Terence Tao, an Australian mathematician and winner of the Fields Medal, said on November 14 that he had read the recent paper by Yitang Zhang on proving the Landau-Siegel zeros conjecture. Tao commented that the basic accuracy of the paper has not yet been confirmed, and there are some printing errors and technical problems that have been forwarded to Zhang for clarification.
Specifically, Tao listed some missing equations on pages 63-67, 70, 98-99 and 109. It can be seen that he not only read the paper from beginning to end, but that he also encountered these obstacles in the process of trying to verify it.
In fact, regarding the imperfection of the thesis, Zhang himself has been very clear. He mentioned this during an online salon organized by the Peking University Alumni Association, saying, “My article is very complicated. In short, I know that I have made it. Of course, there are still some problems with writing it well, so I don’t want to submit it first, but I will post it on arXiv.”
Tao wrote in his comments that these problems, along with some more serious problems, could be corrected, but it will take some time. He did not specify what the “more serious problems” were. In addition, he specifically mentioned that he didn’t want to put pressure on Zhang and hoped he could proofread it carefully instead of uploading the revised version in a hurry, so he suggested that everyone wait patiently.
Tao’s views on the matter have garnered considerable attention. He is considered a master mathematician in nearly 10 important research fields, such as harmonic analysis, partial differential equations, combinatorial mathematics, analytic number theory and algebraic number theory. At the age of 31, he won the Fields Medal, SASTRA Ramanujan Prize and MacArthur Fellow. He is a professor of mathematics at UCLA, where he holds the James and Carol Collins chair.
Tao and Zhang have overlapping research fields, and Tao has made important improvements to Zhang’s last important achievement – the proof of the twin prime number conjecture. Whether Tao will make further contributions has naturally become a question of widespread interest.
As for Chinese scholars, Pan Chengbiao, Zhang’s tutor at Peking University, studied his academic report. In addition to proving the results, Pan also paid special attention to Zhang’s new method of improvement. Zhang replied at that time that “listening to Pan’s affirmation is more valuable than listening to the praise of 10,000 people.”
Roger Heath-Brown FRS, a mathematician in the field of analytic number theory, said he thought that Zhang’s paper was “clearly written and wise in strategy.” In addition to Heath-Brown, Andrew Granville of Montreal University and other scholars in the field of number theory agree that it takes a long time to study this paper well, and it is still too early to draw conclusions.
Regarding his future, 67-year-old Zhang mentioned in a long article on Chinese Q&A platform Zhihu that he had never considered retirement. “If I do leave math, I really don’t know how I should live.”