A mathematician who went from obscurity to luminary status in 2013 for cracking a century-old question about prime numbers now claims to have solved another. The problem is similar to—but distinct from—the Riemann hypothesis, which is considered one of the most important problems in mathematics.
Number theorist Yitang Zhang, who is based at the University of California, Santa Barbara, posted his proposed solution—a 111-page preprint—on the arXiv preprint server on 4 November. It has not yet been validated by his peers. But if it checks out, it will go some way towards taming the randomness of prime numbers, whole numbers that cannot be divided evenly by any number except themselves or 1.
The Landau–Siegel zeros conjecture is similar to—and, some suspect, less challenging than—the Riemann hypothesis, another question on the randomness of primes and one of the biggest unsolved mysteries in mathematics. Although it has been known for millennia that there are infinitely many prime numbers, there is no way to predict whether a given number will be prime; only the probability that it will be, given its size. Solving either the Riemann or Landau-Siegel problems would mean that the distribution of prime numbers does not have huge statistical fluctuations.
“For me in the field, this result would be massive,” says Andrew Granville, a number theorist at the University of Montreal in Canada. But he warns that others, including Zhang, have previously proposed solutions that turned out to be faulty, and that it will take a while for researchers to comb through Zhang’s argument to see if it is correct. “Right now, we’re very far from being certain.”
Zhang did not respond to Nature’s requests for comment. But he did write about his latest work on the Chinese website Zhihu. “As for the Landau–Siegel zeros conjecture, I didn’t think about giving up,” he wrote. He added: “As for my planning of the future, I won’t give away these math problems. I think I probably have to do mathematics all my life. I don’t know what to do without doing mathematics. People have asked questions about my retirement. I’ve said that if I leave math, I really won’t know how to live.” (His comments were translated into English by the website Pandaily.)
Passion for primes
Rumours had been circulating since mid-October that Zhang had made a breakthrough on the Landau–Siegel problem, and the mathematics community is certain to pay attention. Zhang has only one significant result to his name, but it is one for the ages. For years after attaining his PhD in 1991, he was estranged from his thesis adviser, working odd jobs to make ends meet. He then took up a teaching position at the University of New Hampshire in Durham, where he quietly chiselled away at his passion, the statistical properties of prime numbers. He posted a preprint on the Landau–Siegel conjecture in 2007, but mathematicians found problems and it was never published in a peer-reviewed journal.
Zhang’s first big breakthrough came in 2013, when he showed that although the gaps between subsequent primes grow larger and larger on average, there are infinitely many pairs that stay within a certain finite distance of each other. This was the first big step towards solving a major question in number theory—whether there are infinitely many pairs of primes that differ by just 2 units, such as the primes 5 and 7 or 11 and 13. (Number theorist James Maynard at the University of Oxford, UK, won a Fields Medal in July for improving on Zhang’s result, among other achievements.)
The problem Zhang now says he has solved dates back to the turn of the twentieth century, when mathematicians were exploring ways to tame the randomness of prime numbers. One way to count them is to partition them into a finite number of baskets, based on the remainders one gets when dividing a prime by another prime, denoted by p. For example, when divided by p = 5, a prime can give a remainder of 1, 2, 3 or 4. A result from the early nineteenth century shows that—once one considers a large enough statistical sample—these possibilities should ‘eventually’ occur with equal probability. But the big question, Granville explains, was how large the statistical sample should be for the equal-distribution pattern to show up: “What does ‘eventually’ mean? When do they start becoming well distributed?”
The methods known at the time suggested that the samples should be stupendously large, growing exponentially with the size of p. But a German mathematician called Carl Ludwig Siegel found a relatively simple formula that linked to this basket problem, and potentially made the samples much smaller. He showed that if, under certain circumstances, the formula did not yield 0, this was tantamount to proving the conjecture. “He removed all the dead wood out of the way and left just one massive oak to be felled,” Granville says. The problem, also formulated independently by another German mathematician, Edmund Landau, became known as the Landau–Siegel zeros conjecture. What Zhang now claims to have proved is a weaker version of it, but one that would have similar consequences regarding the distribution of primes.
The conjecture is a cousin of the Riemann hypothesis—a way to predict the probability that numbers in a certain range are prime that was devised by German mathematician Bernhard Riemann in 1859.
The Riemann hypothesis will probably remain at the top of mathematicians’ wishlists for years to come. Despite its importance, no attempts so far have made much progress. Only the bravest of mathematicians—often those who already have major accomplishments and prizes under their belts—publicly admit to trying to solve it. “It’s one of those things—you’re not supposed to talk about Riemann,” says Alex Kontorovich, a number theorist at Rutgers University in Piscataway, New Jersey. “People work secretly on it.”
Although progress towards solving the Riemann hypothesis has stalled, the Landau–Siegel problem offers similar insights, he adds. “Resolving any of these issues would be a major advancement in our understanding of the distribution of prime numbers.”
This article is reproduced with permission and was first published on November 11 2022.