Explicit Evaluation of Double Gauss Sums

摘要

The double (quadratic) Gauss sum G(a, b, c; S; p~n) is defined by where a, b, c are integers such that gcd(a,b， c） = 1， p is a prime， n is a positive integer, and S is an integer coprime to p. The sum G(a, c; S; p~n) was first evaluated by Weber [6] for b even, and subsequently refined by Jordan [5] around 1870. More recently, Alaca, Alaca and Williams [1] evaluated the sum G(a, b, c; S;p~n) under the condition 4αc — b~2 ≠ 0.We refine their ideas and present an explicit evaluation of the sum G(a, b, c; S;p~n) without any condition in a uniform manner. Our approach is advantageous as it can naturally be generalized to quadratic form Gauss sums of three or more variables in an explicit fashion.