Hilbert–Pólya Conjecture
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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 : \tfrac12 + it 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 \tfrac+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_=E_^+ \left. \left \langle \varphi^_n \right , V \left , \varphi^_n \right. \right \rangle where E^_n and \varphi^_n 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 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_^ \left (g(k)+\overline-E_^ \right )\,R(x,k)\,dk where R(x,k) is the resolvent kernel, A is a real constant and : g(k)=i \sum_^ \left(\frac-\rho_n \right)\delta(k-n) where \delta(k-n) 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 \rho_n are the "non-trivial" roots of the zeta function \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''. The simplest Hermitian operator corresponding to ''xp'' is :\hat = \tfrac1 (\hat\hat+\hat\hat) = - i \left( x \frac + \frac1 \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
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'' : \frac + i \frac. 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 :\hat = \frac \left (\hat\hat+\hat\hat \right ) \left (1-e^ \right ) 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 \exp(i\gamma) taken over the imaginary parts of the zeros. {{DEFAULTSORT:Hilbert-Polya conjecture Zeta and L-functions Conjectures Unsolved problems in mathematics