The Riemann Zeros and Eigenvalue Asymptotics

@article{Berry1999TheRZ,
  title={The Riemann Zeros and Eigenvalue Asymptotics},
  author={Michael V. Berry and Jonathan P. Keating},
  journal={SIAM Rev.},
  year={1999},
  volume={41},
  pages={236-266}
}
Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn… 

Figures from this paper

A quantum mechanical model of the Riemann zeros

In 1999, Berry and Keating showed that a regularization of the 1D classical Hamiltonian H=xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper, we first

Spectral Spacing Correlations for Chaotic and Disordered Systems

New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can

RIEMANN ZETA ZEROS AND PRIME NUMBER SPECTRA IN QUANTUM FIELD THEORY

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line Re(s) = 1/2. Hilbert and Polya suggested that one possible way to prove the Riemann hypothesis is

Finite Quantum Chaos

This talk considers nite analogs in which the zeta function is the Ihara zetafunction of a graph (and zeros are replaced by poles) and the Riemann hypothesis says the poles lie outside the green circle.

Decidability of the Riemann Hypothesis

The Hamiltonian of a quantum mechanical system has an affiliated spectrum, and in order for this spectrum to be observable, the Hamiltonian should be Hermitian. Such quantum Riemann zeta function

H = xp with interaction and the Riemann zeros

Quantum mechanical potentials related to the prime numbers and Riemann zeros.

The Marchenko approach is applied to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the zeta(s) function, and the multifractal nature of these potentials is demonstrated by measuring the Rényi dimension of their graphs.

A Schrödinger equation for solving the Bender-Brody-Müller conjecture

The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the

ON STRATEGIES TOWARDS THE RIEMANN HYPOTHESIS: FRACTAL SUPERSYMMETRIC QM AND A TRACE FORMULA

The Riemann hypothesis (RH) states that the non-trivial zeros of the Riemann zeta-function are of the form sn = 1/2+iλn. An improvement of our previous construction to prove the RH is presented by

A compact hamiltonian with the same asymptotic mean spectral density as the Riemann zeros

For the classical hamiltonian (x + 1/x)(p + 1/p), with position x and conjugate momentum p, all orbits are bounded. After a symmetrization, the corresponding quantum integral equation possesses a
...

References

SHOWING 1-10 OF 72 REFERENCES

Random matrix theory and the Riemann zeros. I. Three- and four-point correlations

The non-trivial zeros of the Riemann zeta-function have been conjectured to be pairwise distributed like the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) of random matrix theory.

Random matrix theory and the Riemann zeros. I. Three- and four-point correlations

The non-trivial zeros of the Riemann zeta-function have been conjectured to be pairwise distributed like the eigenvalues of matrices in the Gaussian Unitary Ensemble (GUE) of random matrix theory.

Periodic orbit resummation and the quantization of chaos

  • J. Keating
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1992
We study the spectral determinant ∆(E), which has, by construction, zeros at the quantum energy levels of a given system. If the classical motion of the system in question is chaotic then ∆(E) has a

Semiclassical formula for the number variance of the Riemann zeros

By pretending that the imaginery parts Em of the Riemann zeros are eigenvalues of a quantum Hamiltonian whose corresponding classical trajectories are chaotic and without time-reversal symmetry, it

Random matrix theory and the Riemann zeros II: n -point correlations

Montgomery has conjectured that the non-trivial zeros of the Riemann zeta-function are pairwise distributed like the eigenvalues of matrices in the Gaussian unitary ensemble (GUE) of random matrix

Periodic Orbits and Classical Quantization Conditions

The relation between the solutions of the time‐independent Schrodinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the

The phase of the Riemann zeta function

We, offer an alternative interpretation of the Riemann zeta functionζ(s) as a scattering amplitude and its nontrivial zeros as the resonances in the scattering amplitude. We also look at several

Quantum chaos, irreversible classical dynamics, and random matrix theory.

This Letter argues that the statistical quantum properties of the system are intimately related to the irreversible classical dynamics or, more precisely, to the Perron-Frobenius (PF) modes in which a disturbance in the classical probability density of a chaotic system relaxes into the ergodic distribution.

Phase of the Riemann zeta function and the inverted harmonic oscillator.

  • BhaduriKhareLaw
  • Physics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 1995
Analysis of the Riemann zeta function shows analytically that as the real part of the argument is increased toσ>1/2, the memory of the zeros fades only gradually through a Lorentzian smoothing of the density of theZeros.

Calculation of spectral determinants

  • J. KeatingM. Sieber
  • Mathematics
    Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
  • 1994
A method for regularizing spectral determinants is developed which facilitates their computation from a finite number of eigenvalues. This is used to calculate the determinant ∆ for the hyperbola
...