On The Generalized Binomial Exponents And The Sum Of Features On The Lehmer Number Set

The mean value of arithmetical functions plays an important role in the study of number theory. The research on the mean value of some classical arithmetical functions is helpful for studying the properties of the arithmetical functions. It will promote the development of the analysis number theory.In a variety of arithmetical functions, exponential sums and Dirichlet character have a long history and rich content. The various exponential sums not only occupy the significant status in pure theory, but also have very extensive and profound applications in algebraic coding and cryptography.Inspired by many scholars who have furthered the research of the exponential sums, we define two kinds of exponential sums with similar form which have different properties in Chapter Three. We explore their fourth power and get a series of explicit formula, and we also find there exists close relationship between some mean values of the two sums. The concrete content as follows:Firstly, we define the sum as which is analogous to the classical kth general two-term exponential sums We study the fourth power mean value of the exponential sums about the coeffi-cients m, n respectively by the elementary and analytic method. Prom this process of the calculation, we find the difference and connection for the high and low pow-er's coefficients and get some interesting results. Meanwhile, we obtain two simple corollaries directly.Similarly, we define the following sums:Flrom the study of C₂?m,n,k,x?q?, we and that even though the form of C1?m,n,k,x?q?,C₂?m,n,k,x?q?have very wide difference,their fourth power mean value is strikingly similar. Comparing with the results of many scholars on the sum we find that kth general two-term exponential sums with the similar forms have close connections.In Chapter Four we studies the upper bound of the primitive characters on the D.H. Lehmer numbers.