| Let n>1be an integer of canonical form n=pa11pa22··· parr. The integer n iscalled a k full number if n=pa1a1p2In1982, M.V.Subbarao[1]gave the definition of the exponential divisor, i.e.n>1is an integer, and n=mai=1pii, d=mci=1pii, if ci|ai, i=1,2,···, m, thend is an exponential divisor of n. We denote d|en. The properties of the exponentialdivisor attract the interests of many authors(see, for example,[3,4,5,6,7,8,9,10,11,12,13,14,15]). An integer n=pa1a1p22··· parris called exponentially k full if allthe exponents a1, a2,···, arare k full.In addition, he also defined the exponential convolution(e-convolution) ofarithmetic functions. Suppose f and g are two arithmetic functions, thenr. The e-convolution has the identity element μ2, where μis the M¨obius function.Obviously the function μ(e)(n) is multiplicative and μ(e)(pa)=μ(a) for everyprime power pa. Hence μ(e)(n)∈{1,0,1} for every n>1and for every prime p,μ(e)(p)=1, μ(e)(p2)=1, μ(e)(p3)=1, μ(e)(p4)=0,.... Many authors have studied the properties of the exponential divisor functionL.Toth[3]proved that if the Riemann hypothesis(RH) is true, thenμ(e)(n)=A1x+O(x2901M.V.Subbarao[1]and J.Wu[5]studied the asymptotic properties of the sum|μ(e)(n)|. L.Toth[3]proved that if RH is true, thenXiaodong Cao[22]improved the above results under the assumption of theRiemann hypothesis(RH), thenwhere A1and B1are mentioned above, B2is a computable constant.In this paper, we shall study the mean value of exponential divisor functionwhere fk(n) is the characteristic function of k full integers, i.e.When k=2, we have the following: Theorem1For some D>0,In addition, we also improved the above result under the assumption of theRiemann hypothesis(RH).Theorem2If RH is true, thenThen, in this paper, we shall show the asymptotic formula of the function|μ(e)(n)|in short intervals.Finally, we shall prove the mean value of the exponential divisor function overcube full number. We have the following:... |
