Random matrix theory and \(L\)-functions at \(s=1/2\). (English) Zbl 1051.11047

Summary: Recent results of N. M. Katz and P. Sarnak [Random matrices, Frobenius eigenvalues and monodromy, Providence, R. I: Am. Math. Soc. (1999; Zbl 0958.11004), Bull. Am. Math. Soc., New Ser. 36, 1–26 (1999; Zbl 0921.11047)] suggest that the low-lying zeros of families of \(L\)-functions display the statistics of the eigenvalues of one of the compact groups of matrices \(\text{U}(N)\), \(\text{O}(N)\) or \(\text{USp}(2N)\). We here explore the link between the value distributions of the \(L\)-functions within these families at the central point \(s = 1/2\) and those of the characteristic polynomials \(Z(U,\Theta)\) of matrices \(U\) with respect to averages over \(\text{SO}(2N)\) and \(\text{USp}(2N)\) at the corresponding point \(\Theta = 0\), using techniques previously developed for \(U(N)\) in the authors’ preceding paper [Commun. Math. Phys. 214, 57–89 (2000; Zbl 1051.11048) (see below)]. For any matrix size \(N\) we find exact expressions for the moments of \(Z(U,0)\) for each ensemble, and hence calculate the asymptotic (large \(N\)) value distributions for \(Z(U,0)\) and \(\log Z(U,0)\). The asymptotic results for the integer moments agree precisely with the few corresponding values known for \(L\)-functions. The value distributions suggest consequences for the non-vanishing of \(L\)-functions at the central point.


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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