On the mean square of the error term for the two-dimensional divisor problems(II)

TL;DR

This paper investigates the relationship between discrete and continuous mean squares of the error term in the two-dimensional divisor problem, providing insights into their asymptotic behavior.

Contribution

It establishes a connection between the discrete sum and the integral of the squared error term for the two-dimensional divisor problem.

Findings

Derived asymptotic formulas for the mean square of the error term

Established bounds relating discrete and continuous mean values

Enhanced understanding of error term fluctuations in divisor problems

Abstract

Let $\Delta(a,b;x)$ denote the error term of the general two-dimensional divisor problem. In this paper we shall study the relation between the discrete mean value $\sum_{n\leq T}\Delta^2(a,b;n)$ and the continuous mean value $\int_1^T\Delta^2(a,b;x)dx$.