Random matrix theory and \(\zeta(1/2+it)\). (English) Zbl 1051.11048

Keating, Jonathan P.; Snaith, Nina C.

Commun. Math. Phys. 214, No. 1, 57-89 (2000).

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

Keywords:

circular unitary ensemble; Riemann zeta-function; Riemann hypothesis; orthogonal ensemble; circular symplectic ensemble