Zeros of principal L-functions and random matrix theory

1996/01/01 by Zeév Rudnick, Peter Sarnak · 580 citations
Mathematics · #Advanced Algebra and Geometry #Analytic Number Theory Research #Computer science #Eigenvalues and eigenvectors #Mathematics #Matrix (chemical analysis) #Physics #Principal (computer security) #Pure mathematics #Random Matrices and Applications #Random matrix

paper · doi:10.1215/s0012-7094-96-08115-6

published in Duke Mathematical Journal 81(2) (Duke University Press)

openalex publication_date 1996/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Citations

• Toeplitz matrices, translation kernels and a related problem in probability theory
• On Euler Products and the Classification of Automorphic Representations I
• On Certain L-Functions
• Rankin-Selberg Convolutions

Cited by

• Zeros of Polynomials in Derivatives of Automorphic L-functions
• Distribution of Eigenvalues of Matrix Ensembles arising from Wigner and Palindromic Toeplitz Blocks
• A class number formula for Picard modular surfaces
• Discorrelation of multiplicative functions with nilsequences and its application on coefficients of automorphic L-functions
• On the absolute convergence of the spectral side of the Arthur trace formula for GL(n)
• The Set of Prime Numbers
• On the local Langlands correspondence for quasi-split even orthogonal groups
• Formes automorphes et voisins de Kneser des réseaux de Niemeier
• Random Matrices, Intrinsic Freeness, and Sharp Non-Asymptotic Inequalities
• The Riemann-zeta function on vertical arithmetic progressions
• The highest lowest zero and other applications of positivity
• Strong multiplicity one for the Selberg class
• Universality for mathematical and physical systems
• Non-vanishing of L-functions, the Ramanujan conjecture, and families of Hecke characters
• Integral moments of L-functions
• Centered Moments of Weighted One-Level Densities of GL(2) L-Functions
• The correlations of Farey fractions
• Prime pairs and Zeta's zeros
• Low-lying zeros of Hilbert modular L-functions weighted by powers of central L-values
• Joint value distribution of L-functions on the critical line
• Low-lying Zeros of Number Field L-functions
• The second moment of Dirichlet twists of a \textrmGL₄ automorphic L-function
• The second moment theory of families of L-functions
• Selberg orthogonality for half-integral weight modular forms
• A Combinatorial Approach to Mixed Ratios of Characteristic Polynomials
• On Hypothesis H of Rudnick and Sarnak
• Elliptic curves of high rank and the Riemann zeta function on the one line
• Upper bounds on L-functions at the edge of the critical strip
• Some Results in the Theory of Low-lying Zeros: Determining the 1-level density, identifying the group symmetry and the arithmetic of moments of Satake parameters
• On the Effective Non-vanishing of Hecke--Maass L-functions at Special Points
• Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann ζ zeros
• Wheeler-DeWitt wavefunctions for 5d BKL dynamics, automorphic L-functions and complex primon gases
• Weyl's law for the cuspidal spectrum of SL(n)
• Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles
• Selberg's Central Limit Theorem weighted by Linear Statistics of Zeta Zeros
• Low-lying Zeros of Cuspidal Maass Forms
• Quantum Knots and Riemann Hypothesis
• When Kloosterman sums meet Hecke eigenvalues
• Distribution of Eigenvalues of Random Real Symmetric Block Matrices
• Effective Multiplicity One for GL(n)
• The eigenvalue spacing of a random unipotent matrix in its action on lines
• Weak subconvexity for central values of L-functions
• Pair correlation for Dedekind zeta functions of abelian extensions
• Low-lying zeros in families of Maass form L-functions: an extended density theorem
• Properties of scalar partition functions of 2d CFTs
• SUMS OF FOURIER COEFFICIENTS OF CUSP FORMS
• Universality in Random Matrix Theory for orthogonal and symplectic ensembles
• On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices
• Statistics of the zeros of L-functions and arithmetic correlations
• Semiclassical form factor for chaotic systems with spin ½
• Boundedness of automorphic 𝐿–functions in vertical strips
• The quantum Mellin transform
• Applications of Arthur's multiplicity formula to Siegel modular forms
• A relation between the zeros of different two L-functions which have the Euler product and functional equation
• Chebyshev's Bias
• Random matrices, generalized zeta functions and self-similarity of zero distributions
• Nuclei, Primes and the Random Matrix Connection
• A unitary test of the Ratios Conjecture
• One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries
• Low-lying zeros of L-functions with orthogonal symmetry
• The Riemann Zeros and Eigenvalue Asymptotics
• Random Matrix Theory and ζ(1/2+ it)
• Zeroes of zeta functions and symmetry
• Gaps between zeros of GL(2) L-functions
• Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory
• A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros
• A note on the number of coefficients of automorphic L-functions for GL_m with same signs
• Distribution of the Ratio of Consecutive Level Spacings in Random Matrix Ensembles
• Semiclassical universality of parametric spectral correlations
• Unnormalized differences between zeros of L -functions
• On the critical points of random matrix characteristic polynomials and of the Riemann ξ-function
• Hilbert–Pólya conjecture [wikipedia]
• Montgomery's pair correlation conjecture [wikipedia]