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Hamiltonian for the Zeros of the Riemann Zeta Function

zhpd55 添加于 2017/4/10 9:37:24  298次阅读 | 0次推荐 | 0个评论

A Hamiltonian operator ^H is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of ^H is 2xp, which is consistent with the Berry-Keating conjecture. While ^H is not Hermitian in the conventional sense, i^H is PT symmetric with a broken PT symmetry, thus allowing for the possibility that all eigenvalues of ^H are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that ^H is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true.

作 者:Carl M. Bender, Dorje C. Brody, Markus P. Müller
期刊名称: Phys. Rev. Lett.
期卷页: Published 30 March 2017 第118卷 第13期 130201(5)页
学科领域:数理科学 » 数学 » 几何学
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原文链接:https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.118.130201
DOI: 10.1103/PhysRevLett.118.130201
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备 注: 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.
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