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Random matrix theory and $$\zeta(1/2+it)$$. (English) Zbl 1051.11048
Summary: We study the characteristic polynomials $$Z(U, \Theta)$$ 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 $$N\to\infty$$. In the limit, we show that these two distributions are independent and Gaussian. O. Costin and J. L. Lebowitz [Gaussian fluctuation in random matrices, Phys. Rev. Lett. 75, 69–72 (1995)] previously found the Gaussian limit distribution for $$\text{Im}\, \log \mathbb Z$$ 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 $$N\to\infty$$ asymptotics of the moments of $$| \mathbb Z|$$ and $$\mathbb Z/\mathbb Z^*$$. These CUE results are then compared with what is known about the Riemann zeta-function $$\zeta(s)$$ on its critical line $$\text{Re}\,s = 1/2$$, 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 $$\log\zeta(1/2+ iT)$$ in the limit $$T\to\infty$$. They are also in close agreement with numerical data computed by A. M. Odlyzko [The $$10^{20\text{th}}$$ zero of the Riemann zeta-function and 70 million of its neighbors. Preprint (1989)] for large but finite $$T$$. This leads us to a conjecture for the moments of $$\zeta(1/2+ it) |$$. Finally, we generalize our random matrix results to the Circular Orthogonal (COE) and Circular Symplectic (CSE) Ensembles.

MSC:
 11M50 Relations with random matrices 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses 15B52 Random matrices (algebraic aspects) 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
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