Over the last 50 years, there’s been many proposals s regarding the Riemann Hypothesis, but none of them have led to conquering the most famous open problem in mathematics. A new paper in the Proceedings of the National Academy of Sciences (PNAS) 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.
Although the paper falls short of proving the Riemann Hypothesis, its consequences include previously open assertions which are known to follow from the Riemann Hypothesis, as well as some proofs of conjectures in other fields.
The idea for the paper was sparked two years ago by a “toy problem” that Ono presented as a “gift” to entertain Zagier during the lead-up to a math conference celebrating his 65th birthday. A toy problem is a scaled-down version of a bigger, more complicated problem that mathematicians are trying to solve.
The hypothesis is a vehicle to understand one of the greatest mysteries in number theory — the pattern underlying prime numbers. Although prime numbers are simple objects defined in elementary math (any number greater than 1 with no positive divisors other than 1 and itself) their distribution remains hidden.
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.
Despite their work, the results don’t rule out the possibility that the Riemann Hypothesis is false and the authors believe that a complete proof of the famous conjecture is still far off.