Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis
A famous mathematician today claimed he has solved the Riemann hypothesis, a problem relating to the distribution of prime numbers that has stood unsolved for nearly 160 years. In a 45-minute talk on 24 September at the Heidelberg Laureate Forum in Germany, Michael Atiyah, a mathematician emeritus at The University of Edinburgh, presented what he describes as a “simple proof” that relies on a tool from a seemingly unrelated problem in physics. But many experts doubt its validity, especially because Atiyah, 89, has been making mistakes in recent years.
“What he showed in the presentation is very unlikely to be anything like a proof of the Riemann hypothesis as we know it,” says Jørgen Veisdal, an economist at the Norwegian University of Science and Technology in Trondheim who has previously studied the Riemann hypothesis. “It is simply too vague and unspecific.” Veisdal added that he would need to examine the written proof more closely to make a definitive judgement.
The Riemann hypothesis, one of the last great unsolved problems in math, was first proposed in 1859 by German mathematician Bernhard Riemann. It is a supposition about prime numbers, such as two, three, five, seven, and 11, which can only be divided by one or themselves. They become less frequent, separated by ever-more-distant gaps on the number line. Riemann found that the key to understanding their distribution lay within another set of numbers, the zeroes of a function called the Riemann zeta function that has both real and imaginary inputs. And he invented a formula for calculating how many primes there are, up to a cutoff, and at what intervals these primes occur, based on the zeroes of the zeta function.
However, Riemann’s formula only holds if one assumes that the real parts of these zeta function zeroes are all equal to one-half. Reimann proved this property for the first few primes, and over the past century it has been computationally shown to work for many large numbers of primes, but it remains to be formally and indisputably proved out to infinity. A proof would not only win the $1 million reward that comes for solving one of the seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000, but it could also have applications in predicting prime numbers, important in cryptography.
A giant in his field, Atiyah has made major contributions to geometry, topology, and theoretical physics. He has received both of math’s top awards, the Fields Medal in 1966 and the Abel Prize in 2004. But despite a long and prolific career, the Riemann claim follows on the heels of more recent, failed proofs.
In 2017, Atiyah told The Times of London that he had converted the 255-page Feit-Thompson theorem, an abstract theory dealing with groups of numbers first proved in 1963, into a vastly simplified 12-page proof. He sent his proof to 15 experts in the field and was met with skepticism or silence, and the proof was never printed in a journal. A year earlier, Atiyah claimed to have solved a famous problem in differential geometry in a paper he posted on the preprint repository ArXiv, but peers soon pointed out inaccuracies in his approach and the proof was never formally published.
Science contacted several of Atiyah’s colleagues. They all expressed concern about his desire to come out of retirement to present proofs based on shaky associations and said it was unlikely that his proof of the Riemann hypothesis would be successful. But none wanted to publicly criticize their mentor or colleague for fear of jeopardizing the relationship. John Baez, a mathematical physicist at the University of California, Riverside, was one of the few willing to put his name to critical remarks about Atiyah’s claim. “The proof just stacks one impressive claim on top of another without any connecting argument or real substantiation,” he says.
For his part, Atiyah seems unfazed. “The audience there has fearlessly bright youngsters and well-informed golden oldies,” Atiyah wrote in an email before his presentation. “I am throwing myself to the lions. I hope to emerge unscathed.” According to Atiyah, word of his proof and copies of his papers circulated online, prompting him to agree to the presentation. He says in an interview that despite criticism, his work lays down a concrete basis for proving not only the Riemann hypothesis, but other unproven problems in mathematics. “People will complain and grumble,” Atiyah says, “but that’s because they’re resistant to the idea that an old man might have come up with an entirely new method.” In his presentation, Atiyah devoted only a handful of slides to his proof, spending the majority of his time discussing the contributions of two 20th century mathematicians, John von Neumann and Friedrich Hirzebruch, on which he said his proof was based.
The crux of Atiyah’s proof depends on a quantity in physics called the fine structure constant, which describes the strength and nature of electromagnetic interaction between charged particles. By describing this constant using a relatively obscure relationship known as the Todd function, Atiyah claimed to be able to prove the Riemann hypothesis by contradiction.
In the five-page write-up of the proof, Atiyah attributes much of the theoretical work that underpins the proof to a paper of his own that has been submitted to the Proceedings of the Royal Society A. That paper has yet to be published.