On the Sign Changes of a Weighted Divisor Problem

Abstract

Let $S(x;\frac{a_1}{q_1},\frac{a_2}{q_2})={{\sum }^{\prime }}_{mn\le x}\cos \left(2\pi m\frac{a_1}{q_1}\right)\sin \left(2\pi n\frac{a_2}{q_2}\right)$ with $x\ge q_1{q}_2,1\le a_i\le q_i$, and $(a_i,q_i)=1$ ($i=1,2$). We study the sign changes of $S(x;\frac{a_1}{q_1},\frac{a_2}{q_2})$, and prove that for a sufficiently large constant $C$, $S(x;\frac{a_1}{q_1},\frac{a_2}{q_2})$ changes sign in the interval $[T,T+C\sqrt{T}]$ for any large $T$. Meanwhile, we show that for a small constant $c^{\prime }$, there exist infinitely many subintervals of length $c^{\prime }\sqrt{T}{\log }^{-7}T$ in $[T,2T]$ where $\pm S(t;\frac{a_1}{q_1},\frac{a_2}{q_2})>c_5(q_1{q}_2{)}^{\frac{3}{4}}{t}^{\frac{1}{4}}$ always holds.