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.
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。
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.
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:
https://www.ias.edu/video/jointiasnts/2013/0926-YitangZhang
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.
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.
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).
