On the sign changes in a weighted divisor problem
Volume 178 / 2017
Acta Arithmetica 178 (2017), 135-152
MSC: 11N37, 11P21.
DOI: 10.4064/aa8464-2-2017
Published online: 29 March 2017
Abstract
Let $$ S(x; {a_1}/{q_1}, {a_2}/{q_2}) =\sideset{}{^\prime}\sum_{mn\leq x} \cos(2\pi m{a_1}/{q_1})\sin(2\pi n{a_2}/{q_2}) $$ with $x\geq (q_1q_2)^{1+\varepsilon}$, $1\leq a_i\leq q_i$, and $(a_i, q_i)=1$ ($i=1, 2$). We study the sign changes of $S(x; {a_1}/{q_1}, {a_2}/{q_2})$, and prove that for a sufficiently large constant $C$, $S(x; {a_1}/{q_1}, {a_2}/{q_2})$ changes sign in the interval $[T,T+C\sqrt{T}]$ for any large $T$. Moreover, for a small constant $c’$, there exist infinitely many subintervals of length $c’\sqrt{T}\log^{-7}T$ in $[T,2T]$ where $\pm S(t; {a_1}/{q_1}, {a_2}/{q_2}) \gt c_5 (q_1q_2)^{{3}/{4}}t^{{1}/{4}}$ always holds.
