Number Theory Function Of The Mean Estimate And The Equation Contains A Function Of Number Theory To Solve

It is well known Smarandache function, Riemann-zeta function and Euler function, some special functions and sequences play an important role in the study of number theory. It has great significance to study their mean value property and the other properties, they relate to many famous number theoretic problems. 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 properties of some arithmetic functions, and got a series of asymptotic formulae about them; we set up some equations by studying the relation of some functions, and solved them completely; we obtained the identities involving Fibonacci numbers. The main achievements contained in this dissertation are as follows:1. We studied on the mean value of Smarandache multiplicative function, and found that it has the same mean value formula as Smarandache function S(n). And we studied the hybrid mean value of Smarandache function to the least common multiple function L(n), and obtained the interesting asymptotic formula.2. Using the analytic methods, we studied the hybrid mean value problems of functionδ_k(n) to several arithmetical functions, and a series asymptotic formulae had be obtained. Meanwhile, using the elementary methods, we studied the property of the k—power part residue function f_k(n) and got the asymptotic formulae for the reciprocal of function f_k(n) and function e_p(f_k(n)).3. By studying the properties of Smarandache function, Euler function and the square complement numbers, we set up the equation , S(n) = and a(n₁) +a(n₂) +…+a(n_k) = m·a(n₁ +n₂+…n_k), and solve themcompletely, got all positive integer solutions for them.4. Studying properties of the first and the second Chebyshev polynomials, according to the relation of Chebyshev polynomial and Fibonacci numbers, we used the elementary methods to obtain some identities involving Fibonacci numbers in even power.