Characteristics And, Of Kloosterman Sums And Generalized Higher Order Bernoulli Numbers,

The main purpose of this dissertation is to study the mean value problems of some summations in number theory. Firstly, we study mean value problems about inversion of Dirichlet L-function, general Kloosterman sums weighted by incomplete character sums, then we generalize the present results on Cochrane sums and Dedekind sums in incomplete intervals. Next, we study mean value of higher order generalized Bernoulli numbers, Gauss sums and general Kloost-erman sums. Finally, the properties of equations on the Smarandache-Type multiplicative functions are studied. The main achievements contained in this dissertation are as follows:1. Some new estimates for are given, and a new estimate for the mean value of inversion of Dirichlet L-function weighted by incomplete character sums is established, which gener-alizes the result of Wenpeng Zhang. Further, the error term of the obtained estimate is improved by using Igor E. Shparlinski's recent work. Moreover, new estimates for the hybrid mean value of general Kloosterman sums weighted by incomplete character sums are given. It is remarkable that the same method can be applied to sums of 2k-th power mean In addition, mean value properties of the Dedekind sums and Cochrane sums over the incomplete interval [1,P/8] are discussed by using the mean value theroems of the Dirichlet L-functions, a generalization of Zhefeng Xu's result is obtained. 2. Mean values of higher order generalized Bernoulli numbers, Gauss sums and general Kloosterman sums are studied, and a few new asymptotic formu-lae are given. For details, by using the properties of higher order generalized Bernoulli numbers Bn,χ(r) and mean value theroems of the Dirichlet L-functions, the hybrid mean value of higher order generalized Bernoulli numbers Bn,χ(r), Gauss sums and general Kloosterman sums are studied, two new estimates are obtained, some results due to Huaning Liu and Wenpeng Zhang are generalized.3. Using the elementary method, the properties of equations on the Smarandache-Type multiplicative functions are studied, and two interesting identities are given.