A Weighted Divisor Problem

We study a weighted divisor function $\mathop{{\sum}'}\limits_{mn\leq x}\cos(2πmθ_1)\sin(2πnθ_2)$, where $θ_i (0<θ_i<1)$ 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.

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