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Many approaches to the Riemann hypothesis have been proposed over the last 150 years, but none of them has overcome the most well-known problem of mathematics. A new paper in the Proceedings of the National Academy of Sciences (PNASsuggests that one of these old approaches is more practical than what has been achieved so far.
"In a surprisingly short proof, we showed that an old abandoned approach to Riemann's hypothesis should not have been forgotten," says Ken Ono, number theorist at the University. Emory and co-author of the article. "By simply formulating an appropriate framework for an old approach, we have proved new theorems, including much of a criterion that involves the Riemann hypothesis, and our general framework also opens up approaches to Other basic questions without answer. "
The document is based on the work of Johan Jensen and George Pólya, two of the most important mathematicians of the twentieth century. It reveals a method for calculating Jensen-Polya polynomials – a formulation of Riemann's hypothesis – not one at a time, but all at once.
"The beauty of our proof lies in its simplicity," says Ono. "We do not invent any new techniques and do not use any new objects in math, but we provide a new view of Riemann's hypothesis.Any reasonably advanced mathematician can verify our evidence.This does not require an expert in number theory. "
Although the article does not prove Riemann 's hypothesis, its consequences include previously open assertions that are known to arise from Riemann' s hypothesis, as well as some evidence of conjectures in the case of Riemann. other areas.
Co-authors of the article are Michael Griffin and Larry Rolen – two of Emo's former students from Ono graduates who are now professors at Brigham Young University and Vanderbilt University, respectively – and Don Zagier of the University of Ottawa. Max Planck Institute of Mathematics.
"The result set here can be seen as further proof of Riemann's hypothesis and, in any case, it is a beautiful, autonomous theorem," says Kannan Soundararajan, a mathematician at the University of California. Stanford University and expert on the Riemann hypothesis.
The idea of this paper was sparked two years ago by a "toy problem" that Ono presented as a "gift" to entertain Zagier at the math conference celebrating his 65th birthday. A toy problem is a reduced version of a more complex and complex problem that mathematicians are trying to solve.
Zagier described the one that Ono had given him as "a nice problem regarding the asymptotic behavior of some polynomials involving Euler's score function, which is an old love for me and for Ken – and pretty much any classical number theorist ".
"I found the problem insoluble and I did not really expect Don to succeed anywhere," Ono recalls. "But he thought the challenge was great fun and he quickly found a solution."
It is thought that this solution could be transformed into a more general theory. That's what the mathematicians finally realized.
"It's a fun project, a really creative process," Griffin said. "Mathematics at the research level is often more artistic than computational, and certainly was the case here, which required us to examine a nearly century old idea of Jensen and Pólya in a new way. "
The Riemann Hypothesis is one of the seven Millennium Prize issues identified by the Clay Mathematics Institute as the most important open problem in mathematics. Each problem has a million dollar bonus for its resolvers.
The hypothesis was debuted in an 1859 article by the German mathematician Bernhard Riemann. He noticed that the distribution of prime numbers is closely related to the zeros of an analytic function, later called the Riemann zeta function. In mathematical terms, Riemann's hypothesis is the assertion that all the non-trivial zeros of the Zeta function have a real ½ part.
"His hypothesis is disproportionate, but Riemann's motivation was simple," says Ono. "He wanted to count the prime numbers."
The hypothesis is a way to understand one of the greatest mysteries of number theory – the model underlying prime numbers. Although prime numbers are simple objects defined in elementary mathematics (any number greater than 1 without positive divisors other than 1 and itself), their distribution remains hidden.
The first prime number, 2, is the one and the same. The next prime number is 3, but prime numbers do not follow a pattern of all three numbers. Next is 5, then 7, then 11. As you keep counting, prime numbers quickly become less frequent.
"It is well known that there are infinitely many prime numbers, but they become rare, even when you reach 100," says Ono. "In fact, out of the first 100,000 numbers, only 9,592 are prime numbers, or about 9,5%, and they are rapidly dwindling from that point on." The probability of choosing a random number and the number To have as prime number equals zero.This almost never happens. "
In 1927, Jensen and Pólya formulated a criterion to confirm Riemann's hypothesis, in order to unleash his potential to elucidate prime numbers and other mathematical mysteries. The problem with the criterion – establishing the hyperbolicity of Jensen-Polya polynomials – is that it is infinite. Over the past 90 years, only a handful of polynomials in the sequence have been verified, which has led mathematicians to abandon this approach, which is considered too slow and cumbersome.
For the PNAS In this paper, the authors have developed a conceptual framework that combines step polynomials. This method allowed them to confirm the criterion for each degree 100% of the time, thus eclipsing the handle of previously known cases.
"The method has a shocking sense of being universal in that it applies to seemingly unrelated problems," Rolen said. "And at the same time, his proofs are easy to understand and understand.Some of the most beautiful ideas in mathematics are those that have taken a long time to understand, but once you've seen them, they appear simple and clear."
Despite their work, the results do not rule out the possibility that Riemann's hypothesis is false and the authors believe that a complete proof of the famous conjecture is still far away.
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The work was funded by grants from the National Science Foundation and the Asa Griggs Candler Fund.
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