# Yitang Zhang's preprint on Landau-Siegel zeros

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

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Good point. . – Péter Komjáth May 20 '13 at 13:39
The second paragraph on page 2 in Zhang's paper: Although Theorem 2 does not completely eliminate the Landau-Siegel zeros in their original deﬁnition, our results will be sufficient for various applications in both of the analytic number theory and algebraic number theory. – zy_ May 20 '13 at 14:44
@zy: I am not sure what you mean. Still the result(s) would be big news. (Also I'd assume GH was aware of the main results claimed when asking the question.) – user9072 May 20 '13 at 15:09
@zy and @quid: I agree with both of you. – GH from MO May 20 '13 at 16:56
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. – Terry Tao May 21 '13 at 2:06

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

" 问：前几天我去北京遇到葛立明，他说当时你在做个大问题，快做出来了。所以找你去新罕布什尔大学。

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.

Edit:

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:

http://video.ias.edu/jointiasnts/2013/0926-YitangZhang

and a summary can be found at here:

http://www.math.ias.edu/node/5320

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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. – GH from MO Aug 24 '13 at 17:54

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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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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@unknown: My question was about the status of this preprint, i.e. if it had been checked and if it is correct. – GH from MO May 20 '13 at 19:25
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. – user9072 May 20 '13 at 23:47
@quid: you mean "downvote this answer" not "question", right? – Suvrit May 21 '13 at 1:29
@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). – user9072 May 21 '13 at 13:26