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

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

1994-09-16

arxiv.org/abs/chao-dyn/9409004v1

Nonlinear Sciences

Chaotic Dynamics

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

Scientific paper

10.1088/0951-7715/7/4/004

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.

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