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Random Matrix Theory and z (1/2 + it)

Keating, J.P.; Snaith, N.C.


Keyword(s): random matrix theory; value distributions; the Riemann zeta function

Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. We study the characteristic polynomials Z (U, ? ) of matrices U in the Circular Unitary Ensemble (CUE) of Random Matrix Theory. Exact expressions for any matrix size N are derived for the moments of *Z* and Z/Z* , and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log Z as N ? 8. In the limit, we show that these two distributions are independent and Gaussian. Costin and Lebowitz [15] previously found the Gaussian limit distribution for Im log Z using a different approach, and our result for the cumulants proves a conjecture made by them in this case. We also calculate the leading order N ? 8 asymptotics of the moments of *Z* and Z/Z*. These CUE results are then compared with what is known about the Riemann zeta function ? (s) on its critical line Res = 1/2, assuming the Riemann hypothesis. Equating the mean density of the non-trivial zeros of the zeta function at a height T up the critical line with the mean density of the matrix eigenvalues gives a connection between N and T . Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ? (1/2+ iT) in the limit T ? 8. They are also in close agreement with numerical data computed by Odlyzko [29] for large but finite T. This leads us to a conjecture for the moments of * ? (1/2+ it) *. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles. Notes: N.C.Snaith, School of mathematics, University of Bristol, University Walk, Bristol, BS8 1TW UK

33 Pages

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