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.
(Score: 3, Informative) by Hartree on Saturday April 08 2017, @07:17PM
Peter Woit over at Not Even Wrong is, at best, quite skeptical over this one.
Here's his take: http://www.math.columbia.edu/~woit/wordpress/?p=9197#comments [columbia.edu]
(Score: 0) by Anonymous Coward on Saturday April 08 2017, @08:39PM (1 child)
Physicists [xkcd.com]
It's eerily apropos.
(Score: 0) by Anonymous Coward on Saturday April 08 2017, @11:27PM
It's funny because this exact thing happened to me during my undergrad (mathematics/computer science). I was working in the computer vision lab at my university and encountered a fellow student whom I shared many classes with. He (a physics major) is certainly one of the brightest lights I have ever met, however when I explained the problem of model-based 3D object recognition to him he said something to the effect of, "Oh, that should be easy. Just do it like this...". He wouldn't believe me when I told him it was way more complicated than that. I didn't have the time to argue with him, nor the I desire to, so I left it at that.
(Score: 2) by kaszz on Saturday April 08 2017, @08:52PM (4 children)
If the Riemann hypothesis is proven to be false. What consequences would that have? things that would not be..
(Score: 0) by Anonymous Coward on Saturday April 08 2017, @09:00PM
Fortunately, the simulation would reboot automatically, but we'd lose all the data.
(Score: 2) by davester666 on Saturday April 08 2017, @09:52PM
The current universe collapses into a singularity, then another big bang happens and we start this wild ride all over again.
(Score: 0) by Anonymous Coward on Saturday April 08 2017, @10:39PM
Probably the main thing would be your mom.
(Score: 5, Informative) by stormwyrm on Sunday April 09 2017, @01:44AM
A lot of number theory results begin by assuming the Riemann hypothesis is true and go from there. All such results would then have to become approximations and the range of their validity must be re-examined. The Riemann hypothesis states that the non-trivial zeroes of the Riemann zeta function all lie exactly on a critical line with real part ½. A disproof of the Riemann hypothesis would mean that non-trivial zeroes can be found at places away from the critical line, and just how far these non-trivial zeroes can be found away from that line and how many of them exist would tell us just how far we can rely on these results. However, there are some known results that can limit the damage a disproof of the theorem can do. It is known that there are no zeroes outside the critical line for Im(z) < ten trillion. It is also known that there are an infinite number of zeroes on the critical line. And most importantly, it has also been proved that there are no non-trivial zeroes outside the critical strip where 0 < Im(z) < 1. So it then becomes rather more like Asimov’s famous relativity of wrong.
As an example of the effect of a disproof of the Riemann hypothesis, there’s the prime number theorem, which gives us an idea of the distribution of primes. Assuming the truth of the Riemann hypothesis, the number of primes less than a given number x is π(x) = Li(x) + O(√x log x), where Li(x) is the logarithmic integral function. The Riemann hypothesis concerns the the error term given in big-Oh notation: if the Riemann hypothesis were false that term would become more complicated. Without assuming the truth of the Riemann hypothesis, the best bounds known to date are something like O(x exp(-A (log x)^(3/5)/(log log x)^(1/5))).
Numquam ponenda est pluralitas sine necessitate.
(Score: 1) by khallow on Sunday April 09 2017, @01:06PM