Mean Value On Some Arithmetical Functions

Arithmetical function is a real-valued or complex-valued function defined on the set of positive integers. The research on the mean value of arithmetical func-tions is an important subject. In this thesis, we investigate to the mean value on arithmetical functions in three problems.First, we discuss the6th,7th and8th questions about structured sequences in the book of 《Only Problems, Not Solutions》 that written by American-Romanian number theorist Florcntin Smarandachc. Let f(n) is the sum of each digit of the n-th item in this sequence, and let g(n) is the n-th item. We use elementary methods to research the mean value of f(n) and g(n), and get some interesting asymptotic formulas.Then, we research the r-th power additive complement. We Look for some functions which can be compounded or mixed with the r-th power additive com-plement, and use elementary or analyzing methods to research the mean value on the composite and blended functions to get some asymptotic formulas.Finally, we research two arithmetical functions about positive integer n. In similar, we use elementary and analyzing methods to research the mean value of the composite functions to get some asymptotic formulas...