Statistical Properties of the Zeros of Zeta Functions - Beyond the Riemann Case

Abstract

We investigate the statistical distribution of the zeros of Dirichlet $L$--functions both analytically and numerically. Using the Hardy--Littlewood conjecture about the distribution of prime numbers we show that the two--point correlation function of these zeros coincides with that for eigenvalues of the Gaussian unitary ensemble of random matrices, and that the distributions of zeros of different $L$--functions are statistically independent. Applications of these results to Epstein's zeta functions are shortly discussed.

18 pages (Latex), 3 figures by request, Nonlinearity 7 (1994) 1155

Cited by in corpus (9)

• Distribution of Eigenvalues for the Modular Group
• The Riemann Hypothesis for Symmetrised Combinations of Zeta Functions
• Two-point correlation function for Dirichlet L-functions
• Generation of off-critical zeros for hypercubic Epstein zeta-functions
• Averages of ratios of the Riemann zeta-function and correlations of divisor sums
• Zeros of Lattice Sums: 1. Zeros off the Critical Line
• Zeros of Lattice Sums: 2. A Geometry for the Generalised Riemann Hypothesis
• Zeros of Lattice Sums: 3. Reduction of the Generalised Riemann Hypothesis to Specific Geometries
• Further Results on the Riemann Hypothesis for Angular Lattice Sums