A Weighted Divisor Problem

Abstract

We study a weighted divisor function $\underset{mn\le x}{{\sum }^{\prime }}\cos (2\pi m{\theta }_1)\sin (2\pi n{\theta }_2)$, where ${\theta }_i(0<{\theta }_i<1)$ is a rational number. By connecting it with the divisor problem with congruence conditions, we establish the upper bound, mean-value, mean-square and some power-moments.