The Mean Value Of â… (n)

In Chapter 1, we study the mean value about a arithmetic function I(n). Suppose canonical representation of a positive integer n is: n = n=p₁^α1p₂^α2…p_r^αr, the definition of Integral function I(n):Obviously I(n) is multiplicative.In the dissertation for doctoral degree of Wang Xiaoying , Wang provedwhere a₀ is a constant.but she did not give the mean value of (?)I(n).In this paper , Using the convolution method we proved the asymptotic formula of (?)I(n) and further improved the result of (?)I(m)I(n), However we can provethat the following result on short interval.There are three main results in this paper:Theorem 1.1 Let N₀(≥1) be a fixed integer.we havewhere c_i(i≥1) are computable constants.Theorem 1.2 Let N(≥2)is an arbitrary but fixed integer.we havewhere n₀ = [N/2],a_j(j≥0) are computable constants.Theorem 1.3 Let x^{1/5+3ε}≤y≤x,thenwhere a₀ is a constant. In Chapter 2, we study the divisor problem over the set of cube-full numbers. In number theory, there is a famous problem which is about the estimation ofâ–³(x), the error term in the asymptotic formula for (?) d(n), where d(n) is the number ofways n can be written as a product of two factors.The estimation ofâ–³_k(x) is known as the Dirichlet divisor problem in honor of P.G.L.Dirichlet,who showed in the middle of the 19th century by elementary arguments thatâ–³(x) << x^1/2. Up to now, people have got many good results,â–³(x) <1/3 G.Voronoi(1903)â–³(x) <27/82 J.G.van der Corput(1928)â–³(x) <346/1067 G.Kolesnik(1973)â–³(x) <35/108log²x G.Kolesnik(1982)â–³(x) <23/73 log^315/146x M.N.Huxley (1993)â–³(x) <131/416log^26957/8320x MN.Huxley(2003)but there is some distance to our expected goal.Many people believe that for any small positive real numberε, we haveâ–³(x)<1/4+εSimilarly, people have considered the divisor problems on certain conditions. for example Health-Brown, Iwanice stdudied the problem under the condition of arithmetic progression:we study the divisor problem over the set of cube-full numbers. A positive integer n is a k-full number : p is a prime factor of n , then p^k|n.In other words, the numbers whose canonical representation isN=p₁^α1p₂^α2…p_r^αr,(α₁≥k,α₂≥k,…α_r≥k).Letδ_k(n)denote the characteristic function of k-full number. Writethe case of k= 2,Ramaiah.V and Suryanarayana,D proved thatS₂(x)=x^1/2(Alog²x+Blogx+C)+O(x^1/3log⁵x).Zhang,Zhai be further improved to the followingS₂(x)=x^1/2(P₂(oogx))+x^1/3(P₃(logx))+O(x^{δ0+ε}),whereδ₀=16829/66564=0.252...P_k(u)is a polynomial of degree k,εalways denotes a sufficiently small positive constant.In this paper we study about the case k = 3:Theorem2.1whereδ₀=49900988/257718336=0.1936261...,P_k(u)is a polynomial of degree k,εalways denotes a sufficiently small positive constant.