No Landau-Siegel zeros?

A couple weeks ago rumors were circulating that Yitang Zhang was claiming a proof of a longstanding open conjecture in number theory, the “no Landau-Siegel zeros” conjecture. Such a proof would be a very major new result. Zhang was a little-known mathematician back in 2013 when he announced a proof of another very major result, on the twin prime conjecture. Before that, he had a 2007 arXiv preprint claiming a proof of the Landau-Siegel zeros conjecture, but this was never published and known to experts to have problems such that at best the argument was incomplete.

Zhang now has a new paper on the arXiv, claiming a complete proof. The strategy of the proof is the same as in the earlier paper, but he now believes that he has a complete argument. At 110 pages the argument in the paper is quite long and intricate. It may take experts a while to go through it carefully and check it. Note that this is a very different story than the Mochizuki/abc conjecture story: Zhang’s argument use conventional methods and is written out carefully in a manner that should allow experts to readily follow it and check it.

For an explanation of what the conjecture says and what its significance is, I’m not competent to do much more than refer you to the relevant Wikipedia article. For a MathOverflow discussion of the problems with the earlier proof, see here, for consequences of the new proof, see here.

Update: I’m hearing that the above is not quite right, that what Zhang proves is weaker than the conjecture, although strong enough for many of its interesting implications. Perhaps someone better informed can explain the difference…

Update: Davide Castelvecchi at Nature has a news story here.

Update: Via David Roberts on Twitter, Zhang answers some questions about the paper here.

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18 Responses to No Landau-Siegel zeros?

  1. Anon says:

    I believe what he proves is strictly weaker than the non-existence of zeros.

  2. Peter Woit says:

    Anon,
    Thanks! I had asked an expert about this, but there must be some detail I misunderstood. Maybe someone who understands the difference can explain…

  3. Will Sawin says:

    The difference is pretty simple.

    The “no Siegel zeros” conjecture is that the distance of any real zero of L(s,chi_D) from 1 is bounded below by a constant times 1/log D.

    The result Yitang Zhang claims is that this distance is bounded below by a constant times 1/(log D)^2024.

    So both involve the claim that there are no zeroes too close to the point 1, but one is stronger than the other by a large power.

    However, Yitang Zhang’s stated result is a big improvement on what was known before – the only effective estimates had a lower bound for the distance of something close to 1/sqrt(D).

    The situation is very analogous to the twin primes paper where he improved the bound from the previously known infinity to 70,000,000, and it was expected that optimization of the method would rapidly get it down further, though falling short of the conjectural value of 2.

  4. anon says:

    And even in this case he says that the exponent could be optimized further, but the method falls short of reducing it to 1.

  5. Georges E. Melki says:

    Will Sawin,

    Thanks for the explanation! It helps greatly…
    However, I checked the ArXiv paper, and the power of log D is (-2022) and not (-2024).

    Regards

  6. Will Sawin says:

    @Georges E. Melki

    I am referring to Theorem 2 of the paper, on the zeroes, not Theorem 1, on the special value.

  7. That’s unfortunate about Davide C’s article getting the facts wrong. Random on Twitter like me jumping the gun a little pales in comparison with reporter with access to experts.

  8. Peter Woit says:

    David Roberts,
    Not sure what you mean about Davide C’s article getting facts wrong. As far as I know, the only one who had facts wrong here was me, and the problem was the combination of my access to experts and my own ignorance (which led me it seems to misinterpret what one of them was telling me).

    And I should have been following your discussions of this on Twitter, for instance the latest:
    https://twitter.com/HigherGeometer/status/1590871536457879552

  9. Peter Woit says:

    Rereading the Nature article I see it doesn’t make clear that what Zhang claims to have proved is somewhat weaker than the usual Siegel zeros conjecture. Perhaps Castelvecchi made the mistake of reading my blog post and trusting me to get it right…

  10. Yeah, it’s not explicitly said, but the implication is that Zhang proved the LS zeroes conjecture, since his earlier, incomplete preprint on that was mentioned, and there’s a bunch of talk about it and quotes and so on.

    I was a bit enthusiastic and misunderstood the intro, so at some point *I* claimed he’d solved the LS zeroes conjecture. But people set me right pretty quickly.

    But I’m sure a correction will be made before too long, with a little note pointing out the change to the article 🙂

  11. Anonymous says:

    Will Sawin,

    What is the best known ineffective bound previously ? Is it Siegel $1/D^\epsilon$ ?
    So Zhang’s effective bound is even better than the best known ineffective bound ?
    So this is a hugh improvement, if correct.

  12. Davide Castelvecchi says:

    My understanding was that indeed the statement was weaker than the one normally referred to as Landau-Siegel conjecture but that this was strong enough for many of its interesting implications, as Peter put it — and it was those implications that were the focus of my article anyway

  13. @Davide

    and Steven Strogatz just tweeted your article, and based on it he claimed Zhang has proved the L–S zeroes conjecture. A very small edit would make it clear what has and hasn’t been proved, but you know the process and the magazine better than I do.

  14. Jim Eadon says:

    Hi, may an expert please comment about the implication, for the educated layman (MSc Physics), of this new proof for the generalised Riemann hypothesis?

  15. Davide Castelvecchi says:

    @David: I know, it’s just difficult to do on a weekend 🙂

  16. @Jim none, unfortunately, as the implications go the other way. But it might be seen as adding to our confidence that GRH is true.

    @Davide that’s definitely true!

  17. Conrad says:

    As of now, there are some significant issues with the paper that need to be addressed by the author (some seem to be just editing issues, but some seem to be substantive), so there will be some time until one can see if the author is able to address them and the paper to be accepted as valid; see the following https://terrytao.wordpress.com/2021/09/15/the-hardy-littlewood-chowla-conjecture-in-the-presence-of-a-siegel-zero/comment-page-1/#comment-660236 and https://mathoverflow.net/questions/434583/philosophy-behind-zhangs-2022-preprint-on-the-landau-siegel-zero?noredirect=1#comment1119426_434583

  18. Nicola Maiorino says:

    Peter (and others),

    I just came across this fairly readable article in ‘Science, The Wire’ titled, “Riemann Hypothesis: What Yitang Zhang’s Paper Means and Why You Should Care”:

    https://science.thewire.in/the-sciences/yitang-zhang-landau-siegel-zeroes-riemann-hypothesis/

    The article presents a non-expert introduction to the Riemann Hypothesis, and trys to clear up precisely what it is that Zhang has proved (at least, tentatively so), and not proved.

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