A prime sensitive Hankel determinant of Jacobi symbol enumerators

TL;DR

This paper investigates the Hankel determinants of Jacobi symbol enumerators, revealing their vanishing for composite dimensions and explicit polynomial evaluations for prime dimensions, linked to quadratic characters and Gauss sums.

Contribution

It introduces a prime-sensitive Hankel determinant related to Jacobi symbols, providing explicit evaluations and properties based on number theory.

Findings

Determinant vanishes iff dimension is composite.

For prime p, determinant equals a polynomial linked to Legendre sums.

Sign determined by quadratic character of -1 modulo p.

Abstract

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.