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)
Let T N,χ p,k (x) be the characteristic polynomial of the Hecke operator T p acting on the space of cusp forms S k (N, χ). We describe the factorization of T N,χ p,k (x) mod ℓ as k varies, and we explicitly calculate those factorizations for N = 1 and small ℓ. These factorizations are used to deduce the irreducibility of certain T 1,1 q,k (x) from the… (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)
Let T p,k (x) be the characteristic polynomial of the Hecke operator Tp acting on the space of level 1 cusp forms S k (1). We show that T p,k (x) is irreducible and has full Galois group over Q for k ≤ 2000 and p < 2000, p prime.
For each prime p, we determine the distribution of the p th Fourier coefficients of the Hecke eigenforms of large weight for the full modular group. As p → ∞, this distribution tends to the Sato–Tate distribution.
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)
We establish relationships between mean values of products of logarithmic derivatives of the Rie-mann zeta-function near the critical line, correlations of the zeros of the Riemann zeta-function and the distribution of integers representable as a product of a fixed number of prime powers.
We evaluate the real character sum m n m n where the two sums are of approximately the same length. The answer is surprising.