On the mean square of the error term for the asymmetric two-dimensional divisor problem with congruence conditions

TL;DR

This paper derives an asymptotic formula for the mean square of the error term in counting a specialized two-dimensional divisor function with congruence conditions, improving previous results in the field.

Contribution

It establishes a new asymptotic formula for the mean square error of a two-dimensional divisor sum with congruence conditions, advancing prior research by Zhai and Cao.

Findings

Derived an asymptotic formula for the mean square error term.

Enhanced the accuracy of previous estimates.

Provided new insights into divisor problems with congruence conditions.

Abstract

Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in\mathbb{Z}^+$, denote by $\tau_{a,b}(n;\ell_1,M_1,l_2,M_2)$ the asymmetric two--dimensional divisor function with congruence conditions, i.e., \begin{equation*} \tau_{a,b}(n;\ell_1,M_1,l_2,M_2)=\sum_{\substack{n=n_1^an_2^b\\ n_1\equiv\ell_1\!\!\!\!\!\pmod{M_1}\\ n_2\equiv\ell_2\!\!\!\!\!\pmod{M_2}}}1. \end{equation*} In this paper, we shall establish an asymptotic formula of the mean square of the error term of the sum $\sum_{n\leqslant M_1^aM_2^bx}\tau_{a,b}(n;\ell_1,M_1,l_2,M_2)$. This result constitutes an enhancement upon the previous result of Zhai and Cao [16].