
2017; 3(6): 219-224[/b]
http://www.sciencepublishinggroup.com/j/ijtam
doi: 10.11648/j.ijtam.20170306.17
ISSN: 2575-5072 (Print); ISSN: 2575-5080 (Online)
The Proof of the Riemann Hypothesis on a Relativistic
Turing Machine
Yuriy N. Zayko
Department of Applied Informatics, Faculty of Public Administration, The Russian Presidential Academy of National Economy and Public
Administration, Saratov, Russia
Email address:
zyrnick@rambler.ru
To cite this article:
Yuriy N. Zayko. The Proof of the Riemann Hypothesis on a Relativistic Turing Machine. International Journal of Theoretical and Applied
Mathematics. Vol. 3, No. 6, 2017, pp. 219-224. doi: 10.11648/j.ijtam.20170306.17
Received: October 2, 2017; Accepted: November 13, 2017; Published: January 2, 2018
Abstract: In this article, the proof of the Riemann hypothesis is considered using the calculation of the Riemann ζ-function
on a relativistic computer. The work lies at the junction of the direction known as "Beyond Turing", considering the application
of the so-called "relativistic supercomputers" for solving non-computable problems and a direction devoted to the study of
non-trivial zeros of the Riemann ζ-function. Considerations are given in favor of the validity of the Riemann hypothesis with
respect to the distribution of non-trivial zeros of the ζ-function.

[Guest]

"Don't pay attention to "authorities," think for yourself."

-- Richard P. Feynman, a great scientist.

[Guest]

Keywords: Fundamental Theorem of Arithmetic and the Riemann Hypothesis

Basic and Important Fact Confirming RH (Riemann Hypothesis or Right Hypothesis, Re(z) = 1/2) where z is any nontrivial and simple zero of the Riemann Zeta Function, [tex]\sum_{n=1}^{\infty }n^{-z} = 0[/tex]:

For all n > 1, where n is any positive composite number, there exists a prime number, p, such that p|n (p divides n)
where p [tex]≤ n^{1/2}[/tex] (p is less than or equal to the square root of n).

And thus, the exponent of of the expression, [tex]n^{1/2}[/tex], which is 1/2 represents the critical line (Re(z) = 1/2) and confirms RH. And in turn, RH confirms the above basic and important fact. Amen!

The Riemann Hypothesis or the Right Hypothesis is true! Please acknowledge that fact!

Moreover, please remember that basic fact in bold font above when thinking about the meaning and the significance of RH.

David Cole, https://www.researchgate.net/profile/David_Cole29

P.S. I cannot retire from mathematics in peace while there are ‘experts’ of RH who claim RH is still a open problem. They are either in denial for personal or political reasons or they do not understand RH!

[Guest]

"Don't pay attention to "authorities," think for yourself."

-- Richard P. Feynman, a great scientist.

I wonder why.

I wonder why.

I wonder why the Riemann Hypothesis is true!

I wonder why I wonder...

Now, I know why...

The Riemann Hypothesis is true! Amen!

-- Richard P. Feynman and David Cole.

Relevant Reference Links:

'Richard Feynman: A Life in Science'
by John Gribbin,

https://www.goodreads.com/book/show/56165.Richard_Feynman

https://en.m.wikipedia.org/wiki/Richard_Feynman

[Guest]

Keywords: Fundamental Theorem of Arithmetic and the Riemann Hypothesis

Basic and Important Fact Confirming RH (Riemann Hypothesis or Right Hypothesis, Re(z) = 1/2) where the complex variable, z, is any nontrivial and simple zero of the Riemann Zeta Function, [tex]\zeta(z) =\sum_{n=1}^{\infty }\frac{1}{n^{z}} = \sum_{n \ is \ prime}^{\infty }\frac{1}{n^{z}} + \sum_{n \ is \ not \ prime}^{\infty }\frac{1}{n^{z}} = 0[/tex]:

For all n > 1, where n is any positive composite number, there exists a prime number, p, such that p|n (p divides n)
where p [tex]≤ n^{1/2}[/tex] (p is less than or equal to the square root of n).

And thus, the exponent of of the expression, [tex]n^{1/2}[/tex], which is 1/2 represents the critical line (Re(z) = 1/2) and confirms RH. And in turn, RH confirms the above basic and important fact. Amen!

The Riemann Hypothesis or the Right Hypothesis is true! Please acknowledge that fact!

Moreover, please remember that basic fact in bold font above when thinking about the meaning and the significance of RH.

David Cole, https://www.researchgate.net/profile/David_Cole29

P.S. I cannot retire from mathematics in peace while there are ‘experts’ of RH who claim RH is still a open problem. They are either in denial for personal or political reasons or they do not understand RH!

[Guest]

FYI:

Rieman's Zeta Equation is equivalent to Euler's Equation:

[tex]\zeta(z) = \sum_{n=1}^{\infty }\frac{1}{n^{z}}[/tex]
[tex]= 1 \\\ + \sum_{\forall n \ that's \ prime}^{\infty }\frac{1}{n^{z}} \ + \
\sum_{\forall n \ that's \ not \ prime}^{\infty }\frac{1}{n^{z}} = 1 + e^{iπ} = 0[/tex].

[Guest]

FYI:

Rieman's Zeta Equation (RZE) is equivalent to Euler's Equation (ER):

[tex]\zeta(z) = \sum_{n=1}^{\infty }\frac{1}{n^{z}}[/tex]
[tex]\\\ = 1 + \sum_{\forall n \ that's \ prime}^{\infty }\frac{1}{n^{z}} \ + \
\sum_{\forall n \ that's \ not \ prime}^{\infty }\frac{1}{n^{z}} \ \\ = 1 + e^{iπ} \\ = 1 - 1 = 0[/tex].

Hmm. Since RZE has infinitely many unique solutions, we must adapt ER accordingly:

[tex]\zeta(z) = \sum_{n=1}^{\infty }\frac{1}{n^{z}}[/tex]
[tex]\\\ = 1 + \sum_{\forall n \ that's \ prime}^{\infty }\frac{1}{n^{z}} \ + \
\sum_{\forall n \ that's \ not \ prime}^{\infty }\frac{1}{n^{z}} \\ = 1 + e^{i(2k + 1)π} \\ = 1 - 1 = 0[/tex] for all integers, [tex]k \ge 0[/tex].

[Guest]

Minor Update:

[tex]\zeta(z) = \sum_{n=1}^{\infty }\frac{1}{n^{z}}[/tex]
[tex]\\\ = 1 + \sum_{\forall n \ that's \ prime}^{\infty }\frac{1}{n^{z}} \ + \
\sum_{\forall n > 1\ that's \ not \ prime}^{\infty }\frac{1}{n^{z}} \\ = 1 + e^{i(2k -1)π} \\ = 1 - 1 = 0[/tex] for all integers, [tex]k \ge 1[/tex].

[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 Theoreom...

David Cole.

[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 Theoreom...

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.