| Summary: | In this thesis, we have explicitly calculated all lower order terms for the n-correlation of zeros of certain families of L-functions. These calculations follow from Conrey and Snaith's similar work for the Riemann zeta function. Katz and Sarnak have argued that the zero statistics of families of L-functions have an underlying symmetry relating to certain ensembles of random matrices. With this in mind, we have looked at a family with orthogonal symmetry (even twists of the Hasse-Weil L-function of a given elliptic curve) and a family with symplectic symmetry (Dirichlet Lfunctions) . Assuming the ratios conjectures of Conrey, Farmer, and Zirnbauer, we prove a formula which explicitly gives all of the lower order terms in the n-correlation . For the families relating to elliptic curves, this formula agrees with the known results of Huynh, Keating and Snaith for n = 1 and as the conductor tends to infinity the 2-correlation matches that of eigenangles of random orthogonal matrices under Haar measure. The method used in this thesis works by first calculating n-correlation of eigenangles of SO(2N) and USp(2N) via ratios of characteristic polynomials. In a similar manner to Conrey and Snaith's work on U(N), we can identify which terms remain in the n-correlation of eigenangles of random orthogonal or symplectic matrices when restrictions are placed on the support of the test function. It is hoped that this will allow for an easier way of checking results with L-functions match those predicted by random matrix theory.
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