| Title | New Insight Into Proving Math's Million-Dollar Problem: the Riemann Hypothesis | |
| Date | Saturday April 08 2017, @06:46PM | |
| Author | martyb | |
| Topic | ||
| from the maths++ dept. | ||
Researchers have discovered that the solutions to a famous mathematical function called the Riemann zeta function correspond to the solutions of another, different kind of function that may make it easier to solve one of the biggest problems in mathematics: the Riemann hypothesis. If the results can be rigorously verified, then it would finally prove the Riemann hypothesis, which is worth a $1,000,000 Millennium Prize from the Clay Mathematics Institute.
While the Riemann hypothesis dates back to 1859, for the past 100 years or so mathematicians have been trying to find an operator function like the one discovered here, as it is considered a key step in the proof.
"To our knowledge, this is the first time that an explicit—and perhaps surprisingly relatively simple—operator has been identified whose eigenvalues ['solutions' in matrix terminology] correspond exactly to the nontrivial zeros of the Riemann zeta function," Dorje Brody, a mathematical physicist at Brunel University London and coauthor of the new study, told Phys.org.
What still remains to be proven is the second key step: that all of the eigenvalues are real numbers rather than imaginary ones. If future work can prove this, then it would finally prove the Riemann hypothesis.
[...] Riemann's hypothesis was that all of the nontrivial zeros lie along a single vertical line (½ + it) in the complex plane—meaning their real component is always ½, while their imaginary component i varies as you go up and down the line.
Over the past 150 years, mathematicians have found literally trillions of nontrivial zeros, and all of them have a real of component of ½, just as Riemann thought. It's widely believed that the Riemann hypothesis is true, and much work has been done based on this assumption. But despite intensive efforts, the Riemann hypothesis—that all of the infinitely many zeros lie on this single line—has not yet been proved.
More information:
Carl M. Bender, Dorje C. Brody, and Markus P. Müller. "Hamiltonian for the Zeros of the Riemann Zeta Function." Physical Review Letters. DOI: 10.1103/PhysRevLett.118.130201
Wikipedia: imaginary numbers, Eigenvalues and Eigenvectors, and the Riemann zeta function.
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