The Mean Values Of The Exponential Divisor Functions

In1982,M.V.Subbarao gave the definition of the exponential divisor,i.e. n>1is an integer,and then d is an exponential divisot of n.We denote d|e n.In addition,he also investigated the mean value of the exponential divisot function τ(e)(n)=∑d|en1,and obtained where E(x)=O(x1/2).J.Wu improved the above result and got the following result where,M.V.Subbarao also proved for some positive integer r, where,L.Toth proved where P2r-2(t)is a polynomial of degree2r-2in ur=(?). Similarly to the generalization of dk(n) from d(n), we define the function τk(e)(n): Obviously when k=2, that is τ(e)(n). In this paper we investigate the case k=3, i.e. the properties of the function τ3(e)(n). τ3(e)(n) is obviously a multiplicative function. The aim of this paper is showing the asymptotic formula of the function τ3(e)(n) by the exponential sum estimation method.We shall prove the following:Theorem1Suppose x is a positive number, then where,In addition, we also investigate the mean value of τ(e)(n) in square-full number set: n<x,n is square-full number n≤x where f2(n) is the characteristic function of the square-full number set,when r=1,2, we have the following:Theorem2 where P1(t), Q1(t) is a polynomial of degree1in t.Theorem3where P3(t), Q3(t) is a polynomial of degree3in t.