[HN Gopher] Monumental (if correct) advance in number theory pos...
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Monumental (if correct) advance in number theory posted to ArXiv by
Yitang Zhang
Yitang Zhang, the mathematician behind the 2013 breakthrough on
bounded gaps in primes, posted to the arxiv today a result which
(if correct) comes close to proving the nonexistence of Landau--
Siegel zeros: https://arxiv.org/abs/2211.02515. To give a sense of
the scale of this claim: If correct, Zhang's work is the most
significant progress towards the Generalized Riemann Hypothesis in
a century. Moreover, I think this result would not only be a more
significant advance than Zhang's previous breakthrough, but also
constitute a larger leap for number theory than Wiles' 1994 proof
of Fermat's Last Theorem (which was, in my opinion, the greatest
single achievement by an individual mathematician in the 20th
century). Some discussion / explanation of Siegel zeros and
Zhang's claim can be found here:
https://old.reddit.com/r/math/comments/y93a86/eliundergradua...
https://mathoverflow.net/questions/433949/consequences-resul... An
account of Zhang's remarkable story (and his previous breakthrough)
can be found here. Famously, prior to his breakthrough, he worked
at Subway and lived in his car:
https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
Author : gavagai691
Score : 257 points
Date : 2022-11-07 20:58 UTC (2 hours ago)
| EGreg wrote:
| Big if true
| [deleted]
| quickthrower2 wrote:
| Thanks for the ELI-not-a-mathematician. I am super excited for
| those involved, and understand none of it :-).
| mabbo wrote:
| I'm not a mathematician, but the story of Yitang Zhang
| desperately makes me want this paper to be correct.
|
| > Prior to getting back to academia, he worked for several years
| as an accountant and a delivery worker for a New York City
| restaurant. He also worked in a motel in Kentucky and in a Subway
| sandwich shop. A profile published in the Quanta Magazine reports
| that Zhang used to live in his car during the initial job-hunting
| days.
|
| https://en.wikipedia.org/wiki/Yitang_Zhang
| kenjackson wrote:
| Someone needs to make a movie about his life, or at least a
| documentary.
| gavagai691 wrote:
| There is a documentary, but I can't attest to its quality (I
| haven't watched it yet):
| http://www.zalafilms.com/films/countingabout.html.
| kenjackson wrote:
| Hmm... thanks, it looks quite good.
|
| But why is streaming rental for 24 hours? Why can't we do
| rentals for two weeks for videos? Is there a good reason to
| make it so difficult to stream? I don't want to watch it a
| million times. I just want to make it through once, but it
| takes me several sittings typically to finish a movie.
| SantalBlush wrote:
| Sadly, they would probably make it a schmaltzy Oscar-grab of
| a movie, like they did with Ramanujan's story.
| commandlinefan wrote:
| There was a scientist who did his best stuff while he was
| working as a patent clerk.
| iLoveOncall wrote:
| > To give a sense of the scale of this claim: If correct, Zhang's
| work is the most significant progress towards the Generalized
| Riemann Hypothesis in a century.
|
| Thanks, that totally failed to give any sense of scale.
| thechao wrote:
| It's the moral equivalent of making major headway against
| P!=NP; or, proving that there are no global hidden variables in
| QM; or, that there's a clear path ("just engineering") to room-
| temperature semiconductors.
| xxs wrote:
| >room-temperature semiconductors
|
| I suppose superconductors. Semiconductors are well in the
| room temperature regions :)
| XCSme wrote:
| I am pretty confident that I will never in my lifetime fully
| understand stuff like this (not the symbols themselves, but the
| overall meaning of each term and why it is like that):
| https://i.snipboard.io/by4tsH.jpg
| postalrat wrote:
| Maybe because you haven't tried to understand it?
|
| Can't be harder than learning the meaning behind these
| characters: https://www.pandatree.com/book/DiaryofWorm.jpg
| gardenhedge wrote:
| It would take a long path to understand it.. and the path
| would need to be filled with good resources.
| xyzzyz wrote:
| For the meaning, you just have to retrace back to where the
| things were defined, just like in programming. I am a
| mathematician, and I do not understand anything in the linked
| screenshot either (other than big O notation, which many people
| here should actually know!). FWIW, the author's preference for
| Greek letters is rather excessive for my personal taste.
| gavagai691 wrote:
| The Greek characters in the screenshot are standard for the
| subject. The letters "chi" and "psi" (in that order) are the
| preferred letters for denoting Dirichlet characters, and
| zeros of L-functions are always denoted by "rho."
| amelius wrote:
| Ok, but there is a difference between looking up definitions
| and understanding something.
| flobosg wrote:
| Alternative link for the New Yorker article:
| https://archive.is/TVj2A
| doodlebugging wrote:
| Should I be suspicious that the exponent in the equation is the
| current year? LOL
|
| I still need to read the article. The letter from Moh was
| interesting.
| gavagai691 wrote:
| Yes, you should :)
| gavagai691 wrote:
| Two additional notes:
|
| 1. Zhang posted an attempt at solving this problem in 2007 that
| he later more or less admitted was flawed:
| https://mathoverflow.net/questions/131221/yitang-zhangs-2007....
| But speaking with mathematicians who are intimately familiar with
| Zhang's previous work, there seems to be good reason to be
| optimistic nevertheless. First, the idea behind Zhang's proof is
| similar to the zero-repulsion ideas appearing in known results
| about Siegel zeros, and is thus reasonable. Second, Zhang seems
| to have matured late, and unlike the flawed 2007 paper, his 2013
| paper on bounded gaps in primes is meticulously written. He came
| a long way between those two papers, and he may have come even
| further since then.
|
| 2. Zhang is 67 years old. If the paper is correct, then Zhang
| constitutes a strong counterexample to G.H. Hardy's famous claims
| that "mathematics is a young man's game" and nobody alive today
| could say, as Hardy did, that "I do not know an instance of a
| major mathematical advance initiated by a man past fifty."
| markus_zhang wrote:
| I think Zhang's previous result was good enough to rebuff
| Hardy's claims.
|
| Actually I think Math is more or less a young people's game is
| because whence someone be super successful and famous it's
| kinda difficult psychological to retain the previous mental
| state and push out similar results.
| salty_biscuits wrote:
| Plenty of counterexamples to the claim
|
| https://mathoverflow.net/questions/25630/major-
| mathematical-...
| gavagai691 wrote:
| More counterexamples here too: https://old.reddit.com/r/mat
| h/comments/xw8i9w/oldest_person_....
| gavagai691 wrote:
| "I think Zhang's previous result was good enough to rebuff
| Hardy's claims."
|
| I agree.
| keepquestioning wrote:
| Is Yitang Zhang this century's Ramanujan?
| vecter wrote:
| Can someone explain the decimal constants that are used
| throughout the proof? For example, on page 52. It's rare to see
| these kinds of numbers used in mathematical proofs, but I'm sure
| they were chosen for good reasons.
| gavagai691 wrote:
| Unfortunately, nobody can explain anything like this right now.
| The paper was posted today, is 111 pages long, and it will
| likely take even professional mathematicians around a year to
| understand / check it completely.
| keepquestioning wrote:
| Is there a real world outcome to this result?
| gavagai691 wrote:
| There are many, many, many consequences for prime numbers--
| which to me are concrete enough to be interesting (and are way
| more concrete than what 99% of mathematicians work on!).
|
| On the other hand, I doubt this proof will help you to build a
| faster gizmo or something in the real world.. especially since
| it's proving something we really think is true (a consequence
| of the Generalized Riemann Hypothesis), and for real-world
| applications you can just assume the thing that we think is
| true is actually true (even if we haven't been able to prove it
| for a century). (E.g., you don't need to prove that factoring
| is hard to use cryptography for practical purposes..)
| davesque wrote:
| I think it's fair to say that this could lead to certain new
| things becoming known about the distribution of primes. This
| could have implications for cryptographic algorithms that
| depend on prime numbers being hard to find.
| cshimmin wrote:
| Wow, from the 2015 article:
|
| [A journal reviewer of his famous paper says]: "you should be
| careful. This guy posted a paper once, and it was wrong. He never
| published it, but he didn't take it down, either.' " The reader
| meant a paper that Zhang posted on the Web site arxiv.org, where
| mathematicians often post results before submitting them to a
| journal, in order to have them seen quickly. Zhang posted a paper
| in 2007 that fell short of a proof. It concerned another famous
| problem, the Landau-Siegel zeros conjecture, and he left it up
| because he hopes to correct it.
|
| Looks like he might have lived up to that!
| gavagai691 wrote:
| If the proof is right, Zhang is in contention for greatest
| living mathematician with _seven_ papers total, and on the
| basis of _two_ of them (and should win all the major prizes he
| has not won yet and is still eligible for: Abel, Wolf, etc.).
| Would truly live up to Gauss ' motto: "pauca, sed matura!"
| whytai wrote:
| Even more incredible is that his own advisor refused to write
| him letters of recommendation upon graduation [1]
| After graduation, Zhang had trouble finding an academic
| position. In a 2013 interview with Nautilus magazine, Zhang
| said he did not get a job after graduation. "During that
| period it was difficult to find a job in academics. That was
| a job market problem. Also, my advisor [Tzuong-Tsieng Moh]
| did not write me letters of recommendation." ... Moh claimed
| that Zhang never came back to him requesting recommendation
| letters. In a detailed profile published in The New Yorker
| magazine in February 2015, Alec Wilkinson wrote Zhang "parted
| unhappily" with Moh, and that Zhang "left Purdue without
| Moh's support, and, having published no papers, was unable to
| find an academic job". In 2018, responding to
| reports of his treatment of Zhang, Moh posted an update on
| his website. Moh wrote that Zhang "failed miserably" in
| proving Jacobian conjecture, "never published any paper on
| algebraic geometry" after leaving Purdue, and "wasted 7 years
| of his [Zhang's] own life and my [Moh's] time".
|
| 1. https://en.wikipedia.org/wiki/Yitang_Zhang
| gavagai691 wrote:
| Yes, if you want to see something incredible (in both the
| literal sense and the usual sense), read
| https://www.math.purdue.edu/~ttm/ZhangYt.pdf (by Moh).
| m_nyongesa wrote:
| In the earlier version I saw (I guess it consists of the
| non-bold parts), he didn't mention as much negative stuff
| about Zhang. His claim that Zhang "want to be famous all
| the time" I regard with suspicion.
| ummonk wrote:
| Yeah I started reading that from the Wiki citation.
| Yikes. Academia is brutal.
| Firmwarrior wrote:
| Man, it's so weird and pathetic
|
| All of these guys are probably a hundred times smarter
| than me or most of the other code monkeys working for the
| FANGMAN, but they're all squabbling over little 5-figure
| scraps of grant money.
| lordnacho wrote:
| > For some 10 years, I had recommended 100 mainland
| Chinese students to the department and all accepted by
| the department. I am always indebt to the trust of my
| judgements by the department. Only very few of them
| misbehaved, bit the hands which fed them, _none of them
| intended to murder their parents /friends_, almost all of
| them performed well and became well-liked.
|
| No murderers, great success!
| izzygonzalez wrote:
| arXiv https://arxiv.org/abs/2211.02515
|
| Discussion
| https://old.reddit.com/r/math/comments/y93a86/eliundergradua...
|
| https://mathoverflow.net/questions/433949/consequences-resul...
|
| New Yorker article
| https://www.newyorker.com/magazine/2015/02/02/pursuit-beauty
| [deleted]
| gavagai691 wrote:
| I didn't know how to make the links clickable. Thanks!
| dang wrote:
| We'll do that.
| vbezhenar wrote:
| I wish there will be time when "(if correct)" would be nonsense
| because correctness is checked in a fraction of second by
| something like CoQ and this proof is accompanied every known
| theorem.
| gavagai691 wrote:
| I can't resist saying one last thing about Siegel zeros: number
| theorists REALLY would like for this result to be correct because
| the possibility of Siegel zeros is unbelievably annoying. I mean
| mathematicians are supposed to enjoy challenges / difficulties,
| but Siegel zeros are just so recurrently irritating. The
| possibility of Siegel zeros means that in so many theorems you
| want to write down, you have to write caveats like "unless a
| Siegel zero exists," or split into two cases based on if Siegel
| zeros exist or don't exist, etc.
|
| But here is the worst (or "most mysterious," depending on your
| mood..) thing about Siegel zeros. Our best result about Siegel
| zeros (excluding for present discussion Zhang's work), namely
| Siegel's theorem, is ineffective. That is, it says "there exists
| some constant C > 0 such that..." but it can tell you nothing
| about that constant beyond that it is positive and finite (we say
| that the constant is "not effectively computable from the
| proof").*
|
| So then, if you try to use Siegel's theorem to prove things about
| primes, this ineffectivity trickles down (think "fruit of the
| poisoned tree"). For example, standard texts on analytic number
| theory include a proof of the following theorem: any sufficiently
| large odd integer is the sum of three primes. _However_ , the
| proof in most standard texts fundamentally _cannot_ tell you what
| the threshold for "sufficiently large" is, because the proof
| uses Siegel's theorem! In this particular case, it turns out that
| one can avoid Siegel's theorem, and in fact the statement "Any
| odd integer larger than _five_ is the sum of three primes " is
| now known
| https://en.wikipedia.org/wiki/Goldbach%27s_weak_conjecture. But
| it is certainly not always possible to avoid Siegel's theorem,
| and Zhang's result would make so many theorems which right now
| involve ineffectively computable constants effective.
|
| *Why is the constant not effectively computable? Because the
| proof proceeds basically like this. First: assume the Generalized
| Riemann Hypothesis. Then the result is trivial, Siegel zeros are
| exceptions to GRH and don't occur if GRH is true. Next, assume
| GRH is false. Take a "minimal" counterexample to GRH, and use it
| to "repel" or "exclude" other possible counterexamples.
| SantalBlush wrote:
| >I can't resist saying one last thing
|
| Please, keep going. This is good reading.
| gavagai691 wrote:
| In that case, you might find interesting these two short
| explanations I posted to Reddit about Siegel zeros (the
| second is a continuation of the first) :)
|
| https://old.reddit.com/r/math/comments/y93a86/eliundergradua.
| ..
|
| https://old.reddit.com/r/math/comments/ymlacu/professor_yita.
| ..
| m_nyongesa wrote:
| As an older person currently working on a PhD, this guy was and
| is a something of a hero to me. He has an interesting life story.
| He was very into math at an early age, so he's different from
| people like me who got interested in it later in life, but he's
| also different in that his family was sent down to the
| countryside in China. I remember reading a lot about him a few
| years ago and relating to some of the professional difficulties
| he had and also to some of his ways of thinking and approach to
| doing math.
|
| I never expected to see his name in a context like this again.
| I'm glad he's still being himself and working hard on what he
| loves.
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(page generated 2022-11-07 23:00 UTC)