Calculation Of The Mean Value Of The Pseudo Smarandache Function

Analytic number theory is a subject using complicated analytic method as an impor-tant tool.And calculating the m ean value of arithmetical function is an important and indispensable subject of analjtic number theory.For the mean value of the Pseudo Smarandache function,there are some problems are not com,pletely solved,and such problems have attracted more and more researchers' attention.The focus of this paper is to study the computational problem of the mean value and hybrid mean valu.e for the Pseudo Smarandache with analytical and elementary methods.In addition,we research and promote the mean value of related exponential divisor function.The specific research contents and results of this paper are as follows:1 Using the analytic methods to study the mean value of Pseudo Smarandache function on M-power,and two asymptotic formulae for it are presented.2.Studying the hybrid mean value of Pseudo Smarandache function with its maximum prime factor and SL(n)on M-power,and obtain some asymptotic formulae for them.3.According to the analytic methods and the properties of D(n),the asymptotic formulae involving Pseudo Smarandache-Squarefree function and D(n)function are calculate.At the same time,the second type o.f results are extended to the mean of the higher powers,and put foward a conjecture at the end of the chapter.4.A new Pseudo Smarandache function is defined,And three asyrmptotic formulae for this new function are calculated by using elementary methods.5.According to the definition and properties of the exponential divisor function,we study the mean values for exponential divisor function of k=3 on square-full,the mean values for number of exponential divisor of e-square-free of n on r power and the characteristic function of the exponential divisor function with Pseudo Smarand.ache Squarefree function on cube-full.and obtain three asymptotic formulae for them.