On the Values of a Polynomial Associated with Quadratic Residues at Roots of Unity

Let [Formula: see text] be a prime and [Formula: see text] be the polynomial, where [Formula: see text] runs over the quadratic residues modulo [Formula: see text]. In this paper, we establish an explicit formula for evaluating [Formula: see text] at an arbitrary root of unity [Formula: see text] utilizing the special values of Dirichlet [Formula: see text]-functions at [Formula: see text]. We apply our results to provide unified proofs for several conjectures of Z.-W. Sun. We also evaluate various finite trigonometric products over quadratic residues. Finally, we extend this framework to the case where the modulus is a square-free composite integer [Formula: see text].