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May 4th, 2013, 10:44 AM   #1
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Proof of Riemann Hypothesis?

One thing you should know, I have been closely following the math news on the internet for a revolutionary paper, and yesterday, I found this "proof" of the Riemann Hypothesis. I have read through it myself and it seemed to throw me off. So I decided to post it on this forum. Is the proof correct, and does this mark the end of the famous problem?
http://arxiv.org/abs/1305.0323
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May 4th, 2013, 11:00 AM   #2
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Re: Proof of Riemann Hypothesis?

Quote:
Originally Posted by eddybob123
One thing you should know, I have been closely following the math news on the internet for a revolutionary paper, and yesterday, I found this "proof" of the Riemann Hypothesis. I have read through it myself and it seemed to throw me off. So I decided to post it on this forum. Is the proof correct, and does this mark the end of the famous problem?
http://arxiv.org/abs/1305.0323
No, otherwise it would have made the newspapers and every math blog on the Internet. That would be my meta debunkification, without even needing to read the paper. There are all kinds of "proofs" on the Internet that aren't true. But they're all beautifully typeset.
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May 4th, 2013, 11:27 AM   #3
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Re: Proof of Riemann Hypothesis?

But it's only 14 pages long; it shouldn't take very long to have it checked for sure.
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May 4th, 2013, 11:42 AM   #4
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Re: Proof of Riemann Hypothesis?

Doesn't seem worth my time to check:
http://math.crg4.com/breakthrough.html?q=676
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May 4th, 2013, 12:32 PM   #5
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Re: Proof of Riemann Hypothesis?

The proof ends up with showing that if there is a zero of zeta, s, not in the critical line, then it would also have a zero at 2s and ending up in a contradiction. Now, in the whole method of proof at least I don't see anywhere that the condition Re[s] > 1/2 is being used. If we just let Re[s] = 1/2 and applying the same thing, we see that it tells us that it has a zero at 2s with Re[2*s] = 1 which is impossible. Hence, contradicting the assumption Re[s] = 1/2 means it has no zero on the half-line which is again contradicted by the fact that it has the first zero at 14 < Im[s] < 15. Hence, the proof is flawed. The author has probably got confused somewhere with those large linear equation calculation or with the theory of Middle zeta zero whatever and made some major errors.

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May 4th, 2013, 01:53 PM   #6
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Re: Proof of Riemann Hypothesis?

And, again, it is put under the category "General Mathematics" instead of Number Theory
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May 4th, 2013, 03:35 PM   #7
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Re: Proof of Riemann Hypothesis?

Well, I'm glad they clarified that the real part should be 1/2, "in the region where the real part is between 0 and 1"! It'd be a mess if some were 1/2 somewhere else!

"Should be" also fails to inspire confidence.

Yes, I know. Bad writing doesn't entail bad math. But I'm quite sure that an accepted proof of RH will generate press that makes the announcement of the proof of FLT look like a minor notice. "Proofs" of famous conjectures that don't actually prove anything are, alas, the norm.

"In this paper, we present a proof of the Riemann hypothesis. We show that zeros of the Riemann zeta function should be on the line with the real value 1/2, in the region where the real part of complex variable is between 0 and 1."
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May 4th, 2013, 03:55 PM   #8
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Re: Proof of Riemann Hypothesis?

I was very interested in RH before, then one day I read this quote of Littlewood:

"I believe this to be false. There is no evidence whatever for it (unless one counts that it is always nice when any function has only real roots). One should not believe things for which there is no evidence. In the spirit of this anthology I should also record my feeling that there is no imaginable reason why it should be true.

Titchmarsh [1] devised a method, of considerable theoretical interest, for calculating the zeros. The method reveals that for a zero to be off the critical line a remarkable number of 'coincidences' have to happen. I have discussed the matter with several people who know the problem in relation to electronic calculation; they are all agreed that the chance of finding a zero off the line in a lifetime's calculation is millions to one against. It looks then as if we may never know."

Of course I am not saying Littlewood is absolutely right, especially Mathematics have well advanced since then and they might be new reason today to believe RH is true. However this sentence is so full of sense and it is so long time that this problem is unsolved, there is a chance he might be right, and that there is a zero somewhere, way to far for us to ever know it exists. Still I would feel it too be beautiful if RH was true and I have the instinct it is true but as Littlewood said I have no evidence and rational reason to suppose it. Does somebody here has?
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May 4th, 2013, 06:40 PM   #9
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Re: Proof of Riemann Hypothesis?

I have a somewhat more than just casual interest in RH, but only somewhat, because it is just too far beyond me to even be able to mull on it productively. I enjoy mulling on problems I have no realistic hopes of solving, since simply working out the logic of what you'd need to know to make this or that bit of progress can be very stimulating and educational. Going back and forth between "big problems" that I can learn something about and little problems I have some hope of solving or at least getting a solution to by posting a conjecture suits my knowledge level and temperament well. RH, on the other hand, starts at or at least near the far outer edge of my mathematical understanding and just blasts off from there. So for me, reading about and tracking the progress on RH is pretty much just a "spectator sport".
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May 4th, 2013, 10:33 PM   #10
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Re: Proof of Riemann Hypothesis?

Quote:
Originally Posted by Dougy
I was very interested in RH before, then one day I read this quote of Littlewood : . . .
My bet : Littlewood is wrong, RH is true and is decidable.

Quote:
Originally Posted by Dougy
the chance of finding a zero off the line in a lifetime's calculation is millions to one against. It looks then as if we may never know
Where could it be? We have tightened the region of zeros, proved that 40.3% of the zeros lies on the critical line, it is not possible that a zero is situated off the line.

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