Math 2.0

Number theory Conjecture

Hilbert–Pólya conjecture

Builds on Riemann zeta functionRiemann hypothesisHilbert spaceAdjoint of an unbounded operatorEigenvalue

Statement

There is a complex Hilbert space HH with a self-adjoint operator TT on HH that has the following property. Here self-adjoint means T=T∗T = T^*, where T∗T^* is the adjoint of TT. A complex number ρ\rho with 0<Re⁡ρ<10 < \operatorname{Re} \rho < 1 is a zero of the Riemann zeta function if and only if ρ=12+it\rho = \tfrac{1}{2} + i t for an eigenvalue tt of TT.

Remarks

The eigenvalues of a self-adjoint operator are real, so the conjecture implies the Riemann hypothesis: every nontrivial zero ρ=12+it\rho = \tfrac{1}{2} + it then has real part 12\tfrac{1}{2}. The conjecture is open. As stated above, it is in fact equivalent to the Riemann hypothesis, since under that hypothesis a diagonal operator with the required eigenvalues can be written down directly. So the real content of the idea lies in finding a natural operator, for instance one coming from geometry or physics, whose spectrum is given by the zeros. Evidence for the approach comes from random matrix theory. Montgomery's pair correlation conjecture of 1973 and Odlyzko's numerical computations indicate that the imaginary parts of the zeros are spaced like the eigenvalues of large random Hermitian matrices. The idea is attributed to Hilbert and Pólya. It is known through a 1982 letter of Pólya to Odlyzko, in which Pólya recalled suggesting it in Göttingen around 1912 to 1914.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

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Adjoint of an unbounded operator, concept, no evidence levelAdjoint of an unbound…conceptEigenvalue, concept, no evidence levelEigenvalueconceptGamma function, concept, no evidence levelGamma functionconceptHilbert space, concept, no evidence levelHilbert spaceconceptMeromorphic function, concept, no evidence levelMeromorphic functionconceptRiemann zeta function, concept, no evidence levelRiemann zeta functionconceptFunctional equation of the Riemann zeta function, statement, no evidence levelFunctional equation o…statementRiemann hypothesis, conjecture, no evidence levelRiemann hypothesisconjectureHilbert–Pólya conjecture, conjecture, no evidence levelHilbert–Pólya conject…conjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Adjoint of an unbounded operator concept
  2. Eigenvalue concept
  3. Gamma function concept
  4. Hilbert space concept
  5. Meromorphic function concept
  6. Riemann zeta function concept
  7. Functional equation of the Riemann zeta function statement
  8. Riemann hypothesis conjecture
  9. Hilbert–Pólya conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Hilbert%E2%80%93P%C3%B3lya_conjecture
About this page
TagECS1GDJ734
URIm2://ECS1GDJ734
Kindconjecture
WritingW0 stub
MSC11M26, 47B25, 81Q10
StatusThis copy of the site has no build data. Run m2 status ECS1GDJ734.
AboutRiemann zeta function, Riemann hypothesis
UsesRiemann zeta function, Riemann hypothesis, Hilbert space, Adjoint of an unbounded operator, Eigenvalue
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Changed2 November 2026

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