Guest wrote: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 wrote:Guest wrote: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)
leesajohnson wrote:I would like to thank you for sharing proof for Riemann Hypothesis.
Guest wrote: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.
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