D. W. Farmer

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We describe numerical calculations which examine the Phillips-Sarnak conjecture concerning the disappearance of cusp forms on a noncom-pact finite volume Riemann surface S under deformation of the surface. Our calculations indicate that if the Teichmüller space of S is not trivial, then each cusp form has a set of deformations under which either the cusp(More)
We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U (N), O(2N) and U Sp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum,(More)
Differentiation causes the small gaps between zeros of a given real entire function with order 1 to become larger and the larger gaps to become smaller. In this article, we show that for the Riemann Ξ-function, there exist A n and C n with C n → 0 such that lim n→∞ A n Ξ (2n) (C n z) = cos z uniformly on compact subsets of C. With our method, one can prove(More)
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