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(Phys Org2)   Physicist at UCSB has proposed a new approach to the Riemann zeta function which could lead to a proof of the Riemann hypothesis. There will be math   (phys.org) divider line
    More: Interesting, Quantum mechanics, Particle physics, Quantum field theory, String theory, Mathematics, Riemann zeta function, Riemann hypothesis, Theoretical physics  
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674 clicks; posted to STEM » on 19 Jan 2022 at 2:30 PM (18 weeks ago)   |   Favorite    |   share:  Share on Twitter share via Email Share on Facebook



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2022-01-19 1:48:00 PM  
COMSTOCK
 
2022-01-19 1:54:49 PM  
"The Riemann zeta function is this famous and mysterious mathematical function that comes up in number theory all over the place," said Remmen, a postdoctoral scholar at UCSB's Kavli Institute for Theoretical Physics. "It's been studied for over 150 years."

You expect me to believe this coming from a person who's name is almost identical to the "scientist" in question? How do we know they're not in cahoots?

I move for a bad science thingy! Bailiff, fetch the slide rules--no lube this time!
 
2022-01-19 3:25:01 PM  
I'm a reasonably bright guy, I think. Mensa, been on Jeopardy, master's degree.

Me reading this article:
Fark user imageView Full Size
 
2022-01-19 3:25:01 PM  
imagesvc.meredithcorp.ioView Full Size


/What every bitcoin owner is doing right now.
 
2022-01-19 3:59:18 PM  

JessieL: COMSTOCK


I understand that reference.jpg
 
2022-01-19 4:13:22 PM  
Meh.  Kids In The Hall was funnier...
 
2022-01-19 4:56:56 PM  

BKITU: I'm a reasonably bright guy, I think. Mensa, been on Jeopardy, master's degree.

Me reading this article:
[Fark user image 340x408]


Try this:
Visualizing the Riemann zeta function and analytic continuation
Youtube sD0NjbwqlYw
 
2022-01-19 5:24:13 PM  
I don't see any reason why this is likely to help solve the Riemann hypothesis.  Rewriting the zeros as the masses in a quantum field theory is a nice math trick, and mapping one math problem into another has proved helpful in other cases. But in this case, I don't see an argument for how bringing the math of QFT to bear is going to help. As the article says, you'd have to prove that his amplitude describes a consistent QFT ... but work on axiomatic/constructive QFT shows how difficult (so far, usually impossible) it is to formally prove any QFT is consistent. And he's only written down the leading order terms - to prove the Riemann hypothesis you'd have to do this to all orders.  But once again, it's rarely possible to prove things nonperturbatively in QFT; that's why everyone relies on perturbation theory or on numerical approximations, except for a few gauge/string duality type theorems.
 
2022-01-19 5:42:19 PM  

Tyrone Slothrop: BKITU: I'm a reasonably bright guy, I think. Mensa, been on Jeopardy, master's degree.

Me reading this article:
[Fark user image 340x408]

Try this:
[Youtube-video https://www.youtube.com/embed/sD0NjbwqlYw]


I've actually got a decent grip on the Riemann in and of itself, and that countless conjectures in mathematics would be proven in turn by proof of the Riemann (and the inverse) which is why it's basically the Holy Grail of math right now.

It's the possible relationship with quantum field theory at all, that interactions of particles could have any bearing whatsoever with the hypothesis and where its non-trivial zeros lie, and the descriptions of how those connections might exist, that sent me into [BRAIN STATIC].

It's like hearing a convincing argument that P vs. NP could be proven by studying human metabolic rates, and the connection has something to do with Tony Gwynn's career slash line.
 
2022-01-19 5:50:30 PM  
i.imgur.comView Full Size

/ Grand Old Opry
 
2022-01-19 6:01:52 PM  

Ambitwistor: I don't see any reason why this is likely to help solve the Riemann hypothesis.  Rewriting the zeros as the masses in a quantum field theory is a nice math trick, and mapping one math problem into another has proved helpful in other cases. But in this case, I don't see an argument for how bringing the math of QFT to bear is going to help. As the article says, you'd have to prove that his amplitude describes a consistent QFT ... but work on axiomatic/constructive QFT shows how difficult (so far, usually impossible) it is to formally prove any QFT is consistent. And he's only written down the leading order terms - to prove the Riemann hypothesis you'd have to do this to all orders.  But once again, it's rarely possible to prove things nonperturbatively in QFT; that's why everyone relies on perturbation theory or on numerical approximations, except for a few gauge/string duality type theorems.


The issue isn't to solve the the Riemann Hypothesis. It's got solutions up the wazoo. The issue is to prove it. That is, to prove that the Riemann Hypothesis is true in our reality, and using our reality to do so.

I think it's pretty neat.

So, numbers are a thing we invented to count goats, even though goats aren't identical. But electrons are identical, so it's a better starting place.
 
2022-01-19 6:04:48 PM  

PartTimeBuddha: The issue isn't to solve the the Riemann Hypothesis. It's got solutions up the wazoo. The issue is to prove it.


That's what I mean by solve, and my points stand.

That is, to prove that the Riemann Hypothesis is true in our reality, and using our reality to do so.

The truth of the Riemann hypothesis doesn't depend on our laws of physics, and as I said, I don't see reason to believe that mapping it into the math of quantum field theory is going to help.  (And, certainly, the particular QFT being described here has nothing to do with our reality.)
 
2022-01-19 6:38:06 PM  

Ambitwistor: The truth of the Riemann hypothesis doesn't depend on our laws of physics


Eh. I say run with it. We use physics for lots of things. Dropping things, smashing things into each other (which is basically an advanced form of dropping). Some math came from such business.

What you have failed to recognize is that perhaps our laws of physics depend on the Riemann Hypothesis. We would then end up in the hilarity of the Anthropic Principle, proved, because we can only exist in a Universe in which the Riemann Hypothesis is true, which proves, for our Universe, the Riemann Hypothesis.

This is why we are in the best Universe and not the Dark Zone.
 
2022-01-19 6:56:46 PM  

PartTimeBuddha: Ambitwistor: The truth of the Riemann hypothesis doesn't depend on our laws of physics

Eh. I say run with it. We use physics for lots of things. Dropping things, smashing things into each other (which is basically an advanced form of dropping). Some math came from such business.

What you have failed to recognize is that perhaps our laws of physics depend on the Riemann Hypothesis. We would then end up in the hilarity of the Anthropic Principle, proved, because we can only exist in a Universe in which the Riemann Hypothesis is true, which proves, for our Universe, the Riemann Hypothesis.

This is why we are in the best Universe and not the Dark Zone.


I did mention that, yes, much math has arisen from making analogies to physics (Witten's career as a mathematician is built in this, for example). But I gave specific reasons why I think, in this case, the analogy may not help, and may in fact just map onto an even harder problem in physics. In any case, as you say, there's no harm in running with the idea, even if I'm skeptical about it.

As for the anthropic principle, no, theorems follow from axioms independently of any physical laws.
 
2022-01-19 7:09:16 PM  

Ambitwistor: As for the anthropic principle, no, theorems follow from axioms independently of any physical laws.


Well, you say that, but do you have any reason to prove it? You immediately run into Godel showing that infinite truths are true while being mathematically unprovable.

The Riemann Conjecture appearing to hold for the Universe in which we find ourselves is a fair shake.
 
2022-01-19 7:30:11 PM  
This function is already connected with certain ionization levels, so I don't know how new this approach actually is.

I've played with the function a bit and even had some ideas about connections to a curvature of the function and the zeros, but didn't get very far.

Afaik, they've been using the root pole counting theorem forever on this: so far all the zeros on a strip in the first quadrant have real part equal to 1/2.

The theorem isn't hard to use: give it a range, like up to 10^6i and the theorem tells you how many zeros and poles to expect. Then actually find them. So far, nobody has ever found a different number with the real part being 0.5

I think the roots tend to be transcendental, but I really don't know.
 
2022-01-20 12:12:52 AM  

PartTimeBuddha: Ambitwistor: As for the anthropic principle, no, theorems follow from axioms independently of any physical laws.

Well, you say that, but do you have any reason to prove it? You immediately run into Godel showing that infinite truths are true while being mathematically unprovable.

The Riemann Conjecture appearing to hold for the Universe in which we find ourselves is a fair shake.


Hmmm seems to me Ambitwistor thinks it true but just not provable with the logical system of physics and holding out hope that mathematics can do the job. Meanwhile PTB is holding the harder position that it is true but math can't prove it. What I am not seeing is any support for the possibility that the Hypothesis is false.

The Hypothesis has some very deep connections with observed physics. Finding a set of conditions where it breaks has far more revolutionary implications for both physics and mathematics than any confirmation.
 
2022-01-20 2:31:28 AM  
i.insider.comView Full Size
 
2022-01-20 8:38:14 AM  

PartTimeBuddha: Ambitwistor: As for the anthropic principle, no, theorems follow from axioms independently of any physical laws.

Well, you say that, but do you have any reason to prove it? You immediately run into Godel showing that infinite truths are true while being mathematically unprovable.


This has nothing to do with Goedel. The point is that the truth of a proposition depends only on the axioms and logic system being assumed, not on the laws of physics.
 
2022-01-20 8:40:05 AM  

ExYank: PartTimeBuddha: Ambitwistor: As for the anthropic principle, no, theorems follow from axioms independently of any physical laws.

Well, you say that, but do you have any reason to prove it? You immediately run into Godel showing that infinite truths are true while being mathematically unprovable.

The Riemann Conjecture appearing to hold for the Universe in which we find ourselves is a fair shake.

Hmmm seems to me Ambitwistor thinks it true but just not provable with the logical system of physics and holding out hope that mathematics can do the job. Meanwhile PTB is holding the harder position that it is true but math can't prove it. What I am not seeing is any support for the possibility that the Hypothesis is false.

The Hypothesis has some very deep connections with observed physics. Finding a set of conditions where it breaks has far more revolutionary implications for both physics and mathematics than any confirmation.


I'm not taking a position on whether the Riemann hypothesis is true, false, or undecidable. I'm saying it's truth doesn't depend on the laws of physics. Also that mapping it into a QFT won't necessarily make its truth any easier to decide.
 
2022-01-21 1:02:55 AM  
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