Mean Values Of The Exponential Divisor Function T^e（n）

In1982, M.V.Subbarao gave the definition of the exponential divisor, i.e. n>1is an integer, and m, then d is an exponential divisor of n. We denoted|e n. In addition, if all the exponents ai,i=1,2,…, r, are square-free numbers, so n is a e-square-free number. Now we consider exponential divisor of e-square-free of n, if bi|ai,i=1,2,…, r, and bi,i=1,2,…, r, are square-free numbers, such that is exponential divisor of e-square-free of notate:integer1is e-square-free number of n but is not exponential divisor of n.Lett(e)(n) denote the number of exponential divisor of e-square-free of n, it is multiplicative, if we know t(e)(n)=2ω(a1)…2ω(ar),here ω(n) denote the number of different prime factors of n, In general, for any prime numberp, t(e)(p)=1, t(e)(p2)=t(e)(p3)=t(e)(p4)=t(e)(p5)=2, t(e)(p6)=4,…. Dirichlet series of t(e)(n) is when Rs>1/4, it convergents absolutely, Laszlo Toth researched the mean value problems of exponential divisor function t(e)(n) in the paper, he got: here R(x)=O(x1/4+ε),and Heng Liu, Yanru Dong[3]improved the result of Laszlo Toth under the RH, they got δ2(n) denote characteristic function of square-full numbers.In this paper the mean values of δ2(n)t(e)(n) will be researched, obviously, δ2(n)t(e)(n) is a multiplicative function. We now research the mean values of expo-nential divisor function on square-full numbers in general, where δ2(n) is a characteristic function of square-full numbers, as the defination before, we have the following conclusions:Theorem1where R1,k (log x), k=1,2is a polynomial of degree1in log x, D>0is a absolute constant.This article focuses on the use of the exponential sum estimation method to estimate asymptotic formula of δ2(n)t(e)(n) under the RH. The following theorem:Theorem2Let x is a positive number, under the RH, we have where Q1,k(log x), k=1,2is a polynomial of degree1in log x.