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Thursday, May 30, 2019

Famous unsolved math problem sees new progress | Mathematics - Futurity: Research News

A new paper suggests that one of the many proposed solutions to the most famous open problem in mathematics—the Riemann Hypothesis—is unexpectedly practical.

“In a surprisingly short proof, we’ve shown that an old, abandoned approach to the Riemann Hypothesis should not have been forgotten,” says Ken Ono, a number theorist at Emory University and coauthor of the paper in PNAS, according to Emory University and Carol Clark-Emory.

Prime numbers. 
Photo: Getty Images
A commentary on the work by Fields Medalist Enrico Bombiero also appears in the journal.

“By simply formulating a proper framework for an old approach we’ve proven some new theorems, including a large chunk of a criterion which implies the Riemann Hypothesis. And our general framework also opens approaches to other basic unanswered questions.”

The paper builds on the work of Johan Jensen and George Pólya, two of the most important mathematicians of the 20th century. It reveals a method to calculate the Jensen-Pólya polynomials—a formulation of the Riemann Hypothesis—not one at a time, but all at once.

“The beauty of our proof is its simplicity,” Ono says. “We don’t invent any new techniques or use any new objects in math, but we provide a new view of the Riemann Hypothesis. Any reasonably advanced mathematician can check our proof. It doesn’t take an expert in number theory.”...

Combining polynomials
In 1927, Jensen and Pólya formulated a criterion for confirming the Riemann Hypothesis, as a step toward unleashing its potential to elucidate the primes and other mathematical mysteries. The problem with the criterion—establishing the hyperbolicity of the Jensen-Pólya polynomials—is that it is infinite. During the past 90 years, only a handful of the polynomials in the sequence have been verified, causing mathematicians to abandon this approach as too slow and unwieldy.

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

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Additional resources
Original Study 
DOI: 10.1073/pnas.1906804116 

Source: Futurity: Research News