The Study On The Smarandache Series And The Mean Values Estimate Of Smarandache Functions

Arithmetic function is the real or plural value function that defined on thepositive integer, and the study to the various properties of arithmetic functions play agreat important part in number Theory, from ancient times to the present, variousproperties of arithmetic functions were studied by the mathematicians, and a numberof important conclusions are obtained. Professor Smarandache has proposed105unresolved questions and conjecture about arithmetic functions and sequence in hisbook 《Only Problems, Not Solutions》, the study on Smarandache problem hasbecome an active field, the unsolved Problems play an important part in numberTheory, also inspired the study interest of the lovers.In this paper, some properties of sequence and Smarandache functions arestudied by using the elementary, analytic and dividing interval methods, theprogramme of Fixed happy k.th number sequence and the mean value of asymptoticformulas are given at the same time, the contributions of this dissertation are asfollows:1、The Fixed happy4.th number is studied based on the Fixed happy cubenumber, and extend to the Fixed happy k.th number.2、The summation of the sequences of the quartic section of positive integer isstudied on the basis of the summation of the sequences of the cube section ofpositive integer, and two summation formulas are given.3、The mean values of the function W (n)is studied by the method of the meanvalues of function S (n).4、The mean value of the composite function S(W (n))about the smarandachefunction S (n)and W (n)is studied, and the mean value of asymptotic formula is given.5、The mean value of the hybrid function S(n) W (n)about the smarandachefunction S (n)and W (n)is studied, and the mean value of asymptotic formula isgiven.