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Mathematicians enliven the abandoned approach to the Riemann Hypothesis

  Mathematicians Revive Abandoned Approach to the Riemann Hypothesis
The idea for the work was sparked by a "toy problem" that Emory mathematician Ken Ono (left) considered a "gift" to Don Zagier's entertainment (right) presented the Max Planck Institute for Mathematics on the occasion of Zagier's 65th birthday. The toy problem can be seen on the whiteboard behind them. Credit: Emory University

Over the last 1

50 years, many approaches to the Riemann Hypothesis approach have been proposed, but none of them has led to the conquest of the most famous open mathematical problem. A recent publication in the Proceedings of the National Academy of Sciences ( PNAS ) suggests that one of these ancient approaches is more practical than before.

"In a surprisingly brief evidence, we have shown that an old, abandoned approach to the Riemann Hypothesis should not be forgotten," says Ken Ono, number theorist at Emory University and co-author of the paper. "By simply formulating an appropriate framework for an old approach, we have proved some new theorems, including much of a criterion that implies the Riemann Hypothesis, and our general framework also opens up approaches to other unanswered fundamental questions." The paper builds on the works of Johan Jensen and George Pólya, two of the most important mathematicians of the 20th century. It shows a method for calculating the Jensen-Pólya polynomials – a formulation of the Riemann hypothesis – not individually, but at once.

"The beauty of our proof lies in its simplicity," says Ono. "We do not invent new techniques or use new objects in mathematics, but we do offer a new view of the Riemann Hypothesis, and any reasonably advanced mathematician can validate our proof – no expert in number theory is needed." 19659005] Although the paper does not prove Riemann's hypothesis, so far open statements, which are known from the Riemann Hypothesis, as well as some evidence for assumptions in other fields, follow.

Co-authors of the work are Michael Griffin and Larry Rolen – two former Ono emory graduates who now study at Brigham Young University and Vanderbilt University respectively – and Don Zagier from the Max Planck Institute for Mathematics "Kannan Soundararajan Stanford University mathematician and expert on the Riemann Hypothesis is considered as further proof of the Riemann Hypothesis and is in any case a beautiful stand-alone theorem. " othesis.

The idea for the newspaper came about two years ago through a "toy problem" that Ono presented as a "gift" for Zagier in the run-up to a maths conference on the occasion of his 65th birthday. A toy problem is a downsized version of a larger, more complicated problem that mathematicians try to solve.

Zagier described the problem Ono gave him as "a nice problem about the asymptotic behavior of certain polynomials with Euler's partition function." I found the problem unsolvable, and I did not really expect Don to get anywhere with it, "recalls Ono "But he found the challenge super funny and soon found a solution."

Ono's guess was that such a solution could become a more general theory, which is what the mathematicians ultimately achieved.

"It was fun to work on, a really creative process," says Griffin. "Mathematics at the research level is often more art than computation, and that was certainly the case here. We had to study a nearly 100-year-old idea from Jensen and Pólya in a new way. "

The Riemann Hypothesis is one of seven Millennium price problems identified by the Clay Mathematics Institute as the most important open issues in mathematics Each problem brings a $ 1 million bonus to its solvers.

The hypothesis was made in a paper by the German mathematician Bernhard Riemann in 1859. He noted that the distribution of primes is closely related to the zeros of an analytic function Mathematically, the Riemannian hypothesis is the assertion that all non-trivial zeros of the zeta function have a real part ½.

"His hypothesis is a mouthful, but Riemann's motivation was simple," says Ono "He wanted to count primes."

The hypothesis is a means of solving one of the greatest secrets of the Z To understand the theory of counting – the pattern that underlies the primes. Although primes are simple objects defined in elementary mathematics (any number greater than 1 without positive divisors except 1 and themselves), their distribution remains hidden.

The first prime, 2, is the only one straight. The next prime is 3, but primes do not follow a pattern of every third number. The next one is 5, then 7, then 11. If you continue to count up, primes quickly become rarer.

"It's common knowledge that there are infinitely many primes, but they rarely become, even if you retrieve them to the 100s," explains Ono. In fact, of the first 100,000 numbers, only 9,592 are prime numbers, or about 9.5 percent. And from there on, they quickly become rarer. The probability of choosing a random number and calling it a prime number is zero. It almost never happens. "

Jensen and Pólya formulated a criterion in 1927 to validate the Riemann Hypothesis in order to unlock their potential for clarifying primes and other mathematical mysteries." The problem with the criterion of determining the hyperbolicity of the Jensen-Pólya polynomials is that that in the last 90 years only a handful of polynomials in the sequence have been verified, causing mathematicians to abandon this approach as too slow and unwieldy.

For the work PNAS developed the Authors a concept framework that combines the polynomials by degrees, which enabled them to confirm the criterion for each degree 100 percent of the time, overshadowing the handful of cases known to date.

"The method has the shocking feeling of being universal, as it applies to seemingly occurring problems independently, "says Rolen Easy to follow and understand proofs. Some of the finest insights in mathematics have taken a long time, but once you see them, they seem simple and clear. "

Despite their work, the results do not exclude that the Riemann Hypothesis is wrong, and the authors believe that a complete proof of the famous conjecture is still far off.

Is one of the greatest mysteries of mathematics, the Riemann Hypothesis, finally solved?

Further information:
Jensen Polynomials for Riemann's Zeta Function and Other Sequences, PNAS DOI: 10.1073 / pnas.1902572116, https://www.pnas.org/content/early/2019/05/20/1902572116

Provided by
Emory University

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