On the mean square of the error term for the two-dimensional divisor problem (I)

TL;DR

This paper derives an asymptotic formula for the mean square of the error term in the two-dimensional divisor problem, advancing understanding of divisor distribution in number theory.

Contribution

It establishes a new asymptotic formula for the mean square of the error term in the two-dimensional divisor problem for coprime integers.

Findings

Asymptotic formula for the mean square of the error term derived

Improved understanding of divisor distribution in two dimensions

Results applicable to general coprime integer pairs

Abstract

Suppose $a$ and $b$ are two fixed positive integers such that $(a,b)=1.$ In this paper we shall establish an asymptotic formula for the mean square of the error term $\Delta_{a,b}(x)$ of the general two-dimensional divisor problem.