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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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 and its analytic c ...
correspond to
eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of a
self-adjoint operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
. 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 pure ...
, by means of
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operator (mathematics), operators in a variety of mathematical ...
.


History

In a letter to
Andrew Odlyzko Andrew Michael Odlyzko (Andrzej Odłyżko) (born 23 July 1949) is a Polish- American mathematician and a former head of the University of Minnesota's Digital Technology Center and of the Minnesota Supercomputing Institute. He began his career i ...
, dated January 3, 1982,
George Pólya George Pólya (; ; December 13, 1887 – September 7, 1985) was a Hungarian-American mathematician. He was a professor of mathematics from 1914 to 1940 at ETH Zürich and from 1940 to 1953 at Stanford University. He made fundamental contributi ...
said that while he was in
Göttingen Göttingen (, ; ; ) is a college town, university city in Lower Saxony, central Germany, the Capital (political), capital of Göttingen (district), the eponymous district. The River Leine runs through it. According to the 2022 German census, t ...
around 1912 to 1914 he was asked by
Edmund Landau Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopo ...
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 and its analytic c ...
corresponded to
eigenvalues In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
of a
self-adjoint operator In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for al ...
.. 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 and philosopher of mathematics and one of the most influential mathematicians of his time. Hilbert discovered and developed a broad range of fundamental idea ...
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 Dir ...
, but his name has become known for the Hilbert–Pólya conjecture due to a story told by
Ernst Hellinger Ernst David Hellinger (September 30, 1883 – March 28, 1950) was a German mathematician and is primarily known for his works on statistics and probability. His works include Hellinger distance and Hellinger integral which were introduced by him ...
, 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 one of the most influential mathematicians of the twentieth century. His influence is du ...
. 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 A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ...
of a
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
and the
eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...
s 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 th ...
. This so-called
Selberg trace formula In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given ...
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 In mathematics, Montgomery's pair correlation conjecture is a conjecture made by that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is :1-\left(\frac\right)^, which, as Freeman ...
. The zeros tend not to cluster too closely together, but to repel. Visiting at the
Institute for Advanced Study The Institute for Advanced Study (IAS) is an independent center for theoretical research and intellectual inquiry located in Princeton, New Jersey. It has served as the academic home of internationally preeminent scholars, including Albert Ein ...
in 1972, he showed this result to
Freeman Dyson Freeman John Dyson (15 December 1923 – 28 February 2020) was a British-American theoretical physics, theoretical physicist and mathematician known for his works in quantum field theory, astrophysics, random matrix, random matrices, math ...
, one of the founders of the theory of
random matrices In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all of its entries are sampled randomly from a probability distribution. Random matrix theory (RMT) is the ...
. 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 In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the ...
. These distributions are of importance in physics — the
eigenstate In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system re ...
s of a
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian ...
, for example the
energy level A quantum mechanics, quantum mechanical system or particle that is bound state, bound—that is, confined spatially—can only take on certain discrete values of energy, called energy levels. This contrasts with classical mechanics, classical pa ...
s of an
atomic nucleus The atomic nucleus is the small, dense region consisting of protons and neutrons at the center of an atom, discovered in 1911 by Ernest Rutherford at the Department_of_Physics_and_Astronomy,_University_of_Manchester , University of Manchester ...
, 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 Alain Connes (; born 1 April 1947) is a French mathematician, known for his contributions to the study of operator algebras and noncommutative geometry. He was a professor at the , , Ohio State University and Vanderbilt University. He was awar ...
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 pure ...
. This strengthened the analogy with the
Selberg trace formula In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given ...
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 Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions, possibly in some g ...
of
Adele Adele Laurie Blue Adkins (; born 5 May 1988) is an English singer-songwriter. Regarded as a British cultural icon, icon, she is known for her mezzo-soprano vocals and sentimental songwriting. List of awards and nominations received by Adele, ...
classes.


Possible connection with quantum mechanics

A possible connection of Hilbert–Pólya operator with
quantum mechanics Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is ...
was given by Pólya. The Hilbert–Pólya conjecture operator is of the form \tfrac+iH where H is the
Hamiltonian Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified 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 {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...
, or equivalently that V is real. Using
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
to first order, the energy of the ''n''th eigenstate is related to the
expectation value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first moment) is a generalization of the weighted average. Informally, the expected va ...
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 mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
, 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 In mathematics and classical mechanics, canonical coordinates are sets of coordinates on phase space which can be used to describe a physical system at any given point in time. Canonical coordinates are used in the Hamiltonian formulation of cla ...
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 wiktionary: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 ...
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, 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 has criticized this paper, and the authors have responded with clarifications. Moreover, Frederick Moxley has approached the problem with a
Schrödinger equation The Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant landmark in the development of quantum mechanics. It is named after E ...
.


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 about prime numbers Unsolved problems in mathematics