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HP Labs Technical Reports
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Random Matrix Theory and L-functions at s =1/2
Keating, J.P.; Snaith, N.C.
HPL-BRIMS-2000-05
Keyword(s): random matrix theory; number theory; L-functions
Abstract: Please Note. This abstract contains mathematical formulae which cannot be represented here. Recent results of Katz and Sarnak [9,10] 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 U (N ), O(N) or USp (2 N ). 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, ?) of matrices U with respect to averages over SO (2N) and U Sp (2N ) at the corresponding point ? = 0, using techniques previously developed for U (N) in [7]. 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. Notes: N.C. Snaith, School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, UK
20 Pages
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