In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Hilbert–Pólya conjecture states that the non-trivial
zeros of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
correspond to
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of a
self-adjoint operator. It is a possible approach to the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
, by means of
spectral theory.
History
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
:
of the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
corresponded to
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of a
self-adjoint operator.
[.] The earliest published statement of the conjecture seems to be in .
[.]
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
did not work in the central areas of
analytic number theory
In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
, 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
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...
. 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.
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
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
. This so-called
Selberg trace formula bore a striking resemblance to the
explicit formulae, which gave credibility to the Hilbert–Pólya conjecture.
1970s and random matrices
Hugh Montgomery investigated and found that the statistical distribution of the zeros on the critical line has a certain property, now called
Montgomery's pair correlation conjecture. The zeros tend not to cluster 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.
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
eigenstate
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
s 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.
Later developments
In 1998,
Alain Connes formulated a trace formula that is actually equivalent to the
Riemann hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in ...
. This strengthened the analogy with the
Selberg trace formula to the point where it gives precise statements. He gives a geometric interpretation of the
explicit formula of number theory as a trace formula on
noncommutative geometry of
Adele classes.
Possible connection with quantum mechanics
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.
Using
perturbation theory to first order, the energy of the ''n''th eigenstate is related to the
expectation value of the potential:
:
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
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, and the
are the "non-trivial" roots of the zeta function
.
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''
[.] The simplest Hermitian operator corresponding to ''xp'' is
:
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
dilation
Dilation (or dilatation) may refer to:
Physiology or medicine
* Cervical dilation, the widening of the cervix in childbirth, miscarriage etc.
* Coronary dilation, or coronary reflex
* Dilation and curettage, the opening of the cervix and surgic ...
s perhaps the boundary condition ''f''(''nx'') = ''f''(''x'') for integer ''n'' may help to get the correct asymptotic results valid for large ''n''
:
A paper was published in March 2017, written by
Carl M. Bender,
Dorje C. Brody
Dorje C. Brody (born 1970 in Hong Kong) is a British applied mathematician and mathematical physicist.
Career
Dorje C. Brody was born in Hong Kong, but lived in Japan for a number of years. He received his BSc in physics at Niigata University ...
, and
Markus P. Müller, which builds on Berry's approach to the problem. There the operator
:
was introduced, which they claim satisfies a certain modified versions of the conditions of the Hilbert–Pólya conjecture.
Jean Bellissard
Jean Vincent Bellissard (born 1 March 1946, Lyon) is a French theoretical physicist and mathematical physicist, known for his work on C*-algebras, K-theory, noncommutative geometry as applied to solid state physics, particularly, to quantum Hall ef ...
has criticized this paper, and the authors have responded with clarifications. Moreover, Frederick Moxley has approached the problem with a
Schrödinger equation.
References
Further reading
* .
*. 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.
{{DEFAULTSORT:Hilbert-Polya conjecture
Zeta and L-functions
Conjectures
Unsolved problems in mathematics