A Generalized Dirichlet Divisor Problem

Let d(n)denote the Dirichlet divisor function and Δ(x)denote the error term of the sum Σn≤xd(n).There are many papers which study the properties of Δ(x).including its upper bound.Ω-results and moments,etc.In this dissertation,we mainly consider a generalized Dirichlet divisor problem.There exist seven chapters in this dissertation.The following are the main contents:First,we introduce the history and the development of this problem.At the same time,we list the current results and give our main results.Then we apply exponential sums to derive a new upper bound.Combining with the derivative of the Riemann zeta function,we focus on the truncated Vorono? summation formula.Applying the truncated Vorono? summation formula,we consider the mean square estimate and the higher-power moments.And we further improve the third-and the fourth-power moments.At last,we study the sign changes of this problem and consider a special sum involving the generalized Dirichlet divisor function.