On an identity associated with Weil's estimate and its applications

In this paper we study the fourth mean value of generalized quadratic Gauss sums, and get an interesting identity associated with Weil's estimate,∑a=2p−2∑b=1p−1(a2−b2p)(b2−1p)={10−2p,p≡1 (mod 4);2p−6,p≡3 (mod 4). By virtue of the identity, higher-order mean values of generalized quadratic Gauss sums are obtained.