On the divisor problem with congruence conditions

Lirui Jia; Wenguang Zhai; Tianxin Cai

paper

On the divisor problem with congruence conditions

arXiv:1908.05598

Abstract

Let $d (n; r_{1}, q_{1}, r_{2}, q_{2})$ be the number of factorization $n = n_{1} n_{2}$ satisfying $n_{i} \equiv r_{i} (mod q_{i})$ ( $i = 1, 2$ ) and $Δ (x; r_{1}, q_{1}, r_{2}, q_{2})$ be the error term of the summatory function of $d (n; r_{1}, q_{1}, r_{2}, q_{2})$ with $x \geq (q_{1} q_{2})^{1 + ε}, 1 \leq r_{i} \leq q_{i}$ , and $(r_{i}, q_{i}) = 1$ ( $i = 1, 2$ ). We study the power moments and sign changes of $Δ (x; r_{1}, q_{1}, r_{2}, q_{2})$ , and prove that for a sufficiently large constant $C$ , $Δ (q_{1} q_{2} x; r_{1}, q_{1}, r_{2}, q_{2})$ changes sign in the interval $[T, T + C T]$ for any large $T$ . Meanwhile, we show that for a small constant $c^{'}$ , there exist infinitely many subintervals of length $c^{'} T lo g^{- 7} T$ in $[T, 2 T]$ where $\pm Δ (q_{1} q_{2} x; r_{1}, q_{1}, r_{2}, q_{2}) > c_{5} x^{\frac{1}{4}}$ always holds.

arXiv admin note: substantial text overlap with arXiv:1603.04977