In a letter to Andrew Odlyzko, dated January 3, 1982, George Pólya
said that while he was in Göttingen around 1912 to 1914 he was asked by Edmund Landau for a physical reason that the Riemann hypothesis should be true, and suggested that this would be the case if the imaginary parts t of the zeros
David Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture due to a story told by Ernst Hellinger, a student of Hilbert, to André Weil. Hellinger said that Hilbert announced in his seminar in the early 1900s that he expected the Riemann Hypothesis would be a consequence of Fredholm's work on integral equations with a symmetric kernel.[3][4][5][6]
1950s and the Selberg trace formula
At the time of Pólya's conversation with Landau, there was little basis for such speculation. However Selberg in the early 1950s proved a duality between the length spectrum of a Riemann surface and the eigenvalues of its Laplacian. This so-called Selberg trace formula bore a striking resemblance to the explicit formulae, which gave credibility to the Hilbert–Pólya conjecture.
Dyson saw that the statistical distribution found by Montgomery appeared to be the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix. These distributions are of importance in physics — the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics. Subsequent work has strongly borne out the connection between the distribution of the zeros of the Riemann zeta function and the eigenvalues of a random Hermitian matrix drawn from the Gaussian unitary ensemble, and both are now believed to obey the same statistics. Thus the Hilbert–Pólya conjecture now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.[7]
A possible connection of Hilbert–Pólya operator with quantum mechanics was given by Pólya. The Hilbert–Pólya conjecture operator is of the form where is the Hamiltonian of a particle of mass that is moving under the influence of a potential . The Riemann conjecture is equivalent to the assertion that the Hamiltonian is Hermitian, or equivalently that is real.
where and are the eigenvalues and eigenstates of the free particle Hamiltonian. This equation can be taken to be a Fredholm integral equation of first kind, with the energies . Such integral equations may be solved by means of the resolvent kernel, so that the potential may be written as
where is the resolvent kernel, is a real constant and
where is the Dirac delta function, and the are the "non-trivial" roots of the zeta function .
This refinement of the Hilbert–Pólya conjecture is known as the Berry conjecture (or the Berry–Keating conjecture). As of 2008, it is still quite far from being concrete, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections. Berry and Keating have conjectured that since this operator is invariant under dilations perhaps the boundary condition f(nx) = f(x) for integer n may help to get the correct asymptotic results valid for large n
was introduced, which they claim satisfies a certain modified versions of the conditions of the Hilbert–Pólya conjecture. Jean Bellissard has criticized this paper,[12] and the authors have responded with clarifications.[13] Moreover, Frederick Moxley has approached the problem with a Schrödinger equation.[14]
^ abMontgomery, Hugh L. (1973), "The pair correlation of zeros of the zeta function", Analytic number theory, Proc. Sympos. Pure Math., vol. XXIV, Providence, R.I.: American Mathematical Society, pp. 181–193, MR0337821.
^Broughan, K. (2017), Equivalents of the Riemann Hypothesis Volume 2: Analytic Equivalents, p. 192, ISBN978-1107197121
^Dieudonne, J. (1981), History of Functional Analysis, p. 106, ISBN978-0444861481
^Endres, S.; Steiner, F. (2009), "The Berry–Keating operator on and on compact quantum graphs with general self-adjoint realizations", Journal of Physics A: Mathematical and Theoretical, 43 (9): 37, arXiv:0912.3183v5, doi:10.1088/1751-8113/43/9/095204, S2CID115162684
^Simon, B. (2015), Operator Theory: A Comprehensive Course in Analysis, Part 4, p. 42, ISBN978-1-4704-1103-9
^Belissard, Jean (2017), "Comment on "Hamiltonian for the Zeros of the Riemann Zeta Function"", arXiv:1704.02644 [quant-ph]
^Bender, Carl M.; Brody, Dorje C.; Müller, Markus P. (2017), "Comment on 'Comment on "Hamiltonian for the zeros of the Riemann zeta function"'", arXiv:1705.06767 [quant-ph].
^Moxley, Frederick (2017). A Schrödinger equation for solving the Bender-Brody-Müller conjecture. 13Th Imt-Gt International Conference on Mathematics. AIP Conference Proceedings. Vol. 1905. p. 030024. Bibcode:2017AIPC.1905c0024M. doi:10.1063/1.5012170.
Elizalde, Emilio (1994), Zeta regularization techniques with applications, World Scientific, Bibcode:1994zrta.book.....E, ISBN978-981-02-1441-8. Here the author explains in what sense the problem of Hilbert–Polya is related with the problem of the Gutzwiller trace formula and what would be the value of the sum taken over the imaginary parts of the zeros.