# Random Matrix Theory and ζ(1/2+it)

DOI: 10.1007/s002200000261

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

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## 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 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.