We give a correction to the previous paper of S. F. Cao and T. T. Wang (2021). Additionally, provide a conversion formula of the power mean of a certain exponential sums weighted by a Dirichlet character ${\chi }_3$ and some of its applications.

The main purpose of this paper is to use very simple elementary and analytic methods to study a problem related to the primitive root modulo $p$ asked by Brizolis and prove a more general and stronger conclusion.

Classical Gauss sums is one kind sums of the exponential functions weighted by character functions, which occupies an important position in the study and practical application of number theory. In this paper, we focus on the problem of calculating formula of one kind square power mean of special Gauss sums. By using elementary methods, analytical methods and the properties of character sums, we obtain an exact calculating formula involving the classical Gauss sums τ(χ₈) when p is an odd prime with p≡1 mod 8, where χ₈ denotes an eight-order character modulo p. The results not only expand the research content of classical Gauss sums, but also contribute to the further development of relevant research work.