Weighted Quadratic Partitions Over a Finite Field

Abstract

Using some known results on Gauss sums in a finite field, it is shown that the sum (1.3) defined below can either be evaluated explicitly or expressed in terms of a Kloosterman sum. The same result applies to the more general sum S(α, λ, Q) defined in (5.1). The latter sum also satisfies the reciprocity formula (5.5). Some related sums are discussed in §§6, 7.