On the sign changes in a weighted divisor problem

Lirui Jia¹ ; Tianxin Cai² ; Wenguang Zhai³

¹School of Mathematical Sciences Zhejiang University Hangzhou 310027 People’s Republic of China
²School of Mathematical Sciences Zhejiang University Hangzhou 310027, People’s Republic of China
³Department of Mathematics China University of Mining and Technology Beijing 100083, People’s Republic of China

Acta Arithmetica, Tome 178 (2017) no. 2, pp. 135-152

Résumé

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.

Détail

DOI : 10.4064/aa8464-2-2017

Keywords: sideset prime sum leq cos sin geq varepsilon leq leq study sign changes prove sufficiently large constant nbsp changes sign interval sqrt large nbsp moreover small constant there exist infinitely many subintervals length sqrt log where always holds