The recent sensational news on bounded gaps between primes made me wonder: what is the status of Yitang Zhang's earlier arXiv preprint on LandauSiegel 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?

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: http://video.ias.edu/jointiasnts/2013/0926YitangZhang and a summary can be found at here: 


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 wellwritten 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 Note that the paper was first arXived in May 2007, with no updates since then, and it hasn't been withdrawn.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). 


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
Yitang Zhang should at least provide a respective comment at his arxivarticle 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. 

