A new generalized Gauss sums and its fourth power mean

This paper mainly presents a new class of generalized Gauss sums. In order to find the potential relationship between these sums, we start from their definition, and study the fourth means of this new generalized Gauss sums by applying elementary analytic methods and some important qualities related to trigonometric sums. This result extends the classical Gauss sums and Kloosterman sums in analytic number theory, and makes a new contribution to the further development of this field. Finally, by calculating, an important identity is given. At the same time, we also put forward some problems that still exist and have not been solved in this field.