A prime sensitive Hankel determinant of Jacobi symbol enumerators

Details A prime sensitive Hankel determinant of Jacobi symbol enumerators A prime sensitive Hankel determinant of Jacobi symbol enumerators

2008-03-19

arxiv.org/abs/0803.2834v1

Mathematics

Combinatorics

13 pages, to appear in Annals of Combinatorics

Scientific paper

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.

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