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In mathematics, the Hilbert–Pólya conjecture states that the non-trivial zeros of the Riemann zeta function correspond to eigenvalues of a self-adjoint operator. It is a possible approach to the Riemann hypothesis , by means of spectral theory .
Riemann hypothesis: number theory: ⇐Generalized Riemann hypothesis⇐Grand Riemann hypothesis ⇔De Bruijn–Newman constant=0 ⇒density hypothesis, Lindelöf hypothesis See Hilbert–Pólya conjecture. For other Riemann hypotheses, see the Weil conjectures (now theorems). Bernhard Riemann: 24900 Ringel–Kotzig conjecture: graph theory
The connection with random unitary matrices could lead to a proof of the Riemann hypothesis (RH). The Hilbert–Pólya conjecture asserts that the zeros of the Riemann Zeta function correspond to the eigenvalues of a linear operator, and implies RH. Some people think this is a promising approach (Andrew Odlyzko ).
Riemann hypothesis. Critical line theorem; Hilbert–Pólya conjecture; Generalized Riemann hypothesis; Mertens function, Mertens conjecture, Meissel–Mertens constant; De Bruijn–Newman constant; Dirichlet character; Dirichlet L-series. Siegel zero; Dirichlet's theorem on arithmetic progressions. Linnik's theorem; Elliott–Halberstam conjecture
Hilbert–Smith conjecture: if a locally compact topological group has a continuous, faithful group action on a topological manifold, then the group must be a Lie group. Mazur's conjectures [ 162 ] Novikov conjecture on the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold , arising from the fundamental group .
The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Millennium Prize Problems of the Clay Mathematics Institute, which offers US$1 million for a solution to any of ...
Conjectures are qualified by having a suggested or proposed hypothesis. There may or may not be conjectures for all unsolved problems. ... Hilbert–Pólya conjecture;
Hilbert's problems ranged greatly in topic and precision. Some of them, like the 3rd problem, which was the first to be solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative answer.