The Riemann Zeros and Eigenvalue Asymptotics

1999/01/01 by Michael Berry, Jonathan P. Keating · 381 citations
Chemistry · Mathematics · Physics and Astronomy · #Eigenvalues and eigenvectors #Explicit formulae #Hamiltonian (control theory) #Logarithm #Mathematical Dynamics and Fractals #Mathematical analysis #Mathematical physics #Mathematics #Molecular spectroscopy and chirality #Physics #Quantum #Quantum chaos and dynamical systems #Quantum mechanics #Random matrix #Riemann hypothesis #Riemann surface #Riemann zeta function #Semiclassical physics

paper · doi:10.1137/s0036144598347497

published in SIAM Review 41(2), 236-266 (Society for Industrial and Applied Mathematics)

openalex publication_date 1999/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

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 have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl =XP.

Citations

• Phase of the Riemann ζ function and the inverted harmonic oscillator
• Periodic Orbits and Classical Quantization Conditions
• A Brownian-Motion Model for the Eigenvalues of a Random Matrix
• Basic Hypergeometric Series
• The First 50 Million Prime Numbers
• Zeros of principal L-functions and random matrix theory
• Some problems of ‘Partitio numerorum’; III: On the expression of a number as a sum of primes
• Semiclassical theory of spectral rigidity
• The Riemann Zeta-Function and Quantum Chaology

Cited by

• The Saddle Point of Everything
• The Berry–Keating operator on L²(\mathbb R_\gt,\rm dx) and on compact quantum graphs with general self-adjoint realizations
• A quantum mechanical model of the Riemann zeros
• Random Matrix Theory and ζ(1/2+ it)
• Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture
• Hidden symmetries and nonlinear (super)algebras
• A random matrix model for elliptic curveL-functions of finite conductor
• Distribution of xp in some molecular rotational states
• Level spacings and periodic orbits
• The Lerch zeta function IV. Hecke operators
• A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros
• Emergent phase space description of unitary matrix model
• New strings for old Veneziano amplitudes
• Duality between quantum and classical dynamics for integrable billiards
• Unnormalized differences between zeros of L -functions
• Riemann zeros and the KKR determinant
• Hilbert–Pólya conjecture [wikipedia]