Dirichlet characters of the rational polynomials

Denote by $\chi$ a Dirichlet character modulo $q\ge 3$, and $\overline{a}$ means $a\cdot \overline{a}\equiv 1modq$. In this paper, we study Dirichlet characters of the rational polynomials in the form

$${\sum _{a=1}^q}^{\prime }\chi (ma+\overline{a}),$$

where ${\sum _{a=1}^q}^{\prime }$ denotes the summation over all $1\le a\le q$ with $(a,q)=1$. Relying on the properties of character sums and Gauss sums, we obtain W. P. Zhang and T. T. Wang's identity ^[6] under a more relaxed situation. We also derive some new identities for the fourth power mean of it by adding some new ingredients.