The Mean Value Of Function (?)(n) Over Cube-full Numbers

In this paper, we study the mean value of exponential divisor function over cube-full numbers. We establish its asymptotic formulas by using three-dimensional divisor problem and Perron's formula, which extend the properties of exponential divisor function and have o significant meaning for the further research.An integer n=p1?1p2?2…pr?r is called k - full number if all the exponents a1?k,?2> k,…,a??k. When k=3, n is called cube - full integer. Let fk(n) be the characteristic function of k - full integers, i.e.In 1982, M.V.Subbarao gave the definition of the exponential divisor, i.e. n>1 is an integer and n=(?)piai,d=(?)pici,if ci|ai,i=1,2…, r, then d is an exponential divisor of n. We denote d|e n. The smallest exponential divisor of n>1 is its squarefree kernel k(n)=(?) pi.Properties of these functions were investigated by several authors, see [2], [3], [4], [5], [6], [7].Two integers n, m>1 have common exponential divisors if they have the same prime factors and in this case, i.e. for n=(?)piai,m=(?)pibi,ai,bi?1(1?i?r), the greatest common exponential divisor of n and m is Here(1,1)e=1 by convention and(1,m)e does not exist for m>1.The integers n,m>1 are called exponentially coprime,if they have the same prime factors and(ai,bi)=1 for every 1?i?r,with the notation of above.In this case,(n,m)e=?(n)=?(m).1 and m>1 are not exponentially coprime.Let Obviously P(n)is multiplicative and for every p?, here P(p)=p, P(p)=p+p2, P(p3)=2p+p3, P(p4)=2p+p2+p4.Many authors have investigated the properties of the exponential divisor func-tion P(n).Recently L.Toth proved the following result: where the constant c is given by S.Li investigate the mean value of P(n) over square-full numbers. +O(x7/6 exp(-D(log x)3/5(log logx)-1/5)), where which is absolutely convergent for Rs>0.The aim of this paper is to establish the following asymptotic formula for the mean value of the function P(n) over cube- full numbers. Also the results for P(n) over k-full(k= 2,3) under the Riemann Hypothesis can be get.We have the following:Theorem 1. If RH is true, then where is absolutely convergent for Rs>0.Theorem 2. For some D>0,Theorem 3. If RH is true, then+1/6?/(3/5)?(4/5)G(1/5)/?(8/5)x6/5+O(x65/58+?), where G(s):=(?)g(n)/ns=?(1-1/p9s+…),which is absolutely convergent for Rs>1/9+?.