On Certain Sum Involving Quadratic Residue

Abstract

Let p be a prime and Fp be the set of integers modulo p. Let χp be a function defined on Fp such that χp(0)=0 and for a∈Fp\{0}, set χp(a)=1 if a is a quadratic residue modulo p and χp(a)=−1 if a is a quadratic non-residue modulo p. Note that χp(a)=ap is indeed the Legendre symbol. The image of χp in the set of real numbers. In this paper, we consider the following sum ∑x∈Fpχp((x−a1)(x−a2)…(x−at)) where a1,a2,…,at are distinct elements in Fp.