After announcing he had achieved the solution to the Landau-Siegel zeros conjecture in mid-October, Yitang (Tom) Zhang, a Chinese-American mathematician and professor of mathematics at the University of California, Santa Barbara, released a related paper on November 5.
When Zhang gave an academic report for China’s Shandong University, screenshots and full texts of the above paper were spread within an alumni group of the School of Mathematical Sciences at Peking University. The title of the paper is “Discrete mean estimates and the Landau-Siegel Zero.” The full paper is 111 pages and consists of 18 sections, including an introduction, “The Set φ 1,” and “Proof of Proposition.”
In an acknowledgments section, Zhang wrote, “The basic ideas of the present proof were initially formed during my visit to the Institute for Advanced Study [Princeton] in the spring of 2014. I thank the Institute for Advanced Study for providing me with excellent conditions. I also thank Professor Peter Sarnak for his encouragement.” The paper was last revised at 07:50:53 local time on November 4.
Tian Gang, director of the Beijing International Center for Mathematical Research and director of Peking University’s Elite Math Class Committee, confirmed to The Paper that Zhang’s new report has been completed and submitted to arXiv, an open-access repository of electronic preprints and postprints, and is expected to be officially released on November 7. Up to now, the authenticity of this paper has not been publicly confirmed by Zhang.
As early as May 2007, Zhang submitted a paper entitled “On the Landau-Siegel Zeros Concept” to arXiv. The paper consisted of 13 sections and 54 pages, but the argument reportedly contained some defects.
This conjecture studied by Zhang is related to another dilemma in the history of mathematics – the Riemann Hypothesis Problem – which was put forward by mathematician Bernhard Riemann in 1859. Many consider it to be the most important unsolved problem in pure mathematics.
The Landau-Siegel zero conjecture is a type of potential counterexample to the generalized Riemann Hypothesis. According to an introduction by Chinese Science Daily in October this year, if there is a Landau-Siegel zero, the Riemann Hypothesis is wrong, and if the Landau-Siegel zero does not exist, it will not conflict with Riemann Hypothesis.
Has Yitang Zhang’s new paper solved the Landau-Siegel zero conjecture? This is the key issue at present. A number theorist who read the electronic version of the article told The Paper that Zhang’s argument had improved since the one posted to arXiv in 2007. On one hand, the results of Zhang’s new paper are revolutionary improvements compared with previous research results in this field, but on the other hand, it does not contain any truly earth-shattering discoveries.
“The new paper has not completely proved that the Landau-Siegel zero does not exist, so Zhang has not completely solved the Landau-Siegel zero conjecture at this stage. At the same time, judging from Zhang’s paper, its current research route is likely to fail to finally solve the conjecture,” the number theorist said, adding that “although this result can’t prove the conjecture, its strength is enough to exclude the Landau-Siegel zero in a great range. This range is enough for analytic number theorists to apply it to number theory problems and reach a lot of meaningful conclusions.”
Of course, the new paper must await further verification by professionals. The number theorist said that the paper has expanded from 54 pages in the 2007 edition to 111 pages in the 2022 edition, and that reviewing it will be a major task. Top experts will need to probe all details in the paper for several months, so it will be difficult to draw a conclusion quickly. In addition, on November 8, Zhang will give an academic report to Peking University’s teachers and students about the conjecture.