Equations On The Euler Function And Smarandache Function Mean

Number theory has been named the royal crown of mathematics. The mean value and properties of arithmetical functions have a very important role in the study of the number theory. Florentin Smarandache who was a famous mathe-matician raised more than hundreds unsolved arithmetical questions and guesses of the book named "Only questions, Not solutions!" in 1991. For instance, the arithmetical function Smarandache S(n), a few scholars had studied,and them had gained plenty of valuable conclusions. However, we must further explore properties of Smarandache functions because it is more and more applied in many fields.Because of the above reasons, in the dissertation, we mainly apply elemen-tary and analytic means to discuss properties of Smarandache and Euler func-tions on base of previous theory. The main results included in the dissertation are as follows:1. The mean properties of the dual function S**(n) are discussed by using classifying n which is positive integer into odd and even, as well as the conver-gence of special power series and the limitation of certain function. we obtain and2. We discuss the solutions of a few equations involving the Euler and Smarandache functions using the elementary and analytic methods.3. We give for prime p and any integer q. where D is constant.