Study Of The Smarandache Function And The Nature Of The Function Of Classical Number Theory

It is well known that it is an important part of number theory to study the properties of arithmetical functions.For a long period of time,mathematicians have attached great importance to the study of properties of integers, and made a significant contribution.The renowned American-Romanian number theorist Florentin Smarandache presented 105 arithmetical problems and conjectures about special sequences and functions in "Only Problems,Not Solutions". Many researchers studied these sequences and functions,and obtained some important results.In this dissertation,elementary and analytic methods are used to study the relationship between some Smarandache functions and the classical arithmetic functions,and give some related asymptotic formula,inequality and the number of positive integer solutions of some equations.The main achievements contained in this dissertation are as follows:1.The properties of the Smarandache power sequence and the divisor functions are studied.The elementary and analytic methods are used to study the hybrid mean value and the property of the divisor functions involving the Smarandache power sequence.Simultaneously,the conjecture proposed by Felice Russo is also solved.2.The elementary method are used to study the inequality and equation involving the pseudo Smarandache square-free function,and two interesting results are obtained.3.Through the studying the properties of the Smarandache double factorial function,a limit problem related to the Smarandache double factorial function has been completely solved.The solvability of an equation involving the Smarandache double factorial function and Euler function Sdf(n)=φ(n) are discussed,and its all positive integer solutions are given.