Mean Value Of Primitive Character Sums Over Incomplete Intervals

The Dirichlet character sums have an important role in the study of the number theory, the character sums estimation is one of the important research topics in number theory, especially in the analytic number theory. All the time, various types of character sums estimation have been widely concerned by many scholars, domestic and foreign scholars have carried out extensive and in-depth research on this issue, and obtained a wealth of research results. This thesis is a generalization of the previous studies on the mean value of character sums, from the point of the mean value, we study the mean value of primitive character sums over incomplete intervals by using the method of elementary number theory and analytic number theory or some elementary methods in related literatures, meanwhile combining the estimates for character sums and mean value theorems of Dirichlet L-functions. The results are as follows:1. Let q> 3 be an integer and p be a prime with p(?)q and p< q, then for any positive integer k, we study the 2k-th power mean of the even primitive character sums over interval [1, q/p) Finally, the asymptotic formula is given.2. Furthermore, we study the mean value of primitive character sums over incomplete intervals with the weight of Dirichlet L-functions where, N= [1, q/4), [1, q/2) and [1,q/p). Finally, the asymptotic formulae on the corresponding intervals are given.