Proof of Riemann Hypothesis

[Guest]

PROOF OF RIEMANN HYPOTHESIS:

Keywords: Riemann zeta function, Power Series Expansion of the Riemann zeta Function, Fundamental Theorem of Arithmetic, Harmonic Series ([tex]H = \sum_{k=1}^{\infty }1/k = \infty[/tex]), and Complex Conjugate variables and s'.

The Riemann Hypothesis states that the real part of all non-trivial zeta zeros of the Riemann zeta function
[tex](ζ(s) = \sum_{k=1}^{\infty }1/k^{s} = 0[/tex] or [tex]ζ(s') = \sum_{k=1}^{\infty }1/k^{s'} = 0)[/tex] have a real part equal to one-half.

Fact I: The real part of all non-trivial zeros of the Riemann zeta function are located in the critical strip, the closed interval, [0, 1], according to a Riemann Theorem.

Fact II: There are infinitely many non-trivial zeros of the Riemann zeta function whose real part equals one-half according to a Hardy Theorem.

Fact III: The sum of each corresponding complex conjugate pair of non-trivial zeta zeros (s and s') of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and according to the Harmonic Series (H):

If s = a + bi, then s' = a - bi. And s + s' = 1 according to Fact III.
This equation implies 2* a = 1 or a = 1/2, and b - b = 0.

Note: Euler and others have proven that there exists an infinitely many primes in H. And the divergence of H is the key reason for that result.


Fact IV: For all k > 1, where k is a positive integer, there exists a prime number, p, so that p|k such that p = k or p ≤ [tex]\sqrt{k}[/tex] = [tex]k^{1/2}[/tex].

Therefore, a = 1/2 which is the optimal exponent value of the expression, [tex]k^{1/2}[/tex], according to Fact IV.

Thus, Riemann Hypothesis is true!


Author: David Cole
(aka primework123)

[Guest]

PROOF OF RIEMANN HYPOTHESIS:

Keywords: Euclid Theorem (on infinitely many prime numbers), Riemann zeta function, Power Series Expansion of the Riemann zeta Function, Fundamental Theorem of Arithmetic, Harmonic Series ([tex]H = \sum_{k=1}^{\infty }1/k = \infty[/tex]), and Complex Conjugate variables s and s'.

The Riemann Hypothesis states that all non-trivial zeta zeros of the Riemann zeta function
[tex](ζ(s) = \sum_{k=1}^{\infty }1/k^{s} = 0[/tex] or [tex]ζ(s') = \sum_{k=1}^{\infty }1/k^{s'} = 0)[/tex] have a real part equal to one-half.

Fact I: The real part of all non-trivial zeros of the Riemann zeta function are located in the critical strip, the closed interval, [0, 1], according to a Riemann Theorem.

Fact II: There are infinitely many non-trivial zeros of the Riemann zeta function whose real part equals one-half according to a Hardy Theorem.

Fact III: The sum of each corresponding complex conjugate pair of non-trivial zeta zeros (s and s') of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and according to the Harmonic Series (H):

If s = a + bi, then s' = a - bi. And s + s' = 1 according to Fact III.
This equation implies 2* a = 1 or a = 1/2, and b - b = 0.

Note: Euler and others have proven that there exists infinitely many primes in the Harmonic Series (the source of all prime numbers). And the divergence of Harmonic Series is the key reason for that result.


Fact IV: For all k > 1, where k is a positive integer, there exists a prime number, p, so that p|k such that p = k or p ≤ [tex]\sqrt{k}[/tex] = [tex]k^{1/2}[/tex].

Therefore, the real value, a = 1/2 which is the optimal exponent value of the expression, [tex]k^{1/2}[/tex], according to Fact IV.

Thus, Riemann Hypothesis is true!


Author: David Cole
(aka primework123)

[Guest]

Reference Link:

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

Author of 'Proof of Riemann Hypothesis': David Cole (aka primework123)

https://www.linkedin.com/in/davidcole11

[Guest]

In regards to the simple and subtle proof of Riemann Hypothesis, we recall a famous quote of the great Albert Einstein.

"When the solution is simple, GOD is answering." -- Albert Einstein.

[Guest]

FYI: Reference Link -- https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true.

[Guest]

PROOF OF RIEMANN HYPOTHESIS:

Keywords: Euclid Theorem (on infinitely many prime numbers), Riemann zeta function, Power Series Expansion of the Riemann zeta Function, Fundamental Theorem of Arithmetic, Harmonic Series ([tex]H = \sum_{k=1}^{\infty }1/k = \infty[/tex]), and Complex Conjugate variables s and s'.

The Riemann Hypothesis states that all non-trivial zeta zeros of the Riemann zeta function
[tex](ζ(s) = \sum_{k=1}^{\infty }1/k^{s} = 0[/tex] or [tex]ζ(s') = \sum_{k=1}^{\infty }1/k^{s'} = 0)[/tex] have a real part equal to one-half.

Fact I: The real part of all non-trivial zeros of the Riemann zeta function are located in the critical strip, the closed interval, [0, 1], according to a Riemann Theorem.

Fact II: There are infinitely many non-trivial zeros of the Riemann zeta function whose real part equals one-half according to a Hardy Theorem.

Fact III: The sum of each corresponding complex conjugate pair of non-trivial zeta zeros (s and s') of the Riemann zeta function equals one according to the Fundamental Theorem of Arithmetic and according to the Harmonic Series (H):

If s = a + bi, then s' = a - bi. And s + s' = 1 according to Fact III.
This equation implies 2* a = 1 or a = 1/2, and b - b = 0.

Note: Euler and others have proven that there exists infinitely many primes in the Harmonic Series (the source of all prime numbers). And the divergence of Harmonic Series is the key reason for that result.


Fact IV: For all k > 1, where k is a positive integer, there exists a prime number, p, so that p|k such that p = k or p ≤ [tex]\sqrt{k}[/tex] = [tex]k^{1/2}[/tex].

Therefore, the real value, a = 1/2 which is the optimal exponent value of the expression, [tex]k^{1/2}[/tex], according to Fact IV.

Thus, Riemann Hypothesis is true!

Reference Link: https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true


Author: David Cole
(aka primework123)

[Guest]

FYI: Reference Links:

https://www.linkedin.com/in/davidcole11

https://www.gofundme.com/22jz2yk[/size]

[leesajohnson]

I would like to thank you for sharing proof for Riemann Hypothesis.

[Guest]

I would like to thank you for sharing proof for Riemann Hypothesis.

Leesa, you're very welcome! And many cheers to you for appreciating my work too. Best wishes, Dave.

"God made the integers, and all else is the work of man." — Leopold Kronecker.

“Mathematics, rightly viewed, possess not only truth, but supreme beauty, a beauty cold and austere, like that of sculpture.” — Bertrand Russell.

“Of all the great innovations and intellectual achievements of mankind there is nothing that rivals the invention of counting and discovery of the number system.” — Jan C. A. Boeyens.

[Guest]

The Riemann Hypothesis (RH) is true!

It's a SIN to think otherwise.

1. Where are all the primes in the critical strip, [0, 1] ?

2. Where are all the power of primes in the critical strip, [0, 1]?

3. How are prime numbers and the nontrivial zeros of the Riemann zeta function connected?

Hints: Fundamental Theorem of Arithmetic, the Generalized Fundamental Theorem of Algebra, the Harmonic Series, and the Prime Number Theorem

If one can answer the above questions, then one understands why the Riemann Hypothesis is true!

Reference link: https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2.

[Guest]

The Riemann Hypothesis (RH) is true!

It's a SIN to think otherwise.

1. Where are all the primes in the critical strip, [0, 1] ?

2. Where are all the powers of primes in the critical strip, [0, 1]?

3. How are prime numbers and the nontrivial zeros of the Riemann zeta function connected?

Hints: Fundamental Theorem of Arithmetic, the Generalized Fundamental Theorem of Algebra, the Harmonic Series, and the Prime Number Theorem

If one can answer the above questions, then one understands why the Riemann Hypothesis is true!

Reference link: https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2.

[Guest]

Reference link Update: https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2.

[Guest]

Reference links:

https://www.quora.com/What-great-conjectures-in-mathematics-combine-additive-theory-of-numbers-with-the-multiplicative-theory-of-numbers;

https://www.linkedin.com/pulse/prime-work-three-laws-which-govern-general-behaviour-numbers-cole.

[Guest]

Good Reference link:

'The Riemann Hypothesis, Explained',

https://medium.com/@JorgenVeisdal/the-r%20...%20g9bwnzq0t;

[Guest]

Riemann never claimed a proof of his famous and important conjecture, the Riemann Hypothesis.

[Guest]

Good Reference Link:

‘DECOUPLING, EXPONENTIAL SUMS AND THE RIEMANN ZETA FUNCTION’,
http://www.ams.org/journals/jams/2017-30-01/S0894-0347-2016-00860-3/S0894-0347-2016-00860-3.pdf.

[Guest]

FYI: 'Does the nth nontrivial simple zero of the Riemann zeta function indicate the nth prime, p_n, occurs as a prime factor in all multiples of p_n?',

https://www.researchgate.net/post/Does_the_nth_nontrivial_simple_zero_of_the_Riemann_zeta_function_indicate_the_nth_prime_p_n_occurs_as_a_prime_factor_in_all_multiples_of_p_n.

[Guest]

What are the implications of the valid Riemann Hypothesis for mathematics and for physics?

[Guest]

FYI: 'Subtle relations: prime numbers, complex functions, energy levels and Riemann.'
https://www.google.com/url?sa=t&source=web&rct=j&url=http://wwwf.imperial.ac.uk/~hjjens/Riemann_talk.pdf&ved=0ahUKEwigsOWcj-LVAhVOx2MKHRGzBQcQFgg3MAQ&usg=AFQjCNH6WgHZQ6MlbUFCED4XX1mscvrxZw

[Guest]

'A Friendly Introduction to The Riemann Hypothesis' by Thomas Wright

http://www.math.jhu.edu/~wright/RH2.pdf