Guest wrote:FYI: RH (Riemann Hypothesis) means [tex]Re([tex]z_{n}) = 1/2[/tex] for all n, positive integers.
Note: [tex]z_{n}[/tex] is the nth nontrivial zeta zero of the Riemann Zeta Function. And there is the unique prime number, [tex]p_{n}[/tex], associated with it.
https://en.m.wikipedia.org/wiki/Riemann_hypothesis
Keywords: Fundamental Theorem of Arithmetic
Hmm. A prime number, p, divides any integer, m,
if and only if 1) p = m or 2) [tex]2\le p \le m^{1/2}[/tex].
And of course, if 2) ... is false, then 1) p = m is true. Therefore, we shall consider only 2) ...
That exponent, e =1/2, of m in 2) ... is central because it implies the truth of RH.
If we had 0 < e < 1/2, then p does not divide all m according to 2) ...
And if we had 1/2 < e < 1, then p does divide all m according to 2) ... but e is just too big here.
Hence, e = 1/2 is best or optimum which means RH is optimum.
[/quote]Guest wrote:FYI: RH (Riemann Hypothesis) means [tex]Re(z_{n}) = 1/2[/tex] for all n, positive integers.
Note: [tex]z_{n}[/tex] is the nth nontrivial zeta zero of the Riemann Zeta Function. And there is the unique prime number, [tex]p_{n}[/tex], associated with it.
https://en.m.wikipedia.org/wiki/Riemann_hypothesis
Keywords: Fundamental Theorem of Arithmetic
Hmm. A prime number, p, divides any integer, m,
if and only if 1) p = m or 2) [tex]2\le p \le m^{1/2}[/tex].
And of course, if 2) ... is false, then 1) p = m is true. Therefore, we shall consider only 2) ...
That exponent, e =1/2, of [tex]m^{1/2}[/tex] in 2) ... is central because it implies the truth of RH.
If we had 0 < e < 1/2, then p does not divide all m according to 2) ...
And if we had 1/2 < e < 1, then p does divide all m according to 2) ... but e is just too big here.
Hence, e = 1/2 is best or optimum which means RH is optimum.
Please do not delete this post since it corrects the previous post.
Guest wrote:Furthermore, a nontrivial zeta zero off the critical line (Re([tex]z_{n }[/tex]) = 1/2) does not make any sense. There’s no good reason for it. And it’s pure fiction or fake news too think otherwise.
Guest wrote: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 wrote:"Don't pay attention to "authorities," think for yourself."
-- Richard P. Feynman, a great scientist.
Guest wrote: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 wrote: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].
Guest wrote:Guest wrote:FYI:
Rieman's Zeta Equation (RZE) is equivalent to Euler's Equation (EE):
[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 EE 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 wrote: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 wrote:Guest wrote: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.
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