Recap: RIEMANN HYPOTHESIS IS TRUE!

[Guest]

Keywords: Simple Prime Numbers (p), Simple Nontrivial Zeros ([tex]z = \frac{1}{2} \pm i * b[/tex]), Harmonic Series (HS), Riemann Zeta Function (RZF), and Riemann Hypothesis (RH: Re(z) [tex]= \frac{1}{2}[/tex]), Fundamental Theorem of Arithmetic (FTA).

Hmm. The Harmonic Series states all prime numbers are simple or unique according to the Fundamental Theorem of Arithmetic.

And there are infinitely many prime numbers too. (The divergence of HS, Euclid's Theorem, Euler's Theorem, etc.)

Hmm. And since all prime numbers are simple and since there are infinitely many prime numbers, all nonzero trivial zeros of the Riemann Zeta Function (RZF) are simple, and there are infinitely many nonzero trivial zeros of the Riemann Zeta Function. Why?

There exists one and only one simple nonzero trivial zero (RZF) for every simple prime number (RZF). THERE ARE NO EXCEPTIONS!

Moreover, for every positive integer that is not prime, there exists a prime number that divides that integer and that prime number is less than or equal to the square root (RH is best!) of that integer according to the Fundamental Theorem of Arithmetic.

Therefore, the Riemann Hypothesis (RH) is true! THERE ARE NO EXCEPTIONS!


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

(RZF): [tex]\sum_{k=1}^{\infty }\frac{1}{k^{z}} = 0[/tex].

All simple prime numbers and all simple nonzero trivial zeros of RZF are defined by the truth of the Riemann Hypothesis. THERE ARE NO EXCEPTIONS!

Why is the Riemann Hypothesis true? THE RIEMANN HYPOTHESIS IS BEST!

Dave.

P.S. I apologize for returning to the topic of RH... I felt compelled to summarize the work of many mathematicians to settle RH, affirmatively or negatively.

Relevant Reference Links:

'Proof of Riemann Hypothesis',

https://www.math10.com/forum/viewtopic.php?f=63&t=1549;

'Why is RH optimum?',

https://www.math10.com/forum/viewtopic.php?f=63&t=8042.

[Guest]

Hmm. One is a positive integer, and it is not prime in the strictest sense according to FTA. Right!

Oops! Okay! We exclude one there... But one counts (it is important), and therefore, it is counted....

[Guest]

The answer has no sense. It's only a blabla unthinkable. What is the link with the Riemann zeta function ?

[Guest]

The answer has no sense. It's only a blabla unthinkable. What is the link with the Riemann zeta function ?

Okay! We agree to disagree. And I do not understand your question... And you have made an unintelligible argument here.

When the Riemann Zeta Function (RZF) equals zero, [tex]\zeta(z) = \sum_{k=1}^{\infty }\frac{1}{k^{z}} = 0[/tex], one can determine the complex nonzero trivial zeros (z) of RZF.

And the Riemann Hypothesis (RH) states Re(z) = 1/2. My claim and my proof support the truth of RH. The Harmonic Series and RZF are deeply connected...

Moreover, my work here is only a summary.

Please study carefully the contents of the two reference links above for more details. Reading is fundamental and essential...

Why is RH true? RH is best!

Dave.

Remarks: Re(z) means the real part of z where [tex]z = a \pm i * b[/tex] where a and b are real numbers. And [tex]i = \sqrt{-1}[/tex]. Therefore, Re(z) = a. And RH indicates a = 1/2.

Two More Relevant Reference Links:

'The Riemann Hypothesis For Dummies',

https://qntm.org/riemann;

'Riemann Hypothesis',

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

[Guest]

"Everything should be made as simple as possible, but not simpler." -- Albert Einstein, a great scientist.

Hmm. Keep it simple... We shall try.

There's a fundamental one-to-one correspondence between every simple prime and every simple nontrivial zero of the Riemann Zeta Function (RZF),
[tex]\zeta(z) = \sum\limits_{k=1}^{\infty }\frac{1}{k^{z}}[/tex].

The Riemann Hypothesis (RH) states that the one-to-one correspondence can only occur when Re(z) = [tex]\frac{1}{2}[/tex].

Why Re(z) = [tex]\frac{1}{2}[/tex]?
Re(z) = [tex]\frac{1}{2}[/tex] is best or optimum! (This is the best result of the optimization of [tex]\zeta(z) = 0[/tex] in accordance with the Fundamental Theorem of Arithmetic.)

And since there are infinitely many primes, there are also infinitely many nontrivial zeros.

Together, [tex]\zeta(z) = 0[/tex] and RH, indicate the existence of primes and the existence of nontrivial zeros of RZF and their one-to-one correspondence in the best way possible in accordance with the Fundamental Theorem of Arithmetic.

Remark: Simple here indicates both uniqueness and no multiple roots of [tex]\zeta(z) = 0[/tex].

Dave.

[Guest]

Hmm. There are still some or many doubters (mathematicians) among us who still think RH is an open problem. Why is a valid proof of RH not enough to satisfy them?

Beware of disinformation or misinformation! There's a dark side to human nature. May the force be with you always!

FYI: "Finding Prime Locations: The Continuing Challenge to Prove the Riemann Hypothesis (RH).
The 160-year-old Riemann hypothesis has deep connections to the distribution of prime numbers and remains one of the most important unsolved problems in mathematics."

https://www.simonsfoundation.org/2020/05/06/finding-prime-locations-the-continuing-challenge-to-prove-the-riemann-hypothesis/.