Mathematical Physics © Springer-Verlag 2000 Random Matrix Theory and ζ(1/2+ it)

Abstract: We study the characteristic polynomialsZ(U, θ) of matricesU 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 logZ as N →∞. 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 logZ 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 → ∞ asymptotics of the moments of |Z | and Z/Z∗. These CUE results are then compared with what is known about the Riemann zeta function ζ(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 betweenN and T. Invoking this connection, our CUE results coincide with a theorem of Selberg for the value distribution of log ζ(1/2 + iT) in the limit T → ∞. 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 |ζ(1/2 + it)|. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles. 1