The Mean Value Of The Index Of Composition Of An Integer Over The Set Of Square-full Numbers

The mean value of the index of composition of an integer is one of the most important classical problems, its study has pretty good significance in theory. For each integer n≥2,letλ(n) = (?),whereγ(n) =Π_p|np,we writeλ(1) =γ(1) = 1. Many people studied the mean value ofλ(n).De Koninck and Doyon first provedandwhere c =∑_p (?)≈0.75536. These two asymptotic formulas imply that the averger order ofλ(n) is 1.De Koninck and K(?)tai provedandhold for y = x^1/5 log³ x.When y = (?),they proved that for any fixed integer r≥1,there exist computable constants c₁,…,c_r,d₁,…,d_r such thatand Then they provedandwhere c'_j,d'_j(j≥1) are computable constants .Using the Selberg method Zhai Wenguang proved the higher moments ofλ(n) and further improved the results of De Koninck and K(?)tai. For each integer k≥1, he provedwhereIn this paper, we study the high moments ofλ(n) over the set of square-full numbers. A postive integer n is a square-full number : p is a prime factor of n, then p²|n.In other words the numbers whose canonical representation isLet f₂ (n) denote the characteristic function of square-full numbers, there are two main results in this paperTheorem 1 For each integer k≥1,there exits a constantδ> 0,such that whereare computable constants.The exponent (?) in the error term depends on the none-zero domain ofζ(s), which can not be improved now. But under the assumtion of the Riemann Hypothesis, we can seek better upper bound for the error term R_k(x).Using analytic method and exponential sum we getTheorem 2 Under the assumption of the Riemann Hypothesis, then...