A Study On The Mean Value Of Some Famous Summations In Number Theory

It is well known that the mean value problems of some famous summations play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Therefore, any nontrivial progress in this field will contribute to the development of analytic number theory.In this dissertation, the mean value of some famous summations including character sums over a short interval, two-term exponential sums with Dirichlet character, Cochrane sums, Kloosterman sums and their generalized sums, are studied. Besides these, problems concerning an integer and its inverse are also studied. The main achievements contained in this dissertation are as follows:1. The study of character sums over a short interval is one of the hottest issues in analytic number theory. In this dissertation, the hybrid mean value involving the high power mean of character sums over the short interval [1,(?)), general quadratic Gauss sums and general Kloosterman sums is studied, and sev-eral asymptotic formulae are given.2. The study of Dedekind sums enjoys its long history. Cochrane sumsis studied, which is analogous to Dedekind sums, and generalized to the multiple varieties case as hyper Cochrane sums. An upper bound of hyper Cochrane sums is generalized. By using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions, the hybrid mean value between hyper Cochrane sums and hyper Kloosterman sums is discussed.3. There has been much progress in the study of Kloosterman sums in recent years. The hybrid mean value including the general quadratic Kloosterman sums is considered, and some connections between the r-th hyper Kloosterman sums and a problem of D.H. Lehmer are established.4. The study of exponential sums also enjoys its long history. In this dissertation, the hybrid mean value of the two-term exponential sums with Dirichlet character is studied, and several interesting mean value formulae are obtained.5. The study on the problem of an integer and its inverse will help us know more about the distributions of integers. In this dissertation, the high power mean of an integer and its inverse is studied, and an interesting asymptotic formula is given, which is sharpened by Shparlinski later.