Abstract. By using the analytic methods, the mean value of the generalquadratic Gauss sums weighted by the first power mean of character sumsover a short interval is investigated. Several sharp asymptotic formulaeare obtained, which show that these sums enjoy good distributive prop-erties. Moreover, interesting connections among them are established. 1. Introduction and main resultsFor any integer n, the general quadratic Gauss sums G(n,χ;q) is defined asG(n,χ;q) =X qa=1 χ(a)ena 2 q,where e(y) = e 2πiy . This summation is very important, since it is the general-ization of the classical quadratic Gauss sums. But we still know little aboutthe properties of G(n,χ;q), we do not even know how large G(n,χ;q) is. Sincethe value of |G(n,χ;q)| is irregular as χ varies, one can only get some upperbound estimates. For example, for any integer n with (n,q) = 1, from thegeneral result of Cochrane and Zheng [1] we can deduce that|G(n,χ;q)| ≤ 2 ω(q) q 12 ,where ω(q) denotes the number of distinct prime divisors of q. The case thatq is prime is due to A. Weil [2].However, weighted sums [4] involving G(n,χ;q) enjoys many good valuedistribution properties, through which interesting connections among them areestablished. Now we shall use analytic methods to study the mean value of thegeneral quadratic Gauss sums weighted by the first power mean of character