Mean Value Estimates Of Error Term In The Divisor Problem

The divisor problem has always been one of the classical problems in analytic number theory.In this paper,we study the mean value estimates of the error term of the divisor problem Δ2(x)and Δ3(x)and the k-free divisor problem with three-dimensional divisor function Δ3(k)(x).The work of this paper is divided into four chapters which are as follows:In Chapter one,we introduce the research background,the history of development and important results of the divisor problem and the k-free divisor problem.The we give the main results of this paper.In Chapter two,we give some necessary lemmas and the error term of the k-free divisor problem with three-dimensional divisor function.In Chapter three,we give the mean value estimate of Δ24(x)and Δ3(x).Then we get Theorem 1.1 by the technique of exponential sum methods and the estimate of the solutions of the multivariable inequalities.In Chapter four,we give the mean estimate of Δ3(k)(x)which is shown in Theorem 1.2.Then we get the Ω result of Δ3(k)(x)which is shown in Corollary 1.3.The results of this paper are as follows:Theorem 1.1.For any ε>0,we haveTheorem 1.2.For positive integers k≥4,we have where gk(m):=(?)μ(d)d3(n)dk/3,Corollary 1.3.For positive integers k≥4,we have...