On the integral of the error term in the Dirichlet divisor problem

TL;DR

This paper investigates the behavior of the error term in the Dirichlet divisor problem, providing estimates for its integral and mean square, with detailed analysis for the case when k=2.

Contribution

It offers new estimates for the integral and mean square of the error term in the Dirichlet divisor problem, especially focusing on the case k=2.

Findings

Estimates for the integral of the error term _k(x)

Results on the mean square integral of _k(x)

Detailed analysis of the mean square for _2(x)

Abstract

Several results are obtained concerning the function $\Delta_k(x)$, which represents the error term in the general Dirichlet divisor problem. These include the estimates for the integral of this function, as well as for the corresponding mean square integral. The mean square integral of $\Delta_2(x)$ is investigated in detail.