On The Mean Value Of The Index Of Composition Of An Integer

For each integer n≥2, letλ(n):=(?) be the index of composition of n,whereγ(n):=(?)We writeλ(1)=γ(1)=1. The index of composition of aninteger measures the multiplicity of its prime factors. De Koninck and Doyon first studied the mean value ofλ(n). They proved the asymptotic formulasandwhere c=(?)≈0.75536. These two asymptotic formulas imply that theaverage order ofλ(n) is 1.De Koninck and Katai proved a series of results about the mean value ofλ(n). They proved that the asymptotic formulasandhold for y=x^1/5log³x.When y=x^1/2, they proved that for any fixed integer r≥1, there exist computable constants c₁,...c_r,d₁,...d_r such thatand Then they deducedandwhere c-J^',d_J^'(j≥1) are computable constants.Zhai Wenguang improve the results of De Koninck and Katai, using the Selberg Methord studied the higher moments ofλ(n).andwherethe errot x means is (?)+ε.In this paper we mainly improve Zhai Wenguang results, study(?)λ^-k(n)higher moments ofλ(n), and proved the following results,Theorem 1 Let k≥1 be fixed integers, Thenwhere c> 0 is the positive constant, and C_k,j1(j₁=1,2, ...,k),C_k,j2^'= 1,2,..., k-1), are computable constants.Let the upword formula's Theorem 2 If the R. H. is true, then...