On The Arithmetical Properties Of Some Special Sequences In Number Theory

It is well known that the study of arithmetic properties on some special sequences play an important role in the study of number theory, and they related to many famous number theoretic problems. Any nontrivial progress in this field will contribute to the development of number theory.Professor F. Smarandache is a Rumanian famous number-theoretic expert. One of his numerous contributions is the excellent unsolved problems that are presented by him continually. In 1993, Professor F. Smarandache presented 105 unsolved problems in " Only Problems, Not Solutions", it arose great interests for scholars. So does another famous book "Unsolved Problems in Number Theory", written by R.K. Guy from Canada.In this dissertation, we use the elementary methods and analytic methods to study some problems which were given in "Only Problems, Not Solutions" and "Unsolved Problems in Number Theory", especially to study the arithmetic properties on some special sequences, and give several interesting asymptotic formulae. The main achievements contained in this dissertation are as follows:1. Some asymptotic formulae on the mean value of the Dirichlet's divisor function in some special sets are obtained.2. Studied the hybrid mean value on the difference between a R-th residue and its inverse modulo p, and given an asymptotic formula.3. We have studied the hybrid mean value of the r—th hyper-Kloosterman sums and a problem of D.H. Lehmer.4. Studied the solutions of an equation involving the Smarandache dual function S~*(n), and obtain all its positive integer solutions by using elementary method.5. Studied an asymptotic property of the Smarandache LCM sequences and obtained an interesting limit formula.