Skepticism surrounds the attempt of the renowned mathematician's proof of a 160-year-old hypothesis | Science

A famous mathematician said today that he had solved the Riemann hypothesis, a problem related to the distribution of prime numbers that has not been solved for nearly 160 years. Michael Atiyah, a mathematician emeritus at the University of Edinburgh, has presented what he describes as a "simple proof" that relies on a tool of a seemingly physical problem. But many experts doubt its validity, especially because Atiyah, 89, has made mistakes in recent years.

"It is highly unlikely that what he showed in the presentation looks like a proof of Riemann's hypothesis as we know it," says Jørgen Veisdal, an economist at the Norwegian University of Applied Sciences. Science and Technology of Trondheim. "It's just too vague and nonspecific." Veisdal added that he should review the written evidence more closely to make a final judgment.

The Riemann hypothesis, one of the last major unresolved problems in mathematics, was first proposed in 1859 by the German mathematician Bernhard Riemann. This is an assumption about prime numbers, such as two, three, five, seven and eleven, which can only be divided by one or the other. They become less frequent, separated by more and more distant distances on the digital line. Riemann discovered that the key to understanding their distribution lies in another set of numbers, the zeros of a function called the Riemann zeta function, which has both real and imaginary inputs. And he invented a formula for calculating how many prime numbers there are, up to a threshold, and at which intervals these prime numbers occur, based on the zeros of the zeta function.

However, the Riemann formula applies only if it is assumed that the real parts of these zeros are all equal to half. Reimann proved this property for the first prime numbers and in the last century it has been shown that the computation worked for a large number of prime numbers, but it remains to prove it formally and indisputably at infinity. Evidence would not only be a $ 1 million reward for solving one of the millennium problems set by the Clay Mathematics Institute in 2000, but it could also have applications in predicting important prime numbers in cryptography.

A giant in his field, Atiyah has made major contributions to geometry, topology and theoretical physics. He received the two major mathematics awards, the Fields Medal in 1966 and the Abel Award in 2004. But despite a long and prolific career, Riemann's claim follows more recent and failed evidence.

In 2017, Atiyah said The temperature from London, he had converted the 255-page Feit-Thompson theorem, an abstract theory dealing with groups of numbers, first proved in 1963, in an extremely simplified 12-page proof. He sent his evidence to 15 experts in the field and met with skepticism or silence, and the evidence was never printed in a newspaper. A year ago, Atiyah claimed to have solved a famous problem in differential geometry in an article he published on the ArXiv pre-print repository, but his peers quickly reported inaccuracies in his approach and the evidence was never officially released .

Science contacted several Atiyah colleagues. They all expressed their concern about his desire to retire from presenting evidence based on frail associations and stated that it was unlikely that his evidence of Riemann's hypothesis would be successful . But none of them wanted to publicly criticize his mentor or colleague for fear of compromising the relationship. John Baez, a mathematical physicist at the University of California at Riverside, was one of the few who wanted to give his name to critical remarks about Atiyah's claim. "The evidence stacks an impressive claim on top of another, without any connection argument or real justification," he says.

For its part, Atiyah seems unfazed. "The public has brilliantly intrepid young people and well-informed seniors," Atiyah wrote in an e-mail before his presentation. "I throw myself at the lions. I hope to come out unscathed. According to Atiyah, the word of his evidence and copies of his papers circulated online, prompting him to accept the presentation. He says in an interview that despite his criticism, his work poses concrete bases for proving not only Riemann's hypothesis but other unproven problems in mathematics. "People will complain and complain," says Atiyah, "but that's because they're reluctant to the idea that an old man might have invented a whole new method." . In his presentation, Atiyah dedicated only a handful of slides to his evidence, spending most of his time discussing the contributions of two 20th century mathematicians, John von Neumann and Friedrich Hirzebruch, on whom he stated that his evidence was based.

Atiyah's proof node depends on a quantity in physics called the fine-structure constant, which describes the strength and nature of the electromagnetic interaction between the charged particles. In describing this constant using a relatively obscure relation known as Todd's function, Atiyah claimed to be able to prove Riemann's hypothesis by contradiction.

In the five-page summary of the evidence, Atiyah attributes much of the theoretical work that underlies the evidence to an article that was submitted to the Acts of the Royal Society A. This document still needs to be published.