The recent sensational news on bounded gaps between primes made me wonder: what is the status of Yitang Zhang's earlier arXiv preprint On the Landau-Siegel zeros conjecture? If this result is correct, then (in my opinion) it is even bigger news for analytic number theory. Has anyone checked this paper carefully?

Update (November 2022). Yitang Zhang posted a new arXiv preprint Discrete mean estimates and the Landau-Siegel zero.

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    $\begingroup$ Good point. . $\endgroup$ May 20, 2013 at 13:39
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    $\begingroup$ If one wishes to inspect the manuscript carefully, I would focus attention on Lemma 7.1, as this is a crucial lemma whose proof is extremely sketchy, to put it mildly. $\endgroup$
    – Terry Tao
    May 21, 2013 at 2:06
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    $\begingroup$ I'm not sure if you're still interested, but it seems Zhang is going to release a preprint next month with progress on this. $\endgroup$
    – David Roberts
    Oct 16, 2022 at 14:05
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    $\begingroup$ It's released: pan.baidu.com/s/1GM61FrLynSfpSn67SoaHow?pwd=1105 , password: 1105 $\endgroup$
    – athos
    Nov 5, 2022 at 8:25
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    $\begingroup$ @math110 I should point out that the new (2022) claim is weaker than the non-existence of Landau–Siegel zeroes, and that Zhang says the methods seem not strong enough to get that result. $\endgroup$
    – David Roberts
    Nov 5, 2022 at 12:38

3 Answers 3


This is not an answer regarding the paper, but I think should be helpful. During a recent interview (in Chinese), he commented:


答: 那是关于Siegel零点的工作我有一篇网络文章是不完整的。目前我还不敢说我完全做成但是的确有很大进展。孪生质数这个问题我做了三、四年。但希望大家不要误会,这个问题我是想了三、四年,但不是说我所有时间都在做它。一直到去年9月,我因为肯定可以做出来了,才暂时放下别的东西。

The highlighted part can be translated as:

...that is my work on Siegel zeros. I have a paper online, which is incomplete. I cannot say I have finished the work by now, but I did made remarkable progress...

So the paper is unfinished, and we can wait until later when it is officially published.


I guess it is public. But in case OP or other people do not know, he has visited IAS and gave some lectures in public on this topic. The videos are available at here:


and a summary can be found at here:

http://www.math.ias.edu/node/5320 (Wayback Machine - the below is copied from here with the MathJax reinstated)

Let $\chi$ be a primitive real character. We first establish a relationship between the existence of the Landau-Siegel zero of $L(s,\chi)$ and the distribution of zeros of the Dirichlet $L$-function $L(s,\psi)$, with $\psi$ belonging to a set $\Psi$ of primitive characters, in a region $\Omega$. It is shown that if the Landau-Siegel zero exists (equivalently, $L(1,\chi)$ is small), then, for most $\psi \in \Psi$, not only all the zeros of $L(s,\psi)$ in $\Omega$ are simple and lie on the critical line, but also the gaps between consecutive zeros are close to integral multiples of the half of the average gap. In comparison with certain conjectures on the vertical distribution of zeros of $\zeta(s)$, it is reasonable to believe that the gap assertion would fail to hold. In order to derive a contradiction from the gap assertion, we attempt to reduce the problem to evaluating a certain discrete mean; the idea is motivated by the work of Conrey, Ghosh and Gonek on the simple zeros of $\zeta(s)$. We shall describe the coefficient of the main term and provide some numerical evidences. In some special cases, the problem is further reduced to calculating small positive eigenvalues of linear integral equations with Hermitian kernels.

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    $\begingroup$ Thank you! I accept this as official answer, since indeed it is very helpful and I do not hope for a better one in the near future. $\endgroup$
    – GH from MO
    Aug 24, 2013 at 17:54
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    $\begingroup$ @GHfromMO: It seems like he is going to release a new preprint on this problem in early November: pandaily.com/… $\endgroup$ Oct 22, 2022 at 17:18

I started a careful study of the paper but stopped after already stumbling over Lemma 2.3. Let me cite from an email in Jan 2008

I can't follow the proof of Lemma 2.3, which is a key in proving Lemmas 2.4-2.6 and therefore also Prop. 2.7 and therefore also the Theorem: As far as I understand, the paper estimates (see last line on page 8) the Supremum (over the s in Omega_1) of the left sum via standard integral-estimation by L^2 times the supremum (over the w in R_1) of the respected sum. The last equals the sum at a special w in R_1 (Maximum-principle of continuous functions), but this w is (highly) dependent on the psi. Thats why I dont understand how one can then use the great sieve as in page 9 top in order to estimate the initial sum at the left of 2.11, because the great sieve applies only when the coefficients are (of course) independent of psi. Contrary, if the way of proving Lemma 2.3 was actually as sketched above, then I dont see it necessary to go the extra way over the integral, but one could estimate immediately. That's why I think I may have missed a point. ... (I don't think that a variation of the statement could help, since Lemmas 2.4-2.6 are using very precisely the full statement of Lemma 2.3, the same with Prop. 2.7).

Yitang Zhang should at least provide a respective comment at his arxiv-article where/how incomplete the paper is. Frankly put, not being fully transparent about the state of this work and thus have other people spend their time on it (not knowing where it is lacking) is absolutely ridiculous.

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    $\begingroup$ This is just rude. $\endgroup$ Dec 5, 2018 at 14:47
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    $\begingroup$ It is rude, but it is not just rude. It is also true, from the viewpoint of someone, who is interested in the science, and progress, of mathematics. $\endgroup$
    – tyrex
    Dec 6, 2018 at 14:24
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    $\begingroup$ @Bombyxmori I would be happy if he could say somewhere: "As of 2018, I believe the proof is incomplete in Lemma x.x". Or "I believe the proof is complete, but peer-review believes that Lemma x.x is wrong/not sufficiently proven". His stated theorem is a huge result, so if he could just add those 10-20 words somewhere, it would clarify where we stand. $\endgroup$
    – tyrex
    Dec 10, 2018 at 18:55
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    $\begingroup$ @Bombyxmori Given that so many more are interested in his result, the natural thing to do for him would be to give a statement, which addresses those questions. This seems the natural thing to do from a research perspective. The preprint is now 11 years old, it's not like he published it last month. $\endgroup$
    – tyrex
    Dec 10, 2018 at 22:41
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    $\begingroup$ I do not think this make any sense. Why would someone act differently because there are irrelevant spectators paying attention to what he is doing? Also in my personal experience NO ONE remembers proof of a specific lemma in an eleven years old paper. I agree that you have a point; Yitang Zhang should have been more cautious with the paper when he put it on arxiv. But you probably should also realize he could have chosen not to share it with the public to begin with, and there is nothing anyone can say about it. $\endgroup$ Dec 11, 2018 at 3:50

I would've preferred to write a small comment, but I'm not able to do that for some reason. I looked briefly at the paper, and it seems well-written and readable. It's also quite long (54 pages). Regarding the above comments on the second paragraph, I think the paragraph is a fair description of the significance of the result. In particular, if the main result is correct, then this would be a breakthrough, and a nice story. Note that the paper was first arXived in May 2007, with no updates since then, and it hasn't been withdrawn.

The main result states: For any real primitive character $\chi$ of modulus $D$ we have $L(\sigma,\chi)\ne 0$ for $\sigma > 1 − \frac{c_2}{(\log D)^{19} \log\log D}$ where $c_2>0$ is an effectively computable constant. As far as I can tell, there is no estimate of $c_2$ given in the paper, but this seems unimportant since the author already claims that, with extra effort, it's possible remove the power of $\log \log D$, and even some powers of $\log D$.

This is the second major claim in analytic number theory within a couple of weeks (albeit one first made in 2007, but alas given little attention, possibly because announcing a major result in isolation like that tends to have an opposite effect).

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    $\begingroup$ @unknown: My question was about the status of this preprint, i.e. if it had been checked and if it is correct. $\endgroup$
    – GH from MO
    May 20, 2013 at 19:25
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    $\begingroup$ I downvote this question for the formalistic reason that if this answer has positive score it will make this question (formallu) answered while it is not. $\endgroup$
    – user9072
    May 20, 2013 at 23:47
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    $\begingroup$ @quid: you mean "downvote this answer" not "question", right? $\endgroup$
    – Suvrit
    May 21, 2013 at 1:29
  • $\begingroup$ @S. Sra: yes, I downvoted this answer so that the question is not 'formally' answered by having an answer with a positive score. @unknown: the reason you were not able to leave a comment is that you do not yet have 50 (or more points); for various things one can do on the site one needs a certain amount of points first (for details see faq, the section on reputation, link at the top). $\endgroup$
    – user9072
    May 21, 2013 at 13:26

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