On The Properties Of The Pseudo Smarandache Function And The Equations Involving Arithmetical Functions

It is a well-known fact that analytic number theory is a branch of number theory, its studying implement is analysis method. Studying of the properties of functions of number theory is an important problem. Because they related to many famous number theoretic problems, it has great significance to study their properties.For the sake of making progress in number theory's research. Professor F.Smarandache is a Romanian famous number theorist. In 1993, Professor F.Smarandache presented 105 unsolved arithmetical problems in《Only Problems, Not Solutions》,some of them are related to number theory.In this dissertation, we use the elementary methods and analytic methods to study the properties of some special functions which were given in《Only Problems, Not Solutions》, and obtained many important results. We also studied a problem of the Schur. The main achievements contained in this dissertation are as follows:1.For any positive integer n,the famous Pseudo Smarandache function Z(n) isdefined as the smallest integer m such that n divides (?)k.That is,In chapter two, we mainly use the elementary method to study the distribution of properties of the pseudo Smarandache function Z(n) and solve two properties.2. On a problem of the Schur:For any positive integer n,let r be the positive integer such that the set{1,2,3,…,r} can be partitioned into n classes, and no class contain integersx,y,z with x~y=z,Schur asked us to find the maximum r.In chapter three, we use the elementary methods to study this problem, and give a sharp lower bound estimate for r. 3. For any positive integer n,the famous Smarandache k-th power complement functionα_k{n} is defined as the smallest integer m such that mn=u~k.That is,In chapter four, we mainly use the elementary and combinational method to study the equation involving the Euler function and the Smarandache k-th power complement function, and give its all positive integer solutions.