Random Matrix Theory and ζ(1/2+it)
Abstract
We study the characteristic polynomials Z(U, θ) 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>∞. 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>∞ 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 Re s= 1/2, assuming the Riemann hypothesis. Equating the mean density of the nontrivial 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 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.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2000
 DOI:
 10.1007/s002200000261
 Bibcode:
 2000CMaPh.214...57K