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Mathematicians revive abandoned approach to the Riemann Hypothesis

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Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#21

Post by soVeryTired »

ncmncm wrote: post #11

I've seen a paper[1] that claims RH, as stated, is undecideable, and then proposes to prove that an analytic continuation of it has the property required. The idea seems to be that you can choose your continuation to have or not have the property. It's not clear to me what could be undecideable about the original proposition; either the continuation matches at the points in question, or it doesn't, and if it doesn't,…

I've been following the development of the Bender-Brody-Muller approach to the Riemann hypothesis because I knew one of the authors in a previous life. I don't think it's taken that seriously by number theorists [0][1]. The paper was written by physicists, who sometimes play a little fast and loose with mathematical rigour.

Maybe this line of argument fixes the flaws in the original BBM paper in some way - I'm not qualified to comment - but I sort of doubt it.

[0] https://arxiv.org/abs/1704.02644 [1] https://math.stackexchange.com/questions/2211278/riemann-hyp...

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#22

Post by fspeech »

math-fan wrote: post #9
Ragib_Zaman wrote: post #5

Earlier quoted context omitted.

Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…

I've taken math up to calculus and linear algebra and enjoy watching 3B1B proofs. Is that enough to follow the elementary Prime Number Theorem proof? I've been interested in understanding that for a while. Any good links?

You will need to learn some complex analysis which is very beautiful. Ash has a very good chapter on the Prime Number Theorem as a reward for learning complex variables: https://faculty.math.illinois.edu/~r-ash/CV.html You don't need to learn the whole book just some knowledge of contour integral and analytic continuation is sufficient to understand the proof. So you could just start with the last chapter and fill the gaps as you go. (If you go linearly through the book you will encounter things like the normal family not required for your stated goal. Even though the normal family stuff are even more beautifully amazing IMHO the presentation here may be too abstract to grok and could discourage one to soldier on.)

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#23

Post by throwawaymath »

pmiller2 wrote: post #17
twic wrote: post #13

The idea for the paper was sparked two years ago by a "toy problem" that Ono presented as a "gift" to entertain Zagier during the lead-up to a math conference celebrating his 65th birthday. A toy problem is a scaled-down version of a bigger, more complicated problem that mathematicians are trying to solve. Zagier described the one that Ono gave him as "a cute problem about the asymptotic behavior of certain polynomia…

Definitely. Two fairly standard techniques in mathematics are solving a cut down version of a problem and lifting that to a full solution, and solving a more general version and specializing to a particular case. Sometimes, amazingly, the more general problem is easier to solve, perhaps because fewer irrelevant details stand in the way. Both methods have yielded a lot of progress in mathematics. For example, solving…

> Sometimes, amazingly, the more general problem is easier to solve, perhaps because fewer irrelevant details stand in the way.

True, and that's why we like to generalize things. It's usually easier to prove a mathematical object belongs to another class of structure which has certain properties than to prove the object itself has all those properties.

Of course as you've noted it's not always like that at all. It's significantly easier to prove many things about finite-dimensional vector spaces over fields versus infinite-dimensional modules over rings.

Solving mathematical problems is a lot like working a sponge. It's often useful to alternate between expanding and contracting the scope of your problem.

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#24

Post by Ragib_Zaman »

oneepic wrote: post #16
Ragib_Zaman wrote: post #5

Earlier quoted context omitted.

Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…

I agree, I'm not saying a simple proof is ever out of the question either. But my failed attempt makes me wonder how much of a creative burst a typical person would have to have in order to come up with a simple solution in this case.

I didn't mean you were the one implying an elementary proof is unobtainable, that was directed at some of the other replies to your comment like "oh really?". I agree that it's enlightening to give these things a try.

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#25

Post by throwawaymath »

ncmncm wrote: post #11

I've seen a paper[1] that claims RH, as stated, is undecideable, and then proposes to prove that an analytic continuation of it has the property required. The idea seems to be that you can choose your continuation to have or not have the property. It's not clear to me what could be undecideable about the original proposition; either the continuation matches at the points in question, or it doesn't, and if it doesn't,…

I'm not familiar with that particular paper, but I'm aware of a similar line of attack from a decade or so ago. Off the top of my head I can tell you that we know Riemann is provably false if it is false[1]. I'm not aware of any comparable result for the affirmative, but personally I'm a bit skeptical of the undecidability argument.

That being said, it would be kind of narratively satisfying if Riemann is undecidable. There are many novel theorems from the past two decades which are conditionally true assuming Riemann is true. Riemann itself has become a useful technique for conditionally proving new theorem. If Riemann is undecidable, it would imply a great deal of mathematics has been developed which compartmentalizes the undecidability of other theorems. ______________________

1. https://mathoverflow.net/questions/79685/can-the-riemann-hyp...

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#26

Post by User23 »

fspeech wrote: post #22
math-fan wrote: post #9

Earlier quoted context omitted.

I've taken math up to calculus and linear algebra and enjoy watching 3B1B proofs. Is that enough to follow the elementary Prime Number Theorem proof? I've been interested in understanding that for a while. Any good links?

You will need to learn some complex analysis which is very beautiful. Ash has a very good chapter on the Prime Number Theorem as a reward for learning complex variables: https://faculty.math.illinois.edu/~r-ash/CV.html You don't need to learn the whole book just some knowledge of contour integral and analytic continuation is sufficient to understand the proof. So you could just start with the last chapter and fill th…

Visual Complex Analysis by Tristan Needham is an incredibly approachable textbook on the subject.

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#27

Post by throwawaymath »

pfortuny wrote: post #6
Ragib_Zaman wrote: post #5

Earlier quoted context omitted.

Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…

Notice also the decidability of primality in polynomial time, which was proved by some Computer Science graduates (Neeraj Kayal, and Nitin Saxena) in 2011, and led Bombieri (one of the leading experts on Number Theory) to say that Number Theory may have gone astray in theoretical complexity in some fields.

That's very interesting, do you know offhand where Bombieri said that?

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#28

Post by butterthebuddha »

"This method doesn't work and Ken knows that, not sure wtf this article is. At most, this idea can show that the zeroes off the critical line are density zero, which would be a huge result, but could never prove RH. Again, Ken knows this (he says as much when he talks about it) so wtf is this article?"

- Someone on Reddit (https://old.reddit.com/r/math/comments/brgp5z/mathematicians...)

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#29

Post by [deleted] »

oneepic wrote: post #16
Ragib_Zaman wrote: post #5

Earlier quoted context omitted.

Man HR is harsh as hell. While yes, if a proof of RH is found it will probably use advanced tools, ever finding a proof with elementary tools is not totally out of the question. Regarding the Prime Number Theorem, which can be seen as a baby version of RH, experts in the field such as Hardy have been quoted as saying elementary proofs would not exist. Then 50 years after Hadamard and Poussin found the original proofs…

I agree, I'm not saying a simple proof is ever out of the question either. But my failed attempt makes me wonder how much of a creative burst a typical person would have to have in order to come up with a simple solution in this case.

[deleted]

Re: Mathematicians revive abandoned approach to the Riemann Hypothesis

#30

Post by auggierose »

Kurtz79 wrote: post #10
auggierose wrote: post #3

Earlier quoted context omitted.

Oh really? :-D

Well, parent tried to understand the problem, used the tools at his hand to solve it without preconceptions, and realized it was more complex that it seemed. The world would be a better place if more people had the same approach to problem-solving, IMHO.

Not saying that there is anything wrong with it. It is still funny though :-) It is like somebody saying they tried to go to the moon, but after walking for an hour they discovered it was harder to get there than they initially thought.
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