One of the most famous unsolved problems in mathematics likely remains unsolved. At a hotly-anticipated talk at the Heidelberg Laureate Forum today, retired mathematician Michael Atiyah delivered what he claimed was a proof of the Riemann hypothesis, a challenge that has eluded his peers for nearly 160 years.
“Solve the Riemann hypothesis and you become famous. If you are famous already, you become infamous,” Atiyah said during his talk. “Nobody believes any proof of the Riemann hypothesis because it is so difficult. Nobody has proved it, so why should anybody prove it now? Unless, of course, you have a totally new idea.”
Sometimes called the riddle of the primes, the Riemann hypothesis is intimately connected to the distribution of prime numbers, those indivisible by any whole number other than themselves and one. If the hypothesis is proven to be correct, mathematicians would be armed with a map to the location of all such prime numbers, a breakthrough with far-reaching repercussions in the field.
In more practical terms, a correct solution would earn its composer a $1 million prize from the Clay Mathematics Institute, which ranks the hypothesis among its the six unsolved Clay Millennium Problems. The prestige has tempted many mathematicians over the years, none of which has yet been awarded the prize.
Atiyah’s self-described “simple proof” builds on the work of two leading 20th century mathematicians, John von Neumann and Friedrich Hirzebruch. By combining their insights, and assuming the Riemann hypothesis does not hold, Atiyah claims to reach a logical contradiction, implying that the hypothesis must in fact be correct. “It looks miraculous,” says Atiyah, “but I claim that all the hard work was done 70 years ago.”
In his talk, Atiyah gave a history of von Neumann’s and Hirzebruch’s work, along with other noted figures in the history of mathematics. His proof of the Riemann hypothesis was dealt with in just a few slides and claimed a connection with the fine structure constant, a physical parameter that describes the interaction between light and matter and whose status as a constant has come into question.
Born in 1929, Atiyah is one of the UK’s most eminent mathematical figures, having received the two awards often referred to as the Nobel prizes of mathematics, the Fields medal and the Abel Prize. He also, at various times, served as president of the London Mathematical Society, the Royal Society and the Royal Society of Edinburgh.
Atiyah has produced a number of papers in recent years making remarkable claims which have so far failed to convince his peers. While his latest proof has yet to undergo the rigorous peer review process necessary to test its validity, the initial reaction has been one of cautious scepticism. Most mathematicians contacted by New Scientist declined to comment on the work.
“The Riemann hypothesis is a notoriously difficult problem,” says Nicholas Jackson at Warwick University in the UK. “Lots of other top-rate mathematicians have nearly but not quite managed to prove it over the years, only for a subtle flaw in the proof to become apparent.”
If it is borne out, however, Atiyah hopes that his proof will inspire a younger generation to extend his work to more general cases of the Riemann hypothesis, as well as seemingly unrelated areas of mathematics and even physics. “Hopefully some useful insights will come out of Atiyah’s work even if the Riemann hypothesis proof doesn’t quite hold together,” says Jackson.