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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.

##### 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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