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](https://arxiv.org/abs/0705.4306)? 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](https://arxiv.org/abs/2211.02515).



Yitang Zhang's 2007 preprint on Landau–Siegel zeros
This is not an answer regarding the paper, but I think should be helpful. During a recent [interview](https://web.archive.org/web/20140831055114/http://blog.sina.com.cn/s/blog_c24597bf0101ctdp.html) (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](https://web.archive.org/web/20140420120717/http://www.math.ias.edu/node/5320) - 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.
> 
> 
> 


