Hilbert–Pólya conjecture

In mathematics, the Hilbert-Pólya conjecture is a possible approach to the Riemann hypothesis, by means of spectral theory.

Initial hunches

David Hilbert and George Pólya speculated that real number values of t such that

1/2 + it

is a zero of the Riemann zeta function might be the eigenvalues of a Hermitian operator, sometimes called the Riemann operator, and that this would be a way of proving the Riemann hypothesis. This would necessarily be an unbounded operator.

1950s and the Selberg trace formula

At the time, 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 speculation of Hilbert and Pólya.

1970s and random matrices

Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property. The zeros tend not to be too closely together, but to repel. Visiting at the Institute for Advanced Study in 1972, he showed this result to Freeman Dyson, one of the founders of the theory of random matrices, which is of importance in physics — the eigenstates of a Hamiltonian, for example the energy levels of an atomic nucleus, satisfy such statistics.

Dyson saw that the statistical distribution found by Montgomery was exactly the same as the pair correlation distribution for the eigenvalues of a random Hermitian matrix. Subsequent work has strongly borne out this discovery, and the distribution of the zeros of the Riemann zeta function is now believed to satisfy the same statistics as the eigenvalues of a random Hermitian matrix, the statistics of the so-called Gaussian Unitary Ensemble. Thus the conjecture of Pólya and Hilbert now has a more solid basis, though it has not yet led to a proof of the Riemann hypothesis.

Recent times

In a development that has given substantive force to this approach to the Riemann hypothesis through functional analysis, Alain Connes has formulated a trace formula that is actually equivalent to a generalized Riemann hypothesis. This has therefore strengthened the analogy with the Selberg trace formula to the point where it gives precise statements.

Possible connection with quantum mechanics

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

Using perturbation theory to first order, the energy of the $n$'th eigenstate is related to the expectation value of the potential:

$E_{n}=E_{n}^{0}+\langle \phi _{n}^{0}\vert V\vert \phi _{n}^{0}\rangle$

where $E_{n}^{0}$ and $\phi _{n}^{0}$ are the eigenvalues and eigenstates of the free particle Hamiltonian. This equation can be take to be a Fredholm integral equation of first kind, with the energies $E_{n}$. Such integral equations may be solved by means of the resolvent kernel, so that the potential may be written as

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

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

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

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

Possible connection with statistical mechanics

Using the explicit formula for the Chebyshev function setting x=exp(u) we have

$\sum _{n}e^{-\beta E_{n}}=Z(\beta )=e^{u/2}-e^{-u/2}{\frac {d\psi _{0}}{du}}-{\frac {e^{u/2}}{e^{3u}-e^{u}}},$

where Z is a partition function, hence $Z(\beta )=\operatorname {Tr} [e^{-\beta H}]$ is the trace of the exponential of certain Hamiltonian where "beta" is a pure imaginary quantity.

Using the definition of Z in terms of an integral over (x, p) we have the next non-linear integral equation for the potential:

$Z(u)Au^{1/2}=\int _{-\infty }^{\infty }\cos(uV(x))\,dx$ with $-\beta =iu.$

So the Hilbert-Polya operator is a Hamiltonian, whose "energies" are precisely the imaginary part of the numbers satisfying $\zeta (\rho )=0$.

References

• M. V. Berry and J. P. Keating, The Riemann Zeros and Eigenvalue Asymptotics, (1999) SIAM Review Vol. 41, No. 2, pp. 236-266.
• B. Aneva, Symmetry of the Riemann operator, (1999) Physics Letters B450, pp. 388-396.
• Zeev Rudnick and Peter Sarnak, Zeros of Principal L-functions and Random Matrix Theory,(1996) Duke Journal of Mathematics, 81 pp. 269-322.