| It is well known that some special functions and sequences play an important role in the number theory. The research on mean value properties and relationship of these functions is a significant and important problem. Many famous number theoretic problems relate to these functions. Therefore, any virtuality progress in this field will contribute to the development of number theory.In this dissertation, we studied the mean value and the hybrid mean value of some arithmetic functions, and got a series of asymptotic formulae about them. We set up several equations by studying the relationship of some functions, and studied the solvability of these equations, then gave all their positive integer solutions. The main achievements contained in this dissertation are as follows:1. We studied on the mean value of a new square complements function, and gave a interesting asymptotic formula for ln(6(n!)).2. Using the elementary methods, we studied mean value properties of the integer part of the k-th root of a positive integer and Euler function, and obtained two interesting asymptotic formulae.3. By studying the property of near pseudo Smarandache function defined by A.W.Vyawahare , we studied the hybrid mean value of near pseudo Smarandache function, Dirichlet divisor function and Euler function, and two interesting asymptotic formulae had been obtained. Meanwhile, using the elementary methods, we studied the mean value of the near pseudo Smarandache function on the simple numbers, and gave two interesting asymptotic formulae.4. We set up two equations involving a kind of sum of natural numbers' products, the cubic sum of natural numbers and Smarandache primitive function by studying the property of Smarandache primitive function. By studying the property of Lucas numbers, we also set up an equation involving Smarandache primitive function and Lucas numbers. Then we used the elementary methods to study the solvability of these equations and gave all their positive integer solutions.
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