Mean Value Properties On Some Arithmetical Functions And Some Equations Involving Arithmetical Functions

It is well known analytic number theory is a component of number theory,its studying implement is analytic method.Mean value problems of some arithmetical functions play an important role in the study of analytic number theory,and they relate to many famous number theoretic problems.Therefore,it has great significance to study their properties.Professor F.Smarandache is a Rumanian famous number theoretic expert.In 1993,Professor F.Smarandache presented 105 unsolved problems in《Only Problems, Not Solutions》,some of them are related to number theory.In this dissertation,I mainly USe the elementary methods and analytic methods to study the properties of some special functions which were given in《Only Problems,Not Solutions》,by studying their properties we set up some equations, and obtain all their positive integer solutions,I also study the infinite series about the Smarandache function.The main achievements contained in this dissertation are as follows:1.For any positive integer n and k≥2,we call a_k(n)as a k-power complementary number if a_k(n)denotes the smallest positive integer such that na_k(n)is a perfect k-power.That isa_k(n)=min{l:l·n=m^k,l,m∈N}.In Chapter three,we mainly according to the definition and property of the k-power complement to study the solutions of an equation involving the k-power complementary numbers,and using the elementary methods to give its all positive integer solutions.2.By studying the properties of Smarandache-type function D_m(n),we set up the equationD_m^t+r(n)+D_m^t+r-1(n)+…+D_m^t+1(n)=n,r＞1 and solved it completely.3.About the Smarandache-type function,we used the analytic methods to study its convergence of some infinite series,and obtained some important identities.