A Study On The Mean Value Of Some Famous And

As well known that the mean value problem of some famous sums which are related to many theoretic number problems is very important in the field of ana-lytic number theory. Therefore, the fact that we can get any nontrivial progress in this field will contribute to the development of analytic number theory.In this dissertation, some famous sums include Dedekind sums, Cochrane sums and Kloosterman sums, of which the mean values over different intervals are studied. Besides, the solution of the equation including the function named Euler is also studied. The main achievements contained in this dissertation are as follows:1. The study of the mean value of classical Dedekind sums. The weighted mean values over the whole interval and a short interval [1,p], [1,p/2)(p is a odd prime number) are studied, which resulted in several asymtotic formulae.2. The mean value of generalized Kloosterman sums is studied. The hybrid mean values between generalized Kloosterman sums and generalized Dedekind sums, as well as generalized Cochrane sums and generalized Kloosterman sums, are studied. Several interesting asymtotic formulae are obtained.3. The solutions of an equation about Euler function are solved. With the distributions of fundamental methods, the study on the solutions of an equation about Euler function are discussed, and all the positive integral solutions are given.