On The Study To The Mean Values Of Some Smarandache Special Functions

What is the number theory? The number theory is one mathematics branch to studynumbers rules, especially, on the integer nature, as same as geometry、The number theoryis one of the most ancient and always active mathematical research field. In1993, thefamous expert, Professor Florentin Smarandache put forward105unsolved mathematicalproblems and conjectures about arithmetic function and special series in his book “onlyproblems, not solutions!”This paper is using the elementary and analytic methods, thinking and studyingSmarandache problem and several arithmetical functions will be used throughout thecourse of the study, some of the well-known formula of number theory, such as theperron formula, Euler product formula, Euler-summation formula and Abelidentity andso on. The dissertations also use elementary and analytical methods to study their meanvalues.The contents of the article including the following aspects:The first chapter is introduction; first it introduces the background of subject topicand implications for research. Then it tells the main results of this dissertation. Last itintroduces the basics of number theory about introduction of number theory, numbertheory branch and the application of number theory as preliminaries.The second chapter is text, including the definition and nature of the twonumber-theoretic functions to promote them and construct a new number theory functionwith recursive, inductive, guess elementary method to study its nature. The mean valueproperties of method to study the sequence of simple numbers, and give severalasymptotic formulas.The third chapter is text, including the definition and nature, combined with a simp serie Smarandache function, Smarandache function of integral and differential functionto study the collection of the power function of the mean Abel identity. At last, we canachieve several progressive formulas.