Equations And Mean Values Estimate Involving The Smarandache Function

American-Romanian number theorist Florentin Smarandache published a book“Only problems，Not solutions!”In this book，he presented105unsolved arithmeticalproblems and conjectures about arithmetical functions．Many researchers studied thesessequences and functions from this book．In this dissertation，we use elementary methodand analytic method to study the solvability of the equation involving the pseudoSmarandache function and dual function,obtained all the positive integer solutions ofthese equations,and studied mean value estimate of some arithmetic function.The mainachievements contained in this dissertation are as follows：1. To study the solvability of the equationsΖ （n）=（n）a ndΖ（n）+1=2^ω（n）, andobtained all the positive integer solutions of two equations.2. To study the solvability of the equation involving pseudo Smarandache functionand dual function, and obtained all the positive integer solutions of the equations byusing the elementary method.3. To study the solvability of the equation involving dual function of Smarandachedouble factorial function, discussed the solvability of this equation, and obtained all thepositive integer solutions of the equations.4. To study mean value estimate involving Smarandache pseduo-mltiples of5sequence,and obtained two asymptotic formulas by using analytic method.