Computational Problems, The Hybrid Mean Value Of The Smarandache Function And The Infinite Series

Number theory is the branch of mathematics which deals with properties of the whole number, and also plays an unique part in mathematics. The mean value of number theoretic function is one of the important issues in number theory, and also is an indispensable tool to study various number theory problems, many well-known number theory problems are closely related to the mean value of number theoretic function.In《Only Problems, Not Solutions》, American-Romanian famous number theoretic expert F.Smarandache professor presented 105 unsolved problems, many of them are related to number theory. It is the theoretical significance to study these questions and solve some of them in a certain extent.In this dissertation, on the basis of my interesting in the Smarandache function, we used the elementary methods and analytic methods to study the properties of some special number theoretic functions, and further get mean value estimate related to these functions. In details, the main achievements contained in this dissertation are as follows:1. Studying the properties of the Pseudo-Smarandache function Z(n), andgetting the asymptotic formula of p(n)/Z(n) by using the elementary methods andanalytic methods. That iswhere p(n) denotes the smallest prime divisor of n, and a_i (i = 2, 3,…,k) are computable constants.2. By studying the properties of Smarandache Least Common Multiple function and a new arithmetic function (?)(n), we can get the asymptotic formula of (?)(n)SL(ri) by using the elementary methods and analytic methods. That iswhere d_i (i = 1,2,…, k) are computable constants, when n = p₁^α1p₂^α2…p_s^αs isthe factorization of n into prime powers, this time (?)(n) =α₁p₁+α₂p₂+…+α_sp_s.3. Using the analytic methods and elementary methods, we study one type infinite series involving Smarandache function, then obtained this type infinite series is divergent.