Involving The Smarandache Function And Sequence Of The Mean Estimate

It is well known that studying the properties of arithmetical functions is an important part of Number Theory.Mean value problems of arithmetical functions play an important role in the study of analytic number theory.By studying mean value problems of arithmetical functions we can not only understand the nature of arithmetical functions,but solve a number of mathematical problems. Therefore,any nontrivial progress in this field will contribute to the development of analytic number theory.In this dissertation,we mainly use elementary methods and analytic methods to study the mean value problems of some arithmetical functions and sequences which were given in《Only problems,Not solutions!》by the professor F.Smarandache,relation with some important functions.The main achievements contained in this dissertation are as follows:1.By studying the properties of function Z_w(n),Z(n) and S(n),we use the prime number theorem and Abel identity to study the mean value of Z_w(Z(n)),Z_w(S(n)) and Z_w(n)S(n),and obtain some interesting asymptotic formulae.2.We use the elementary methods to study the properties of Smarandache sequences p_d(n) and q_d(n),and get the asymptotic formulae ofΩ(p_d(n)) andΩ(q_d(n)).3.We use the congruences theory to study the properties of Smarandache combinatorial sequence and solve a problem which was given by Murthy.