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There are six outstanding issues that, if resolved, will bring you a reward of $ 1 million. Monday, a reputable mathematician said in a lecture that he had perhaps proved the most famous of these problems, called Riemann's hypothesis. But there are reasons to be skeptical.
Many people claim have solved the biggest problems in physics and mathematics (and they often send their ideas by email to active physicists and science journalists), but few of these solutions are scrutinized. Yesterday's announcement was different: the accomplished British and Lebanese mathematician Michael Atiyah claimed that he had proved 160-year-old Riemann's hypothesis, trying to understand another problem in particle physics. But the last five-page essay, based on another unpublished article, left its mark.
The Riemann hypothesis surrounds a mathematical object called the Riemann zeta function. It was studied for the first time by the famous mathematician Leonhard Euler on real numbers only; Bernhard Riemann then extended it to complex numbers and studied the consequences. If you remember algebra in high school, there is a whole branch of mathematics surrounding complex numbers – those that include real numbers as well as imaginary numbers, or those multiplied by I, the square root of a negative. The zeta function takes one of these complex numbers and extracts another one. Riemann hypothesized, but did not prove, that this function would return a value of "zero" only if you connect a negative even number (-2, -4, etc.) or a complex number whose part actual was ½ (½, ½ + i, ½ + .66i, etc.).
Whether the Riemann hypothesis is true or not is of fundamental importance for mathematics – for example, the hypothesis could explain why prime numbers (those which are only divisible by one and by themselves, such as 3, 5 , 7 and 2).77,232,917-1) are distributed as they are on the number line. This is one of the most important open mathematical problems. This is one of the six unresolved issues of the "Millennium Award", which means that the person proving or refuting the Riemann hypothesis will win a million dollar award from the Clay Mathematics Institute.
Michael Atiyah announced his evidence last week and delivered it yesterday at a conference at the Heidelberg Prize Winners Forum in Germany. Atiyah himself is perhaps one of the greatest living mathematicians. He won the two most prestigious mathematics awards, the Fields Medal and the Abel Award, for his work in differential geometry, so that others on the field did not immediately cancel his announcement.
But the end result left something to be desired. He presented a five-page document containing a seven-line test in which he explained that most of the work was based on another mathematical object he was studying, called the Todd function, developed in another unpublished article. Using this Todd function, he hoped to explain something in particle physics called the fine-structure constant, which quantifies the strength of the electromagnetic interaction between particles.
"Unfortunately, it seems that Atiyah's article on Riemann's hypothesis suffers from some internal inconsistencies such as they are currently written – the proof as it is incorrect or, at best, far from complete, "said Daniel Litt, a mathematician at the Institute for Advanced Study. Gizmodo. "That does not mean that it has no value, of course! Given the many incredible results of Atiyah, it is reasonable to hope that something of value can be extracted from work. "
Some particle physicists were puzzled by Atiyah's attempt to obtain the constant of fine structure in the same way as he did. It is true that at zero energy there is a constant number (a number like pi) which serves to explain the force of the electromagnetic force. This constant is equal to 1/137. But this number changes to higher energies. And at these higher energies, the electromagnetic force becomes one and the same fundamental force called the weak nuclear force. When they are united as an electroweak force, two share a different fine structure constant. Perhaps the electroweak force unites with another of the fundamental forces, the strong nuclear force, at an even higher energy. The zero-energy fine structure constant of electromagnetism is not as interesting to particle physicists as this higher fine-structure constant.
"[Atiyah’s proof] seems to be the opposite of how most particle physicists think about the problem, "physicist Matthew Buckley told Gizmodo. "It does not mean it's wrong, but it does mean it's suspicious. He ignores the problems of [the fine-structure constant] change as the energy changes. "
Mathematicians have important contributions to make to the field of particle physics, of course, and pursuing the problem from the lowest energies instead of the higher ones could yield something interesting, Buckley said. But in this case, he stated that he did not feel that Atiyah had presented a satisfactory answer.
This is not the first time in recent history that Atiyah has presented evidence judged incorrect by other mathematicians. He presented another incomplete proof dealing with geometry in October 2016.
Strong claims require strong evidence. Atiyah has submitted his Todd paper to a prestigious journal, the Proceedings of the Royal Society A, Science reports, where he faces criticism from other mathematicians. But it is fair to say that until the end of the exam, the problem of the Millennium Prize is still unresolved.
