Mean square of the error term in the asymmetric many dimensional divisor problem

TL;DR

This paper derives asymptotic formulas for the mean square of the error term in the asymmetric multidimensional divisor problem, providing new insights especially for cases where k=2 and 3.

Contribution

It establishes an asymptotic formula for the mean square of the error term in the divisor problem and offers unconditional results for specific cases.

Findings

Asymptotic formula for the mean square of the error term under certain conditions

Unconditional asymptotic formulas for cases k=2 and 3

Enhanced understanding of error behavior in multidimensional divisor problems

Abstract

Let $\ba=(a_1,a_2,\ldots,a_k)$, where $a_j \ (j=1,\ldots,k)$ are positive integers such that $a_1 \leq a_2 \leq \cdots \leq a_k$. Let $d(\ba;n)=\sum_{n_1^{a_1}\cdots n_k^{a_k}=n}1$ and $\Delta(\ba;x)$ be the error term of the summatory function of $d(\ba;n)$. In this paper we show an asymptotic formula of the mean square of $\Delta(\ba;x)$ under a certain condition. Furthermore, in the cases $k=2$ and 3, we give unconditional asymptotic formulas for these mean squares.