Abstract
Let $q>8$ be an odd integer and $p$ a prime with $p$ a prime with $p<q$ and $p\nmid q$. The main purpose of this paper is to study the mean value properties of the character sums over interval $\left[1, \frac{q}{8}\right)$ by using the mean value theorems of the Dirichlet L-functions, and give some interesting mean value formulae.
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