# Why is RH optimum?

### Why is RH optimum?

FYI: RH (Riemann Hypothesis) means $$Re(z_{n}) = 1/2$$ for all n, positive integers.

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) $$2\le p \le m^{1/2}$$.

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 implies the truth of RH.

If we had 0 < e < 1/2, p will not divide all m according to 2) ...

And if we had 1/2 < e < 1, p will divide all m according to 2) ... but e is just too big for our needs here.

Hence, e = 1/2 is best or optimum which means RH is optimum.
Guest

### Re: Why is RH optimum?

Guest wrote:FYI: RH (Riemann Hypothesis) means $$Re([tex]z_{n}) = 1/2$$ for all n, positive integers.

Note: $$z_{n}$$ is the nth nontrivial zeta zero of the Riemann Zeta Function. And there is the unique prime number, $$p_{n}$$, 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) $$2\le p \le m^{1/2}$$.

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.

Please do not delete this post since it corrects the previous post.
Guest

### Re: Why is RH optimum?

FYI: RH (Riemann Hypothesis) means $$Re(z_{n}) = 1/2$$ for all n, positive integers.

Note: $$z_{n}$$ is the nth nontrivial zeta zero of the Riemann Zeta Function. And there is the unique prime number, $$p_{n}$$, 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) $$2\le p \le m^{1/2}$$.

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^{1/2} 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]

Please do not delete this post since it corrects the previous post.[/quote]
Guest

### Re: Why is RH optimum?

Guest wrote:FYI: RH (Riemann Hypothesis) means $$Re(z_{n}) = 1/2$$ for all n, positive integers.

Note: $$z_{n}$$ is the nth nontrivial zeta zero of the Riemann Zeta Function. And there is the unique prime number, $$p_{n}$$, 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) $$2\le p \le m^{1/2}$$.

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 $$m^{1/2}$$ 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.
[/quote]
Guest

### Re: Why is RH optimum?

Furthermore, a nontrivial zeta zero off the critical line (Re($$z_{n }$$) = 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

### Re: Why is RH optimum?

Guest wrote:Furthermore, a nontrivial zeta zero off the critical line (Re($$z_{n }$$) = 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.

Yes! Any attempt to refute the famous and important Riemann Hypothesis, Re($$z_{n}$$) = 1/2, is a very futile exercise since the hypothesis is true! Please do something worthwhile.

And I can give you n (from one to infinity) reasons why the Riemann Hypothesis is true! Amen!

Moreover, it is unsound (incomplete) to discuss the properties of the nontrivial and simple zeta zeros of the Riemann Zeta Function without discussing the properties of primes.

They are are interdependent or interrelated. Amen!

Note: The nth nontrivial and simple zeta zero exists if and only if the nth prime exists.
Guest

### Re: Why is RH optimum?

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:

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 $$≤ n^{1/2}$$ (p is less than or equal to the square root of n).

And thus, the exponent of of the expression, $$n^{1/2}$$, 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!

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

P.S. I cannot retired from mathematics in peace while there are ‘experts’ of RH who claimed RH is still a open problem. They are either in denial or they do not understand RH!
Guest

### Re: Why is RH optimum?

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

-- Richard P. Feynman, a great scientist.
Guest

### Re: Why is RH optimum?

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, $$\sum_{n=1}^{\infty }n^{-z} = 0$$:

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 $$≤ n^{1/2}$$ (p is less than or equal to the square root of n).

And thus, the exponent of of the expression, $$n^{1/2}$$, 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

### Re: Why is RH optimum?

Guest wrote:"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.

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

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

### Re: Why is RH optimum?

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, $$\zeta(z) =\sum_{n=1}^{\infty }\frac{1}{n^{z}} = \sum_{n \ is \ prime}^{\infty }\frac{1}{n^{z}} + \sum_{n \ is \ not \ prime}^{\infty }\frac{1}{n^{z}} = 0$$:

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 $$≤ n^{1/2}$$ (p is less than or equal to the square root of n).

And thus, the exponent of of the expression, $$n^{1/2}$$, 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

### Re: Why is RH optimum?

FYI:

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

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

### Re: Why is RH optimum?

Guest wrote:FYI:

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

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

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

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

### Re: Why is RH optimum?

Guest wrote:
Guest wrote:FYI:

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

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

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

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

### Re: Why is RH optimum?

Minor Update:

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

### Re: Why is RH optimum?

A Recap On Why The Riemann Hypothesis Is True:

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

$$\zeta(z) = \sum_{k=1}^{\infty }\frac{1}{k^{z}}$$,

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:

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

where $$z_n$$ is the nth nontrivial and simple zero of the Riemann Zeta Function, and $$p_n$$ 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

### Re: Why is RH optimum?

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,

$$\zeta(z) = \sum_{k=1}^{\infty }\frac{1}{k^{z}}$$,

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:

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

where $$z_n$$ is the nth nontrivial and simple zero of the Riemann Zeta Function, and $$p_n$$ 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,

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

is derived from the complex Riemann Zeta Function.
Guest

### Re: Why is RH optimum?

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,

$$\zeta(z) = \sum_{k=1}^{\infty }\frac{1}{k^{z}}$$,

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:

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

where $$z_n$$ is the nth nontrivial and simple zero of the Riemann Zeta Function, and $$p_n$$ 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,

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

is derived from the complex Riemann Zeta Function.
Guest

### Re: Why is RH optimum?

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

### Re: Why is RH optimum?

Given the values, $$z_{n}$$ and $$p_{n}$$, how do we correctly derive equation two,

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

from equation one,

1. $$\zeta(z) = \sum_{k=1}^{\infty }\frac{1}{k^{z}}$$ = 0?

We hope that our latest question is a big question which leads us eventually to deeper insights about Riemann's great work on prime number theory: his lofty quest to prove the Prime Number Theorem and his very lofty quest to compute most accurately the number of primes less than or equal to any appropriate real value, x.
Guest

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