# 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…

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## References

SHOWING 1-10 OF 72 REFERENCES

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

- Mathematics
- 1995

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

- Mathematics
- 1995

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

- MathematicsProceedings 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

- Mathematics
- 1988

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

- Mathematics
- 1996

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

- Physics
- 1971

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

- Physics, Mathematics
- 1997

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.

- PhysicsPhysical review letters
- 1996

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.

- PhysicsPhysical 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

- MathematicsProceedings 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…