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Mathematicians revive the abandoned approach of the Riemann hypothesis

Many ways of approaching the Riemann Hypothesis have been proposed for the last 150 years, but none of them has led to conquer the most famous open problem of mathematics. A new role in the Procedures of the National Academy of Sciences (PNAS) suggests that one of these old approaches is more practical than what was done previously.

"In a surprisingly short test, we have shown that an old and abandoned approach to the Riemann hypothesis should not have been forgotten," says Ken Ono, an Emory University theorist and co-author of the paper. "By simply formulating an appropriate framework for an old approach, we have tested some new theorems, which include a large part of a criterion involving the Riemann hypothesis, and our general framework also opens up approaches to other basic unanswered questions."

The document is based on the work of Johan Jensen and George Pólya, two of the most important mathematicians of the 20th century. It reveals a method for calculating the Jensen-Pólya polynomials, a formulation of the Riemann Hypothesis, not one at a time, but all at once.

"The beauty of our test is its simplicity," says Ono. "We do not invent any new technique or use any new object in mathematics, but we provide a new view of the Riemann Hypothesis, and any reasonably advanced mathematician can verify our proof, it does not take an expert in number theory."

Although the document fails to demonstrate the Riemann hypothesis, its consequences include previously open statements that are known to emerge from the Riemann hypothesis, as well as some evidence of conjecture in other fields.

The article's co-authors are Michael Griffin and Larry Rolen, two of Emory's former graduate students in Ono who are now part of the faculty of Brigham Young University and Vanderbilt University, respectively, and Don Zagier of the Max Mathematical Mathematics Institute .

"The result set here can be seen as additional evidence of the Riemann Hypothesis, and in any case, it is a beautiful independent theorem," says Kannan Soundararajan, a mathematician at Stanford University and an expert in the Riemann Hypothesis.

The idea for the newspaper began two years ago for a "toy problem" that Ono presented as a "gift" to entertain Zagier during the run-up to a math conference celebrating his 65th birthday. A toy problem is a reduced version of a larger and more complicated problem that mathematicians are trying to solve.

Zagier described what Ono gave him as "a nice problem about the asymptotic behavior of certain polynomials related to the partition function of Euler, which is an old love of mine and Ken, and of almost any classical number theorist".

"I found the problem difficult to solve and I really did not expect Don to get anywhere with him," Ono recalls. "But he thought the challenge was super fun and soon he had created a solution."

Ono's premonition was that a solution of this kind could become a more general theory. That is what mathematicians finally achieved.

"It's been a fun project to work on, a really creative process," says Griffin. "Mathematics at the research level is often more of an art than a calculation and that was certainly the case here, it forced us to see Jensen and Pólya's idea of ​​almost 100 years in a new way."

The Riemann Hypothesis is one of seven problems of the Millennium Prize, identified by the Clay Mathematics Institute as the most important open problems in mathematics. Each problem carries a reward of $ 1 million for its solvers.

The hypothesis debuted in an article of 1859 by the German mathematician Bernhard Riemann. He noticed that the distribution of prime numbers is closely related to the zeros of an analytic function, which came to be called the Riemann zeta function. In mathematical terms, the Riemann Hypothesis is the claim 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. "I wanted to count the prime numbers."

The hypothesis is a vehicle to understand one of the biggest mysteries of number theory: the pattern that underlies the prime numbers. Although prime numbers are simple objects defined in elementary mathematics (any number greater than 1 without positive divisors, except 1 and in itself), their distribution remains hidden.

The first prime number, 2, is the only pair. The next prime number is 3, but prime numbers do not follow a pattern of every third number. The next one is 5, then 7, then 11. As you continue counting up, the prime numbers quickly become less frequent.

"It is well known that there are infinite prime numbers, but they become rare, even when you reach 100," explains Ono. "In fact, of the first 100,000 numbers, only 9,592 are prime numbers, or about 9.5 percent, and they quickly get rarer from there.The probability of choosing a number at random and having it as a prime is zero. . "

In 1927, Jensen and Pólya formulated a criterion to confirm the Riemann Hypothesis, as a step towards unleashing its potential to elucidate prime numbers and other mathematical mysteries. The problem with the criterion, which establishes the hyperbolicity of the Jensen-Pólya polynomials, is that it is infinite. Over the past 90 years, only a handful of the polynomials in the sequence have been verified, which makes mathematicians abandon this approach as too slow and unwieldy.

For him PNAS On paper, the authors devised a conceptual framework that combines polynomials by degrees. This method allowed them to confirm the criteria for each degree 100 percent of the time, eclipsing the handful of cases that were previously known.

"The method has a shocking sense of being universal, since it applies to problems that apparently have no relationship," says Rolen. "And at the same time, your tests are easy to follow and understand." Some of the most beautiful ideas in mathematics are those that took a long time to realize, but once you see them, they seem simple and clear.

Despite their work, the results do not rule out the possibility that the Riemann Hypothesis is false and the authors believe that a full proof of the famous conjecture is still far away.


The work was supported by grants from the National Science Foundation and the Asa Griggs Candler Fund.

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