A prime sensitive Hankel determinant of Jacobi symbol enumerators
Abstract
Description
We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p-1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of -1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums and orthogonality of trigonometric functions.
13 pages, to appear in Annals of Combinatorics
13 pages, to appear in Annals of Combinatorics