A year ago April, the editors of the Annals of Mathematics, a journal published by the Institute and Princeton University, received an email with a submission by an unknown mathematician. “Bounded Gaps Between Primes” by Yitang Zhang, an adjunct professor at the University of New Hampshire, immediately caught the attention of the editors as well as Professors in the School of Mathematics. It was refereed by mathematicians who were visiting the Institute at the time and was accepted three weeks later, an unusually expedited pace.
“He is not a fellow who had done much before,” says Peter Sarnak, Professor in the School of Mathematics. “Nobody knew him. Thanks to the refereeing process, there were a lot of vibes here at the Institute long before the newspapers heard of it. His result was spectacular.”
A month after he submitted his paper, Zhang’s result was reported in the New York Times, “Solving a Riddle of Primes,” and in subsequent publications. Zhang’s theorem relates to the twin primes conjecture, which asserts that there are an infinite number of prime numbers that are only two numbers apart. Such pairs are more frequent at the beginning of the number line and less so among large numbers.
Zhang’s result does not prove that there are an infinite number of twin primes; rather, it gives a finite upper bound––70 million––for which the gaps between pairs of primes persist infinitely often. His work is dependent on findings by Institute Faculty and Members, in particular the Bombieri-Vinogradov theorem named in part for Enrico Bombieri, Professor Emeritus in the School. Zhang was immediately invited to give a lecture, “Distribution of Primes in Arithmetic Progressions with Applications,” last fall, and he accepted an invitation to come as a Member for the spring term.
A deep extension of the Bombieri-Vinogradov theorem had been developed through the efforts of Bombieri and former Institute Members––Henryk Iwaniec and John Friedlander––along with Étienne Fouvry. “But the extension was not flexible enough to be used spectacularly,” says Sarnak. “Zhang made it technically flexible, allowing for its application to bounded gaps in a striking way.”
With ingenious and sustained effort, Zhang combined the ideas of Pierre Deligne, Professor Emeritus in the School, with this deep extension of the Bombieri-Vinogradov theorem and work by former Members Daniel Goldston, János Pintz, and Cem Yildirim on bounded gaps.
“I knew this [twin prime] problem very early, when I was an elementary school student in Shanghai,” says Zhang. “I was very interested in many math problems, not only this one, but many number theory problems and, of course, primes.”
During Mao’s Cultural Revolution, Zhang was raised by his grandmother, an illiterate factory worker. “During that time, it was difficult to find a person who had a college education,” says Zhang. “It was difficult to find a book.” He did not attend middle school or high school, and instead taught himself mathematics from books that he had collected from a local high school prior to the revolution.
He went on to attend Peking University where he was a star student, earning bachelor’s and master’s degrees. “I met his adviser in China, who is very proud of him, and he said that Zhang was the most promising student in the year that he finished,” says Sarnak. “He was always, I think, considered very talented, but what is unusual about him is that he is not doing incremental stuff. He is very fixated on mathematics, and he is not distracted by other things. He is extremely focused.”
Zhang earned his doctorate from Purdue University, and, in 1999, he moved to New Hampshire where a few of his classmates from Peking University were on the Faculty and helped him get a job. He had spent periods working as an accountant and at a Subway sandwich shop.
“I was born for math,” says Zhang. “For many years, the situation was not easy, but I didn’t give up. I just kept going, kept pushing. Curiosity was of first-rank importance––it is what makes mathematics an indispensible part of my life.”
Zhang spent three years working on the bounded gap problem. On July 3, 2012, while visiting a friend’s house in Colorado, he made his crucial breakthrough. “I tried to really make it a vacation. I didn’t bring any book, notes, sheets, or my computer. I didn’t use a pen,” says Zhang. “But still I couldn’t get rid of this completely. Sometimes I still tried to think about this point, this one small gap. How can we cross it?” As he was waiting to leave for a symphony concert that his friend was conducting, Zhang went into the backyard and started looking for some deer. “There are many deer sometimes,” says Zhang. “I didn’t see any deer, but I got the idea.”
It took him a little over eight months before he submitted it to the Annals. He says he didn’t feel very excited; he felt peaceful. “I spent a lot of time checking all of the details and simplifying many, many points,” says Zhang. “I was asked by somebody, ‘Could you sleep during that time?’ And I said, ‘Yes, I slept very well.’”
No one he knew understood this work, so there was no one aside from himself who could check it for him. He waited two months after finishing it to submit it. “I told myself I should be very careful and double check all things,” says Zhang. “That took a long time.”
He had not expected that his paper would be accepted so quickly. The day after its acceptance, he received many emails, followed by invitations. “I accepted some invitations,” says Zhang, “but what I want to do is try to just keep quiet and live a very quiet, very peaceful life.”
At the Institute, Zhang has been working on a very difficult problem related to the spacing of the zeros of the Riemann zeta function, which would have spectacular applications, if solved. “Many people have worked on it and have thought they have solved it, but it is very elusive,” says Sarnak. “Zhang has worked on it for many years. He has proved that he is able to stick with something in a very stubborn way, and that is what it takes to do something like this. He never gives up. He likes being left alone to work, and the Institute is the ideal environment for him to do that.”
In the meantime, Zhang’s bounded gap theorem has been proven in a much more elementary way by James Maynard, a postdoc at the University of Montreal. His proof is based primarily on the Selberg sieve of the late Atle Selberg, Professor in the School. Maynard visited IAS in the spring to give a seminar, “Small Gaps between Primes,” on his result. “So this was another shock,” says Sarnak. “As far as gaps between primes, Maynard’s work is just as dramatic at a different level, although Zhang’s was spectacular. The first is always the most important. Zhang’s breakthrough, which was first used in this bounded gap context, will be used in many other ways. It is a very fundamental theorem.”
Recommended Reading: Read Thomas Yin's interview with Yitang Zhang, "After Prime Proof, an Unlikely Star Rises," in Quanta Magazine (April 2, 2015): www.quantamagazine.org/20150402-prime-proof-zhang-interview.