Conjecture of Hilbert-Pólya

In Mathematical, the conjecture of Hilbert-Pólya is a possible approach of the Hypothèse of Riemann, using the spectral Théorie.

First ideas

Hilbert and Pólya speculated that the values of T such as 1/2 + it is one zero of the Fonction Zeta of Riemann must be the eigenvalues of a square operator, and this would be a way to show the assumption of Riemann.

The Fifties and formulas of the traces of Selberg

At this time, it was a small base for such a speculation. Nevertheless Selberg with the beginning of the year 1950 showed a duality between the Length of the spectrum of a Surface of Riemann and the eigenvalues of sound Laplacien. This, which one calls the Formule of the traces of Selberg advances a resemblance striking to the explicit formulas, gave a certain credibility to the speculation of Hilbert and Pólya.

The random Seventies and matrices

Hugh Montgomery sought and found that the statistical distribution of the zeros on the critical line has a certain property. The zeros do not tend to being too firmly together, but pushing back themselves. By visiting the Institute for Advanced Study in 1972, it showed this result with Freeman Dyson, one of the founders of the theory of the random matrices, which are very important in physics - the clean states of a Hamiltonien, for example the energy levels of a Atomic nucleus, satisfy such statistics.

Dyson saw that the statistical distribution found by Montgomery was exactly the same one as the distribution of the pairs of correlations for the eigenvalues of a square Matrice random. Posterior work strongly raised this discovery, and the distribution of the zeros of the Fonction zeta of Riemann is now recognized to satisfy the same statistics as the eigenvalues of a random square matrix, the statistics of what one calls the Gaussian together unit. Thus, the conjecture of Pólya and Hilbert have a more solid base now, although it did not lead yet to a demonstration of the Hypothèse of Riemann.

Recent developments

In a development which gave an appreciable force with this approach of the assumption of Riemann through the analyzes functional, Alain Connes stated a formula of trace which is currently equivalent to the generalized Hypothèse of Riemann. This, consequently, reinforced the analogy with the formula of trace of Selberg at the point where it gives precise results.

Possible connection with quantum mechanics

A possible connection of the operator of Hilbert-Pólya with the quantum Mécanique was given by Pólya. The operator of Hilbert-Pólya is form $1/2+iH\; \backslash ,$ where $H\; \backslash ,$ is the Hamiltonien of a particle of mass $m\; \backslash ,$ i.e. moving under the influence of a potential $V\; (X)$. The conjecture of Riemann is equivalent to the assertion which the Hamiltonian is Hermitien, or in an equivalent way that $V$ is real.

By using the first order perturbation theory, the energy of the $n$ième clean state is connected to the hoped value of the potential:

$E\_\; \{N\}\; =E\_\; \{N\}\; ^\; \{0\}\; +\; \backslash \; langle\; \backslash \; phi^\; \{0\}\; \_n\; \backslash \; green\; V\; \backslash \; green\; \backslash \; phi^\; \{0\}\; \_n\; \backslash \; rangle$

where $E^\; \{0\}\; \_n\; \backslash ,$ and $\backslash \; phi^\; \{0\}\; \_n\; \backslash ,$ are the eigenvalues and the clean states of the Hamiltonian of the free particle. This equation can be taken to be a integral equation of Fredholm of first species, with energies $E\_n$. Such integral equations can be solved by means of the Noyau solving, so that the potential can be written in the form

$V\; (X)\; =A\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{\backslash \; infty\}\; (G\; (K)\; +\; \backslash \; overline\; \{G\; (K)\}-\; E\_\; \{K\}\; ^\; \{0\})\; \backslash ,\; R\; (X,\; K)\; \backslash ,\; dk$

where $R\; (X,\; K)$ is the solving core, $A$ is a real constant and

$G\; (K)\; =i\; \backslash \; sum\_\; \{n=0\}\; ^\; \{\backslash \; infty\}\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{2\}\; -\; \backslash \; rho\_n\; \backslash \; right)\; \backslash \; delta\; (kN)$

where $\backslash \; delta\; (kN)\; \backslash ,$ is the function delta of Dirac, and the $\backslash \; rho\_n\; \backslash ,$ are the " roots; non-triviales" function zeta $\backslash \; zeta\; (\backslash \; rho\_n)\; =0\; \backslash ,$.

Possible connection with statistical mechanics

By using the Formule clarifies for the Fonction of Tchebychef, by fixing x=exp (U), we have

$\backslash \; sum\_\; \{N\}\; e^\; \{-\; \backslash \; beta\; E\_\; \{N\}\}\; =Z\; (\backslash \; beta)\; =e^\; \{u/2\}\; -\; e^\; \{-\; u/2\}\; \backslash \; frac\; \{D\; \backslash \; psi\; \_\; \{0\}\}\; \{of\; the\}\; -\; \backslash \; frac\; \{e^\; \{u/2\}\}\; \{e^\; \{3u\}\; -\; e^\; \{U\}\},$

where Z is a Fonction of partition, consequently $Z\; (\backslash \; beta)\; =\; \backslash \; operatorname\; \{Tr\}\; H\}$ is the trace of exponential of a certain Hamiltonian where " beta" is an imaginary quantity pure.

By using the definition of Z in terms of an integral on ( X ,   p ), we have the nearest non-linear integral for the potential:

$Z\; (U)\; Au^\; \{1/2\}\; =\; \backslash \; int\_\; \{-\; \backslash \; infty\}\; ^\; \{\backslash \; infty\}\; \backslash \; cos\; (UV\; (X))\backslash ,\; dx$ with $-\; \backslash \; beta\; =iu.$

Therefore, the opetator of Hilbert-Pólya is a Hamiltonian, of which " énergies" are precisely the imaginary parts of the numbers satisfying $\backslash \; zeta\; (\backslash \; rho)\; =0\; \backslash ,$.

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