Yitang Zhang, a Chinese-American mathematician, reportedly disclosed in an online salon organized by the Peking University Alumni Association on October 15 that he has proven the longstanding Landau-Siegel zeros theory. This finding is related to the Riemann hypothesis, a formula for the distribution of prime numbers that has remained unsolved for more than a century. However, the claim has not yet been fully verified, and it is reported that a relevant article of more than 100 pages will be sent to a preprint website in early November.
The Landau-Siegel zeros topic has represented one of the most difficult problems in number theory this century. It is a weak form of the Riemann hypothesis, which studies the existence of zeros in the DirichletL-function (a function defined on the whole complex plane). A century of research has shown that the Landau-Siegel zeros can be more difficult to solve than the Riemann hypothesis. Therefore, if Zhang Yitang has really proven that Landau-Siegel zeros exist, the Riemann hypothesis would be wrong. But for now, many people are more inclined to believe that Zhang proved the opposite result.
Regarding the news, a well-known Chinese blogger stated that “if Yitang Zhang proves the existance of Landau-Siegel zeros, then the Riemann conjecture could ‘die.’ Zhang will be the greatest mathematician in the past and future 50 years, no one else.” Others commented, “If Zhang can prove Landau-Siegel zeros, the probability can be equivalent to a person being struck by lightning twice.”
Zhang previously mentioned many times that he was paying attention to the Riemann hypothesis. As early as 2007, he submitted an article entitled “On the Landau-Siegel Zeros Conjecture” in arXiv, but argued later that there were some bugs. Since then, he has been hoping to fix it. In 2019, he said that he had made some gratifying progress on this conjecture.
In April, 2013, Zhang published Bounded Gaps Between Primes in The Annals of Mathematics, in which he proved that there are infinite pairs of prime numbers with gaps of less than 70 million, thus making a qualitative breakthrough in the number theory problem of twin prime number conjectures. This manuscript was reviewed and approved in The Annals of Mathematics, which is regarded for its strictness, in just over a month. This broke the record for the fastest review since the journal’s inception, stirring up huge waves in the global mathematics field, and making Zhang a famous number theorist.
In a review directed to The Annals of Mathematics, Henryk Iwaniec, a Polish mathematician who is one of the top analytic number theorists today, suggested accepting this paper, saying that “the author successfully proved a landmark theorem in the field of prime number distribution. We studied the paper carefully, but found no flaws.”
Throughout more than 40 years of research, Zhang has published few papers. In addition to a paper about prime conjecture in 2013, there were two other articles published in the Duke Mathematical Journal and Acta Mathematica Sinica, both of which were related to the Riemann hypothesis.