On the general trigonometric sums weighted by character sums

The main purpose of this paper is using the estimates for character sums and the mean value properties of Dirichlet L-function to study the hybrid mean value properties of the general trigonometric sums, which is weighted by the 2k  th and first power mean of character sums in [1,p4). The two sharp asymptotic formulae are derived and some interesting connections between these sums are established.