# On the divisor problem with congruence conditions

## TL;DR

This paper investigates the behavior of the error term in a divisor problem with congruence conditions, proving sign change results and the existence of large fluctuations in short intervals.

## Contribution

It establishes sign change intervals and large fluctuation results for the error term in a divisor problem with congruence conditions, extending understanding of its oscillatory nature.

## Key findings

- Sign changes occur within intervals of length proportional to ^{1/2}.
- Infinitely many short intervals exhibit large deviations of the error term.
- The error term's magnitude can be bounded below by a constant times x^{1/4} in certain subintervals.

## Abstract

Let $d(n; r_1, q_1, r_2, q_2)$ be the number of factorization $n=n_1n_2$ satisfying $n_i\equiv r_i\pmod{q_i}$ ($i=1,2$) and $\Delta(x; r_1, q_1, r_2, q_2)$ be the error term of the summatory function of $d(n; r_1, q_1, r_2, q_2)$ with $x\geq (q_1q_2)^{1+\varepsilon}, 1\leq r_i\leq q_i$, and $(r_i, q_i)=1$ ($i=1, 2$). We study the power moments and sign changes of $\Delta(x; r_1, q_1, r_2, q_2)$, and prove that for a sufficiently large constant $C$, $\Delta(q_1q_2x; r_1, q_1, r_2, q_2)$ changes sign in the interval $[T,T+C\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constant $c'$, there exist infinitely many subintervals of length $c'\sqrt{T}\log^{-7}T$ in $[T,2T]$ where $\pm \Delta(q_1q_2x; r_1, q_1, r_2, q_2)> c_5x^\frac{1}{4}$ always holds.