Communications in Mathematical Physics

, Volume 214, Issue 1, pp 57–89

Random Matrix Theory and ζ(1/2+it)


  • J. P. Keating
    • School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
  • N. C. Snaith
    • BRIMS, Hewlett-Packard Laboratories, Filton Road, Stoke Gifford, Bristol BS34 6QZ, UK

DOI: 10.1007/s002200000261

Cite this article as:
Keating, J. & Snaith, N. Commun. Math. Phys. (2000) 214: 57. doi:10.1007/s002200000261


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 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 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.

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© Springer-Verlag Berlin Heidelberg 2000