On The Hybrid Mean Value Of Some Sums And The Properties Of Some Special Sequences

Some famous sums and special sequences play an important role in the study of analytic number theory, and many nontrivial achievements in this field were made by using them. So it is valuable and meaningful to investigate the relationships among them.In this dissertation, we study the hybrid mean value of some sums and the arithmetical properties of some special sequences. A series of hybrid mean value formulae on character sums, Gauss sums, Kloosterman sums, Cochrane sums and Hurwitz zeta-function are given. We study the arithmetical property of D.H.Lehrner number that the variable is confined in some special sets and the estimates of sums related to D.H.Lehmer problem. We also work on some special sequences and arithmetical functions, and obtain some sharp asymptotic formulae involving them. The main achievements contained in this dissertation are as follows:1. By studying the hybrid mean values involving Kloosterman sums , Hurwitz zeta-function and Kloosterman sums, we disclose the relationships among them , and some sharper asymptotic formulae are given.2. By transforming Character sums over short interval into some types of Dirichlet L-functions and Gauss sums, we study three kinds of hybrid mean value between a sum analogous to character sums and Gauss sums, and give a series of asymptotic formulae .3. Using the estimates for triangle sums and Kloosterman sums, we investigate k-free D.H.Lehmer number and its generalization, and obtained some asymptotic formulae about them. By using the properties of primitive character and the mean value theorems of Dirichlet L-function, we also study the estimates of summation related to D.H.Lehmer problem.4. We use the complex integral to study the arithmetical property of sequence {e_q(n)} and its hybrid mean value with divisor function. We also obtain a sharper asymptotic formula involving sequence {e_q(n)} by using the elementary method.