Hilbert–Pólya conjecture

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

${\displaystyle {\tfrac {1}{2}}+it}$

of the Riemann zeta function corresponded to eigenvalues of a self-adjoint operator.[1] The earliest published statement of the conjecture seems to be in Montgomery (1973).[1][2]

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]

A possible connection of Hilbert–Pólya operator with quantum mechanics was given by Pólya. The Hilbert–Pólya conjecture operator is of the form ${\displaystyle {\tfrac {1}{2}}+iH}$ where ${\displaystyle H}$ is the Hamiltonian of a particle of mass ${\displaystyle m}$ that is moving under the influence of a potential ${\displaystyle V(x)}$. The Riemann conjecture is equivalent to the assertion that the Hamiltonian is Hermitian, or equivalently that ${\displaystyle V}$ is real.

Using perturbation theory to first order, the energy of the nth eigenstate is related to the expectation value of the potential:

${\displaystyle E_{n}=E_{n}^{0}+\left.\left\langle \varphi _{n}^{0}\right|V\left|\varphi _{n}^{0}\right.\right\rangle }$

where ${\displaystyle E_{n}^{0}}$ and ${\displaystyle \varphi _{n}^{0}}$ 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 ${\displaystyle E_{n}}$. Such integral equations may be solved by means of the resolvent kernel, so that the potential may be written as

${\displaystyle V(x)=A\int _{-\infty }^{\infty }\left(g(k)+{\overline {g(k)}}-E_{k}^{0}\right)\,R(x,k)\,dk}$

where ${\displaystyle R(x,k)}$ is the resolvent kernel, ${\displaystyle A}$ is a real constant and

${\displaystyle g(k)=i\sum _{n=0}^{\infty }\left({\frac {1}{2}}-\rho _{n}\right)\delta (k-n)}$

where ${\displaystyle \delta (k-n)}$ is the Dirac delta function, and the ${\displaystyle \rho _{n}}$ are the "non-trivial" roots of the zeta function ${\displaystyle \zeta (\rho _{n})=0}$.

Michael Berry and Jonathan Keating have speculated that the Hamiltonian H is actually some quantization of the classical Hamiltonian xp, where p is the canonical momentum associated with x[9] The simplest Hermitian operator corresponding to xp is

${\displaystyle {\hat {H}}={\tfrac {1}{2}}({\hat {x}}{\hat {p}}+{\hat {p}}{\hat {x}})=-i\left(x{\frac {\mathrm {d} }{\mathrm {d} x}}+{\frac {1}{2}}\right).}$

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

${\displaystyle {\frac {1}{2}}+i{\frac {2\pi n}{\log n}}.}$[10]

A paper was published in March 2017, written by Carl M. Bender, Dorje C. Brody, and Markus P. Müller,[11] which builds on Berry's approach to the problem. There the operator

${\displaystyle {\hat {H}}={\frac {1}{1-e^{-i{\hat {p}}}}}\left({\hat {x}}{\hat {p}}+{\hat {p}}{\hat {x}}\right)\left(1-e^{-i{\hat {p}}}\right)}$

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]

• Aneva, B. (1999), "Symmetry of the Riemann operator" (PDF), Physics Letters B, 450 (4): 388–396, arXiv:0804.1618, doi:10.1016/s0370-2693(99)00172-0, S2CID 222175681.

Wolf, M. (2020), "Will a physicist prove the Riemann hypothesis?", Reports on Progress in Physics, 83 (4): 036001, arXiv:1410.1214, doi:10.1088/1361-6633/ab3de7, PMID 31437818, S2CID 85450819.

• Elizalde, Emilio (1994), Zeta regularization techniques with applications, World Scientific, Bibcode:1994zrta.book.....E, ISBN 978-981-02-1441-8. 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 ${\displaystyle \exp(i\gamma )}$ taken over the imaginary parts of the zeros.