  We study a weighted divisor function $\mathop{{\sum}'}\limits_{mn\leq
x}\cos(2\pi m\theta_1)\sin(2\pi n\theta_2)$, 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.

Number Theory