Random Matrix Theory and LFunctions at s= 1/2
Abstract
Recent results of Katz and Sarnak [8,9] suggest that the lowlying zeros of families of Lfunctions display the statistics of the eigenvalues of one of the compact groups of matrices U(N), O(N) or USp(2N). We here explore the link between the value distributions of the Lfunctions within these families at the central point s= 1/2 and those of the characteristic polynomials Z(U,θ) of matrices U with respect to averages over SO(2N) and USp(2N) at the corresponding point θ= 0, using techniques previously developed for U(N) in [10]. 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 Lfunctions. The value distributions suggest consequences for the nonvanishing of Lfunctions at the central point.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2000
 DOI:
 10.1007/s002200000262
 Bibcode:
 2000CMaPh.214...91K