A mathematician who rose from obscurity to prominence in 2013 when he solved a century-old question about prime numbers now claims to have solved another. The problem is similar to—but different from—the Riemann hypothesis, which is considered one of the most important problems in mathematics. Number theorist Yitang Zhang, who works at the University of California, Santa Barbara, posted his proposed solution—a 111-page preprint—to the arXiv preprint server on November 41. His peers have yet to confirm this. But if it opts out, it tames the randomness of prime numbers, integers that cannot be evenly divided by any number except itself or 1.
The Landau-Siegel zero conjecture is similar to—and some believe less challenging than—the Riemann hypothesis, another question about the randomness of prime numbers and one of the greatest 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, given its size. Solving Riemann or Landau-Siegel problems would imply that the distribution of prime numbers does not have large statistical fluctuations.
“For me in the field, this result would be huge,” says Andrew Granville, a number theorist at the University of Montreal in Canada. But he cautions that others, including Zhang, have previously proposed solutions that have turned out to be flawed, and that it will take time for researchers to examine Zhang’s argument to see if it is correct. “We’re not sure right now.
Zhang did not respond to Nature’s requests for comment. But he wrote about his latest work on the Chinese website Zhihu. “As for the Landau-Siegel estimate of zeros, I didn’t think about giving up,” he wrote. He added: “As for my future planning, I will not reveal these mathematical problems. I think I probably have to do math all my life. I don’t know what to do without math. People asked me about my pension. I said that if I leave mathematics, I really won’t know how to live.”
Passion for prime numbers
Rumors have been circulating since mid-October that Zhang has made a breakthrough in the Landau-Siegel problem, and the math community is certainly paying attention. Zhang only has one major result to his name, but it’s one for the ages. After receiving his doctorate in 1991, he was estranged from his thesis advisor for several years and worked odd jobs to make ends meet. He then took a teaching position at the University of New Hampshire in Durham, where he quietly pursued his passion, the statistical properties of prime numbers. He published a preprint of the Landau-Siegel2 problem in 2007, but mathematicians found problems and it was never published in a peer-reviewed journal.
Zhang’s first major breakthrough came in 2013, when he showed that even as the gaps between successive primes get larger and larger on average, there are infinitely many pairs that remain at some finite distance 3 . This was the first major step toward solving a major question in number theory—whether there are infinitely many pairs of primes that differ by only 2 units, such as the primes 5 and 7 or 11 and 13. (Number theorist James Maynard at the University of Oxford in Great Britain he won the Fields Medal in July for improving Zhang’s result.)
The problem Zhang has now 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 divide them into a finite number of bins based on the remainders we get when dividing a prime number by another prime number, denoted p. For example, when p = 5 is divided, the prime number can give a remainder of 1, 2, 3, or 4 .A result from the early nineteenth century shows that—once we consider a sufficiently large statistical sample—these possibilities should “eventually” occur with equal probability. But the big question, Granville explains, was how large the statistical sample would have to be for the pattern of equal distribution to show up: “What does ‘eventually’ mean? When will they start to distribute well?”
Methods known at the time suggested that the samples should be enormously large and should grow exponentially with the size of p. But the German mathematician Carl Ludwig Siegel found a relatively simple formula that related this basket problem and potentially made the samples much smaller. He showed that if, under certain circumstances, this formula did not yield 0, it was tantamount to proving the conjecture. “He cleared all the dead wood out of the way and only had one massive oak tree cut down,” says Granville. The problem, also independently formulated by Edmund Landau, became known as the Landau–Siegel null conjecture.
Unsolved problem
This conjecture is related to the Riemann hypothesis—a way of predicting the probability that numbers in a certain range are prime, devised by the mathematician Bernhard Riemann in 1859. The Riemann hypothesis is likely to remain at the top of mathematicians’ wish lists for years to come. In 2000, the Clay Mathematics Institute, now based in Oxford, included the hypothesis on its list of seven millennium problems and offered a US$1 million prize to anyone who could solve it. But despite its importance, no attempts so far have made much progress.
Only the bravest of mathematicians—often those who already have great achievements and prizes under their belts—publicly admit that they are trying to solve it. “It’s one of those things — you shouldn’t talk about Riemann,” says Alex Kontorovich, a number theorist at Rutgers University in Piscataway, New Jersey. “People are secretly working on it. Although progress toward solving the Riemann hypothesis has stalled, the Landau-Siegel problem offers similar insights, he adds. “Resolving any of these problems would represent a major advance in our understanding of the distribution of prime numbers.”
Source Reference: https://www.nature.com/articles/d41586-022-03689-2
