On the Mean Values of Certain Character Sums

Abstract

Let $q\ge \mathrm{5}$ be an odd number. In this paper, we study the fourth power mean of certain character sums ${\sum }_{\chi \text{mod}q,\left(\chi -1\right)=-1}^{\mathrm{*}}{\left|{\sum }_{1\le a\le q/4}a\chi \left(a\right)\right|}^4$ and ${\sum }_{\chi \text{mod}q,\left(\chi -1\right)=1}^{\mathrm{*}}{\left|{\sum }_{1\le a\le q/4}a\chi \left(a\right)\right|}^4$, where ${{\sum \mathrm{‍}}^{\mathrm{*}}}_{}$ denotes the summation over primitive characters modulo $q$, and give some asymptotic formulae.