On the mean square of the error term for the asymmetric two-dimensional divisor problem with congruence conditions
Zhen Guo, Jinjiang Li, Linji Long, Min Zhang
TL;DR
The paper studies the mean-square behavior of the error term in the asymmetric two-dimensional divisor problem with congruence conditions, for fixed coprime integers $a,b$ with $1\le a<b$. It develops a truncated Voronoï-type formula for $\Delta_{a,b}(x;\ell_1,M_1,\ell_2,M_2)$ using Heath-Brown's $\psi(u)$ expansion and Vinogradov-type exponential-sum techniques, decomposing the error into oscillatory main and off-diagonal components. The authors extract a leading term involving a convergent series $\sum g^{*}_{a,b}(n)$ and prove a precise mean-square asymptotic with an explicit constant $\mathfrak{c}^{*}_{a,b}$, thereby extending prior work of Zhai and Cao. The results provide sharp second-moment information for the congruence-filtered divisor problem in the asymmetric setting, with potential implications for fluctuations and distribution of divisor sums under congruence constraints.
Abstract
Suppose that $a$ and $b$ are positive integers subject to $(a,b)=1$. For $n\in\mathbb{Z}^+$, denote by $τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$ the asymmetric two--dimensional divisor function with congruence conditions, i.e., \begin{equation*} τ_{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 $\sum_{n\leqslant M_1^aM_2^bx}τ_{a,b}(n;\ell_1,M_1,l_2,M_2)$. This result constitutes an enhancement upon the previous result of Zhai and Cao [16].