On The Mean Of Arithmetic Functions

Mean value problems of arithmetical functions play an important role in the study of analytic number theory, and they relate to many famous number theoretic problems. Any nontrivial progress in this field will contribute to the development of analytic number theory.In this dissertation, we study the mean value problems of some arithmetical functions. A series of mean value formulae on the cubic exponential sums, Gauss sums, generalized Bernoulli numbers, Kloosterman sums and Cochrane sums are given. We study the problem of an integer and its inverse, and D.H.Lehmer problem. The multiple varieties and high dimensions D.H.Lehmer problem is denned, and some high dimensions mean value and hybrid mean value are proposed. We also work on some special sequences and functions, and give a few sharp asymptotic formulae. The main achievements contained in this dissertation are as follows:1. Mean value problems of Dirichlet L-function is very important in analytic number theory. There exist close relationships among .L-function and many arithmetical functions. The transform relations among generalized Bernoulli numbers, the problem of an integer and its inverse, and D.H.Lehmer problem are worked out. Some new mean value formulae on L-function are given, which is useful to the study of some mean value problems.2. Exponential sums, Gauss sums and Kloosterman sums enjoy their long history. We study the relation among them, and give exact formulae for the fourth power mean of the cubic exponential sums and general cubic Gauss sums respectively. The hybrid mean values between Gauss sums and generalized Bernoulli numbers, Cochrane sums and high dimensions Kloosterman sums, and the high power mean of Hardy sums are studied, and a few asymptotic formulae are given.3. The study on the problem of an integer and its inverse, and D.H.Lehmer problem, will help us to know more properties of the distributions of integers. In this dissertation, we use the generalized Bernoulli numbers to study the high dimensions mean value of the problem of an integer and its inverse, and the hybrid mean value of this problem and Kloosterman sums. We generalize the problem of an integer and its inverse, and D.H.Lehmer problem. The multiple varieties and high dimensions D.H.Lehmer problem is defined, and some hybrid mean value are studied.4. Some special sequences and functions are studied. We generalize the Mobius inverse formula, improve the asymptotic formulae on the distributions of the' number of squarefree primitive roots, squarefull primitive roots, obtain two interesting mean value formulae, discuss a problem relates to partitions, give an exact formula on digits product, study primitive numbers and their mean value.factorial quotients, Smarandache double factorial numbers and their hybrid mean value with Mangoldt functions.