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.
Magnetic resonance imaging (MRI) was conducted with use of the spin-echo technique (0.35 Tesla) in 22 patients with a variety of congenital and cardiovascular anomalies and in 16 normal volunteers. Electrocardiographic (ECG) synchronization of the data acquisition produced transverse, parasagittal, and coronal tomograms that were used to define size and… (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)
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.