Jk(T ) is clearly de­ ned for all k > 0, and, on the additional assumption that all the zeros are simple, for all k < 0. It has previously been studied by Gonek (1984, 1989, 1999) and Hejhal (1989), and is discussed in Odlyzko (1992, x 2.12) and Titchmarsh (1986, x 14). The model proposed by Keating & Snaith (2000) is the characteristic polynomial of an N £ N unitary matrix U with eigenangles 3...

Theorem 1. Let A be a n × n matrix, and let p(λ) = det(λI − A) be the characteristic polynomial of A. Then p(A) = 0.

Note: The expression for L(s,W ) requires some care, since q v is a complex number acting on a p-adic vector space. What has to be proved is that the characteristic polynomial for the action of Frobv on W i has rational coefficients (and is independent of p), so evaluation using q v makes sense (and the Riemann Hypothesis ensures absolute convergence of the product in a suitable right half-plan...

We describe the set of characteristic polynomials of abelian varieties of dimension 4 over finite fields.

The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the l-adic cohomology of the arrangement (as a variety). We describe the eigenva...

for the polynomial planar vector fields with a hyperbolic or nilpotent critical point at the origin, the monodromy problem has been solved, but for the strongly degenerate critical points this problem is still open. when the critical point is monodromic, the stability problem or the center- focus problem is an open problem too. in this paper we will consider the polynomial planar vector fields ...

We calculate the discrete moments of the characteristic polynomial of a random unitary matrix, evaluated a small distance away from an eigenangle. Such results allow us to make conjectures about similar moments for the Riemann zeta function, and provide a uniform approach to understanding moments of the zeta function and its derivative.

We investigate the joint moments of the 2k-th power of the characteristic polynomial of random unitary matrices with the 2h-th power of the derivative of this same polynomial. We prove that for a fixed h, the moments are given by rational functions of k, up to a well-known factor that already arises when h = 0. We fully describe the denominator in those rational functions (this had already been...

For n-by-n Hermitian matrices A(> 0) and B, define

This paper de nes and develops cycle indices for the nite classical groups. These tools are then applied to study properties of a random matrix chosen uniformly from one of these groups. Properties studied by this technique will include semisimplicity, regularity, regular semisimplicity, the characteristic polynomial, number of Jordan blocks, and average order of a matrix.