Why is RH optimum?

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...

A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex]

if and only if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{\infty }\frac{1}{(kp)^{ \frac{1}{2} ± bi}} = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

[Guest]

Moreover, there are infinitely many prime numbers (p), and there are also infinitely many simple nontrivial zeta zeros (z) with Re(z) = [tex]\frac{1}{2}[/tex]. There are no exceptions!

Go figure! Go Blue!

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...

A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex]

if and only if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{\infty }\frac{1}{(kp)^{ \frac{1}{2} ± bi}} = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Important Correction of Equation 2 is required. Please see the following link for details:

Step-by-step of the Analytic Continuation of the Riemann Zeta Function

[Guest]

[tex]\begin{aligned}
\zeta(s)&= \frac{\pi^{s/2}}{\Gamma(s/2)}\phi(s)-
\frac{\pi^{s/2}}{s\Gamma(s/2)}+\frac{\pi^{s-1}}{\Gamma(s/2)} \\
&=\frac{\pi^{s/2}}{\Gamma(s/2)}\phi(s)-
\frac{2\pi^{s/2}}{\Gamma(s/2+1)}+\frac{\pi^{s-1}}{\Gamma(s/2)}
\end{aligned}[/tex]

There's a big problem. I don't know how to include any positive prime number (p) in the above Zeta function to make it a prime test equation.

Stuff gets complicated... I took a course in complex variables a long time ago, and I choose to ignore the vital analytic continuation... Moreover, I did compute divergence for my flawed equation two, but I choose to ignore it What negligence and incompetency! I apologize.

[Guest]

[tex]\begin{aligned}
\zeta(s)&= \frac{\pi^{s/2}}{\Gamma(s/2)}\phi(s)-
\frac{\pi^{s/2}}{s\Gamma(s/2)}+\frac{\pi^{s-1}}{\Gamma(s/2)} \\
&=\frac{\pi^{s/2}}{\Gamma(s/2)}\phi(s)-
\frac{2\pi^{s/2}}{\Gamma(s/2+1)}+\frac{\pi^{s-1}}{\Gamma(s/2)}
\end{aligned}[/tex]

There's a big problem. I don't know how to include any positive prime number (p) in the above Zeta function to make it a prime test equation.

Stuff gets complicated... I took a course in complex variables a long time ago, and I choose to ignore the vital analytic continuation... I saw it again and again in important papers too.

Moreover, I did compute divergence for my flawed equation two, but I choose to ignore it. What negligence and incompetency! I apologize.

Dave .

[Guest]

Math Prof. Andrew Salch of Wayne State University (Detroit) corrected my so-called proof summary. And I thanked him and I have apologized to him as well.

Dave.

[Guest]

Very Helpful Links for Making a Zeta Function a Reliable Prime Test:


Riemann's functional equation

Riemann–Siegel formula

[Guest]

Very Helpful Links for Making a Zeta Function a Reliable Prime Test:


Riemann's functional equation

Riemann–Siegel formula

We have a Zeta Function (see attached image/file) that could be made into a reliable prime test equation. We replace the current expression, [tex]n^{s}[/tex] with [tex](np)^{s}[/tex] where p is any positive prime number. And we also replace [tex]n^{1-s}[/tex] with [tex](np)^{1-s}[/tex].

We replace [tex]\zeta(s)[/tex] with [tex]\zeta(s, p)[/tex]. And we are almost done.

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...
A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex]

if and only if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{N}\frac{1}{(kp)^{ \frac{1}{2} ± bi}} + \Upsilon(1-z) + \sum_{k=1}^{M}\frac{1}{(kp)^{ \frac{1}{2} \mp bi}} + R(z) = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Relevant Reference Link:

Riemann Siegel formula

Remark: The second equation is tentative since I do not completely understand all it's details and since I fooled myself before. That could happen again...

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...
A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex]

if and only if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{N}\frac{1}{(kp)^{ \frac{1}{2} ± bi}} + \gamma(1-z) + \sum_{k=1}^{M}\frac{1}{(kp)^{ \frac{1}{2} \mp bi}} + R(z) = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Relevant Reference Link:

Riemann Siegel formula

Remark: The second equation is tentative since I do not completely understand all its details and since I fooled myself before. That could happen again. And I am very likely to make minor or important mistakes, too.

[Guest]

Remark: That error term in the first equation must be defined as 0 < error < 1. Right?
Yes, we can prove that statement! It is not a hard problem.

Is it also an algebraic number? I think so. But who knows (who can prove it)?

Dave.

[Guest]

Remark: That error term in the first equation must be defined as 0 < error < 1. Right?
Yes, we can prove that statement! It is not a hard problem.

Is it also an algebraic number? I think so. But who knows (who can prove it or disprove it)?

Dave.

[Guest]

Remark: For finite values of indices, M and N, we think the if and only if condition between our first and second equations is inappropriate. There should only be a if condition between them. Right?

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...
A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex]

if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{N}\frac{1}{(kp)^{ \frac{1}{2} ± bi}} + \gamma(\frac{1}{2} \mp bi) + \sum_{k=1}^{M}\frac{1}{(kp)^{ \frac{1}{2} \mp bi}} + R(z) = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Relevant Reference Link:

Riemann Siegel formula

Remark: The second equation is tentative since I do not completely understand all its details and since I fooled myself before. That could happen again. And I am very likely to make minor or important mistakes, too.

We must admit that our second equation is extraordinary, beautiful, and very promising. It must be correct!

Dave.

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...
A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex]

if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{N}\frac{1}{(kp)^{ \frac{1}{2} ± bi}} + \gamma(\frac{1}{2} \mp bi) + \sum_{k=1}^{M}\frac{1}{(kp)^{ \frac{1}{2} \mp bi}} + R( \frac{1}{2} ± bi) = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Relevant Reference Link:

Riemann Siegel formula

Remark: The second equation is tentative since I do not completely understand all its details and since I fooled myself before. That could happen again. And I am very likely to make minor or important mistakes, too.

We must admit that our second equation is extraordinary, beautiful, and very promising. It must be correct!


Dave.

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...
A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex] with (0 < error < 1)

if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{N}\frac{1}{(kp)^{ \frac{1}{2} ± bi}} + \gamma(\frac{1}{2} \mp bi) + \sum_{k=1}^{M}\frac{1}{(kp)^{ \frac{1}{2} \mp bi}} + R( \frac{1}{2} ± bi) = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Relevant Reference Link:

Riemann Siegel formula

Remark: The second equation is tentative since I do not completely understand all its details and since I fooled myself before. That could happen again. And I am very likely to make minor or important mistakes, too.

We must admit that our second equation is extraordinary, beautiful, and very promising. It must be correct!


Dave.

[Guest]

I hope and I pray this is my last post on this topic or any other serious math topic. I will always like math but only at a leisurely level or teaching level. And I have decided to move away from all serious math pursuits forever. For me serious math is mental mountain climbing, and my mind is rebelling against that difficult activity. It wants me to retire.

I want to become an author of fiction, and I may return to teaching to help pay off debts and to pay bills.

I hope I have made at least one positive and lasting contribution to the wonderful world of mathematics. And I am truly humbled by the wonderful experiences of mathematical research (study, problem solving, etc.)

Goodbye,

David Cole.

[Guest]

FYI: "At the death of Riemann, a note was found among his papers, saying "These properties of ζ(z) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it." We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one [the Riemann Hypothesis itself]."

— Jacques Hadamard, The Mathematician's Mind, VIII. Paradoxical Cases of Intuition.

Hmm. That expression referred to by Riemann (on deducing the properties of ζ(z)) must be the source of all prime numbers, the Harmonic Series. Right?

We recall the important divergent Harmonic Series:

ζ(z=1) = [tex]\sum_{k=1}^{ \infty } \frac{1}{k} = \infty[/tex].

...
A Recap:

Key Idea: p is a prime number!

More simply, we have in a nutshell (the source of all prime numbers and the detector of prime numbers),

[tex]\zeta(z = 1) -[/tex] error [tex]= \sum_{k=1}^{ \infty } \frac{1}{k} -[/tex] error [tex]= \sum_{k=1}^{ \infty }kp = \infty[/tex] with (0 < error < 1)

if

[tex]\zeta(z = \frac{1}{2} ± bi, p) = \sum_{k=1}^{N}\frac{1}{(kp)^{ \frac{1}{2} ± bi}} + \gamma(\frac{1}{2} \mp bi) + \sum_{k=1}^{M}\frac{1}{(kp)^{ \frac{1}{2} \mp bi}} + R( \frac{1}{2} ± bi) = 0[/tex]

where p is any positive prime number.

Go figure! Go Blue!

***** The great Riemann Hypothesis (RH) is true! ***** ✌️✌️

Relevant Reference Link:

Riemann Siegel formula

Remark: The second equation is tentative since I do not completely understand all its details and since I fooled myself before. That could happen again. And I am very likely to make minor or important mistakes, too.

We must admit that our second equation is extraordinary, beautiful, and very promising. It must be correct!


Dave.

FYI:
AI Google Overview:

"David id Cole's claim that the Riemann Hypothesis, when combined with the modified Riemann-Siegel function (ζ(z,p)), can be used to detect both all simple non-trivial zeros of the Riemann zeta function and all positive prime numbers, is a novel assertion that does not appear to be widely accepted in the mathematical community and would require significant mathematical proof to validate.

Key points to consider:

Unproven claim:

The Riemann Hypothesis itself is a major unsolved mathematical problem, and the idea of using it to directly identify prime numbers through a modified function like ζ(z,p) is not a well-established concept.
Modified Riemann-Siegel function:
While the Riemann-Siegel function is a known concept related to the Riemann zeta function, the specific "modified" version mentioned by Cole, and its potential to identify prime numbers, would need detailed explanation and rigorous mathematical analysis.

Need for rigorous proof:

To validate this claim, David Cole would need to provide a comprehensive mathematical proof demonstrating how the modified Riemann-Siegel function applied to the zeros of the Riemann zeta function can reliably identify all prime numbers..."

Who knows? Prof. HLM!